What is a Tensor?

Most people don’t encounter Tensors (the higher level advanced version of Matrices) until they reach senior undergraduate, or even graduate level.

What is a Tensor?

The best explanation I have ever seen, comes from this video by the author of A Student’s Guide to Vectors and Tensors, Daniel A. Fleisch. Using children’s blocks and laymen language, he explains what is a tensor clearly and succinctly in a way that is unbelievably crystal clear.

This YouTube video is watched over 200,000 times, a very commendable achievement for a math video!

Official Definition by Wikipedia

Tensors are geometric objects that describe linear relations between vectors, scalars, and other tensors. Elementary examples of such relations include the dot product, the cross product, and linear maps. Vectors and scalars themselves are also tensors. A tensor can be represented as a multi-dimensional array of numerical values. The order (also degree) of a tensor is the dimensionality of the array needed to represent it, or equivalently, the number of indices needed to label a component of that array. For example, a linear map can be represented by a matrix (a 2-dimensional array) and therefore is a 2nd-order tensor. A vector can be represented as a 1-dimensional array and is a 1st-order tensor. Scalars are single numbers and are thus 0th-order tensors. The dimensionality of the array should not be confused with the dimension of the underlying vector space.

If you have some programming knowledge, you may view tensors as a type of multidimensional array. A more mathematical abstract way can be achieved by defining tensors in terms of elements of tensor products of vector spaces, which in turn are defined through a universal property.

Cool? The word “tensor” really strikes me as a word that is really sophisticated and complicated!

How to find the distance of a plane to the origin

Given the equation of a plane: ax+by+cz=D, or in vector notation $\mathbf{r}\cdot \left( \begin{array}{c} a\\ b\\ c\\ \end{array}\right)=D$ , how do we find the (shortest) distance of a plane to the origin?

(When a question asks for the distance of a plane to the origin, by definition it means the shortest distance.)

One way to derive the formula is this:

Derivation

Let X be the point on the plane nearest to the origin.

$\overrightarrow{OX}$ must be perpendicular to the plane, i.e. parallel to the normal vector $\mathbf{n}=\left(\begin{array}{c}a\\b\\c\\\end{array}\right)$ .

Furthermore, X lies on the plane, hence we have $\boxed{\overrightarrow{OX}\cdot\mathbf{n}=D}$

Using the formula for dot product, we can get $|\overrightarrow{OX}\cdot\mathbf{n}|=|\overrightarrow{OX}||\mathbf{n}|\cos \theta=D$

Since $\overrightarrow{OX}$ is parallel to $\mathbf{n}$ , $\theta$ is either 0 or 180 degrees, hence $\cos \theta$ is either 1 or -1.

Thus, we have $|\overrightarrow{OX}||\mathbf{n}|=|D|$ .

The shortest distance from the point X to the origin is then $\displaystyle|\overrightarrow{OX}|=\frac{|D|}{|\mathbf{n}|}=\frac{|D|}{\sqrt{a^2+b^2+c^2}}$

Ans: Shortest distance from point to plane is $\displaystyle\boxed{\frac{|D|}{\sqrt{a^2+b^2+c^2}}}$

H2 Maths Condensed Notes and Prelim Papers

If you are looking for a short summarized H2 Maths Notes, with Prelim Papers to practice, do check out our Highly Condensed H2 Maths Notes!

GEP Questions: The Gauss Trick

We will continue our series on GEP Questions. To learn more about Recommended Books for GEP, to practice GEP Questions, visit the link here.

Today, we will discuss the quintessential GEP Question: The Gauss Trick. This GEP question illustrates the fact that giftedness can be trained to a large extent.

Question:

Find the sum of 1+3+5+7+…+95+97+99.

Solution will be below after this text, scroll down after you are ready to see the answer!

If a 9 year child can solve this on his/her first attempt (i.e. see the question for the very first time), then the child is actually at the level of Carl Friedrich Gauss, the legendary mathematician! When Gauss was a young kid, his teacher set the class a difficult question, 1+2+3+…+99+100, hoping to keep the children quiet and busy while he could have some time to relax. Little did he expect Gauss to come up with the right answer minutes later.

Most 9 year old kids would not be able to solve this on their first try. I wouldn’t be able to solve it correctly even if given an hour when I was a kid! If the children have seen it before and practiced, that is a different story, as it becomes so easy even for a 7 year old kid. Hence, practicing GEP Questions actually leads to an indirect boost of IQ in this manner! The transition from ignorance to knowledge, leading to increased intelligence, can be accomplished by practice! To practice more of these GEP Questions, check out the Math Olympiad Recommended Books page. As a tutor, I know that this Gauss Trick is a “must-know” question for students aiming high for school Math / GEP Selection Test, since almost every kid knows this nowadays, and it is highly popular in tests.

In fact, this technique (Gauss Trick) commonly tested in GEP Questions can be used all the way to JC and beyond! In JC it is covered under the topic of AP/GP (Arithmetic Progression and Geometric Progression).

Solution:

Now, for the solution. This sum is the famous arithmetic progression, where each term differs from the next by a fixed constant. In this case, the constant is 2.

Some solutions use a number pair matching, which can be problematic to explain when the number of terms is odd, but still works nevertheless. We can use another method of writing the sum backwards.

First we note that there are 50 terms in this series. We can know that either by noting that they are the odd numbers from 1 to 100, and half of the numbers are odd, hence there are 100/2=50 terms. Alternatively, we can note that 1=2(1)-1, 3=2(2)-1, 5=2(3)-1, …, 99=2(50)-1, where the number in the brackets acting like a counter.

1+3+5+7+…+95+97+99
99+97+95+…+5+3+1

Note that each pair, 1+99, 3+97, 5+95 actually add up to the same thing, i.e. 100.

Adding up the two expressions above, we get 100×50=5000.

Dividing that by two (since we double counted), we will get 5000/2=2500.

Ans: 2500

Hope you enjoyed solving this question!

Check out this amazing book on Gauss: The Prince of Mathematics: Carl Friedrich Gauss

Gauss is a true Math genius, and you can read more about his life in this interesting biography! This historical narrative will inspire young readers and even curious adults with its touching story of personal achievement.

JC Ranking Points Automatic Calculator

There is a great deal of confusion over how to calculate your JC Ranking Points. It would be calculated automatically after your exams, but it is good to learn how to calculate it by yourself!

New: JC Ranking Points Calculator App!

I have just written an automatic JavaScript App to calculate JC Ranking or UAS Score. To use, just make sure your JavaScript is enabled in your browser, and you are ready to calculate!

I also have written an article (one of my most popular posts) on how to calculate JC Ranking Points:

Calculate JC Ranking Points

What is the JC Ranking Point System?

Basically, the JC Ranking Point system is a numerical system where students take their three best H2 subjects and assign a numerical value to their grade (A: 20, B: 17.5, C: 15 and so on in intervals of 2.5). Then, their next best H2 subject (or H1 subject) would be assigned half the points (A: 10, B: 8.75, C: 7.5, and so on). PW (Project Work) and GP (General Paper) will also be following the H1 point scheme. (A: 10, B: 8.75, C: 7.5, and so on)

If a student takes Mother Tongue (MTL), it will also follow the H1 point scheme, and the total points will be scaled to 90 points. (by dividing by 100 and then multiplying by 90).

This system gives a numerical way to distinguish between students for admission to university. E.g. just by grades, it is hard to tell who did better: a student who got ABC/A, or ABB/B, but ranking points may provide a distinguishing factor.

H2 Math Notes

If you are still in JC or even doing O Levels, and checking this Rank Point System out, you still have time to change your destiny! Do check out the Highly Condensed Math Notes bundled with Exam Papers for practice!

You may also want to read some Motivational Books to motivate yourself to study. If you are in JC, you are one of the brightest in Singapore already, but however one still needs immense motivation to push through the two intensive years of JC and emerge victorious.

This Is How Long Your Teen Needs to Spend on Homework to Be Better at Math and Science

So it is official, scientists have discovered that a minimum of 1 hour of homework per day is needed to be better at Math!

TI-84 Battery Leak (How to repair your TI-84 Plus Pocket SE)

Today I had a bad shock when my trusted TI-84 Plus Pocket SE Calculator could not be turned on. I just purchased it last year, and TI-84 Calculators are not exactly cheap as every JC student in Singapore would know.

On Amazon, the Texas Instruments TI-84 Plus Silver Edition Graphing Calculator, Silver sells for a whopping $174.99, which is almost like half the price of a smart phone. Students in Singapore can buy it at a slightly lower Student Price, but however as I am no longer a student, I would probably need to pay the full price, unless I get one of my cousins to help me buy one.

How to Repair the TI-84 (Cannot even switch on)

First, I did the common sense approach of changing all the AAA and even the button LR44 battery. No luck…

I noticed something strange happened to one of the batteries. A layer of yellow crust has coated the battery case. (I was using Ikea batteries)

I followed the 5 steps I found on the TI-84 troubleshooting site:

Try a new set of AAA batteries.
Adjust the contrast by pressing and releasing [2ND] followed by pressing and releasing [▲]; repeat as necessary. To adjust or lighten the display, use the down arrow.
Remove one of the AAA batteries. Press and hold [CLEAR]. While holding [CLEAR], reinsert the AAA battery and press [ON]. The calculator should display the message “RAM Cleared“. Release [CLEAR] and then press it one more time to remove the message.
Remove one of the AAA batteries. Press and hold [DEL]. While holding [DEL], reinsert the AAA battery and then press [ON]. The calculator should display “Waiting…Please install calculator software now“. Follow the instructions for the TI-83 family or TI-84 Plus family to reinstall the calculator operating system.
Remove all of the batteries including the round lithium battery for 5 minutes. After the 5 minute period, reinsert all of the batteries and turn the calculator on. Adjust the contrast if necessary.

After trying all the 5 steps, I still couldn’t get the TI-84 to switch on.

I then used a piece of tissue to wipe of the battery contacts which were coated with the battery leak. Then, I tried step 5 again.

Thank God, the TI-84 sprung back to life again, and I can save $200 for now…

H2 Math Condensed Notes

For those taking H2 Maths, do check out my H2 Maths Condensed Notes with Bundled Prelim Papers!

GEP Screening Test Question Sample: The Tap Question

The Tap Question is another one of those questions that only involve fractions and whole numbers, and hence technically within the grasp of a 9 year old kid sitting for the GEP Screening Test.

However, looks are highly deceiving, and whoever tries the Tap Question for the very first time is highly likely to get stuck. (I was one of them years ago!) The Tap Question is highly popular as a challenging question, due to its psychological nature it is a hard question to grasp. This is a question you wouldn’t want to meet for the first time in the GEP Screening Test. However, if you know how to solve it, it is easy as a piece of cake, and you will be able to solve it during the GEP Screening Test no matter how they twist the question.

GEP Screening Test Sample Math Question (The Tap Question):

A fish tank is connected to three taps.
Tap A can fill the tank in 2 hours.
Tap B can fill the tank in 3 hours.
Tap C can drain the tank in 6 hours.
If all three taps are turned on at the same time, how long would it take to fill the empty fish tank?

Do try out the question before looking at the answer below!

There is a huge difference solving a question for the first time, and solving a question that one has seen before. To familiarize yourself with GEP Screening Test questions that can come out, do check out our Recommended GEP Books. Reading one of those books would increase your child’s repertoire of questions, and hence boost your child’s IQ indirectly. Also, do check out Math Olympiad books, as it is well known that GEP Screening Test Math questions do incorporate some Math Olympiad questions.

A book like The Original Collection of Math Contest Problems: Elementary and Middle School Math Contest problems would be suitable for children training for GEP Screening Test and Math Olympiad simultaneously at the same time.

Solution:

The key insight is to find out what each tap can do in one hour.

Tap A can fill 1/2 of the tank in 1 hour.
Tap B can fill 1/3 of the tank in 1 hour.
Tap C can drain 1/6 of the tank in 1 hour.

Hence, if all of them are turned on simultaneously,
$\displaystyle\frac{1}{2}+\frac{1}{3}-\frac{1}{6}=\frac{2}{3}$ of the tank can be filled in 1 hour.

2/3 of the tank takes 1 hour to fill.
Multiplying this statement by 3,
2 tanks takes 3 hours to fill.
1 tank takes 3/2=1.5 hours to fill.

Ans: 1.5 hours

Hope you had fun solving this!

Also check out the following GEP Screening/Round 2 questions:

By the way, for my foreign readers who are curious what is GEP Screening Test, it is a test conducted in Singapore for entry to the GEP (Gifted Education Programme). The screening and selection tests are conducted at the end of Primary 3, equivalent to Grade 3 or age 9. The official website on GEP Screening Test is available at: http://www.moe.gov.sg/education/programmes/gifted-education-programme/faq/general/

The GEP Screening Math questions can be viewed as the epitome of the Singapore Math system, as it features highly challenging Math questions that are technically in syllabus but few students know how to solve!

Buy H2 Maths Notes / Buy H2 Maths Exam Papers

After a long time, the Highly Condensed H2 Maths Notes is finally ready!

Are you looking for a short, summarized notes for H2 Vectors, H2 Complex Numbers, or even H2 Statistics? Do you remember the formula for the sum of a Geometric Progression?

Purchase this Highly Condensed H2 Maths Notes to quickly review and get ready for your exam!

Free Exam Papers included:
Numerous H2 Maths (Prelim) Free Exam Papers in PDF Format, with Solutions. Schools include ACJC, AJC, HCI, NJC, all the way to YJC, and more.

Note: The set of notes is 9 pages long (2 columns per page, total of 18 columns), i.e. highly condensed and summarized short notes.

Do check it out at our page of Math Resources for Sale!

buy h2 maths exam papers — We specially drew this beautiful graph of a Hyperbola using Geogebra for this Highly Condensed H2 Maths Notes.

GEP Sample Question: The Worker Question

Here is a type of a typical GEP Exam Question that can come out. Technically, this question is in syllabus since it only involves Whole Numbers. However, in practice, this is an extremely tough GEP Exam question for students who have not seen the Worker Question before.

GEP Test Question Sample (Worker Question):

6 men working 8 hours a day can paint a house in 2 days. In how many more days will 4 men, working 3 hours a day at the same rate, complete the same job?

Before you scroll down to check the answer, do give it a try! From personal experience as a tutor, even a 16 year old typical Secondary 4 student cannot solve this question if it is the first time they see it. However, once I explain the solution to them, it is extremely easy and students will get it immediately, even for primary school kids. Someone who has seen these types of questions before can solve it under a minute!

This shows the immense advantage one has if he/she has been exposed to certain types of questions. This is same for the GEP General Ability Test (GAT), a type of “IQ test”, which is basically pattern recognition. If a child has been exposed to books like Match Wits With Mensa: The Complete Quiz Book, words cannot describe the huge advantage he/she has over someone who has not seen a logic pattern puzzle before.

Solution to GEP Sample Question (Worker Question):

There are many types of solutions to this question, but my favorite is using the man-days concept. Man-days is a unit for the amount of work that is needed for something. E.g. If building a house needs 10 man-days, it can be accomplished by either 1 man x 10 days = 10 man-days, or 5 men x 2 days = 10 man-days, etc.

For this question, we will use the unit of man-hours instead.

6x8x2=96 man-hours are needed to paint the entire house.

4 men working 3 hours a day would lead to 12 man-hours a day. Hence 96/12=8 days are needed.

Warning: This is where they trap the careless students! The question asks for how many more days. Hence, the answer is 8-2=6 more days.

Ans: 6 days

Do also check out the Chicken and Rabbit GEP Math Question, which is another type of popular GEP Selection Test and Screening Test question, and can be practiced beforehand as a GEP Mock Test.

Xinmin Secondary 2010 Prelim Paper I Q24 Solution (Challenging/Difficult Probability O Level Question)

Just to reblog this earlier post on a really challenging Probability O Level Question.

Also, do check out my other related posts on Probability:

Coursera Exciting New Probability Course

List of really useful Probability Formulae

Probability is becoming a really important branch of mathematics. One of the most famous Singaporean mathematicians is Professor Louis Chen Hsiao Yun who has a theorem named after him! (Stein-Chen method of Poisson approximation) Professor Chen researches on Probability.

To begin your journey in Probabilty, Introduction to Probability, 2nd Edition is a good book to start learning from. An intuitive, yet precise introduction to probability theory, stochastic processes, and probabilistic models used in science, engineering, economics, and related fields. You may also wish to refer to our comprehensive list of Recommended Undergraduate Books.

(One of the best books to begin your journey in studying the mysterious topic of Probability)

Mathtuition88

A bag A contains 9 black balls, 6 white balls and 3 red balls. A bag B contains 6 black balls, 2 white balls and 4 green balls. Ali takes out 1 ball from each bag randomly. When Ali takes out 1 ball from one bag, he will put it into the other bag and then takes out one ball from that bag. Find the probability that

(a) the ball is black from bag A, followed by white from bag B,
(b) both the balls are white in colour,
(c) the ball is black or white from bag B, followed by red from bag A,
(d) both the balls are of different colours,
(e) both the balls are not black or white in colours.

Solution:

(a) $latex displaystylefrac{9}{18}timesfrac{2}{13}=frac{1}{13}$

(b) Probability of white ball from bag A, followed by white ball from bag B=$latex displaystyle=frac{1}{2}timesfrac{6}{18}timesfrac{3}{13}=frac{1}{26}$

Probability of white from B, followed…

View original post 105 more words

Lee Kuan Yew was the Best Student in Mathematics in Raffles College

We sincerely wish our ex-Prime Minister Mr Lee Kuan Yew all the best, and may he have a good recovery.

Many people know that Lee Kuan Yew was a lawyer and politician, but few knew that he liked and was very good at Mathematics as a student. We can know these facts from the book The Singapore Story: Memoirs of Lee Kuan Yew, Vol. 1, which is an interesting book to read.

Mathematical Excerpts from the book (Autobiography, so “I” means Lee Kuan Yew):

Each student had to take three subjects. I (Lee Kuan Yew) read English, which was compulsory for all arts students, and concentrated on it to improve my command of the language, and to help me study law later; mathematics, because I liked it and was good at it; and economics.
After the first year, a student had to choose one subject as his major field of study. I chose mathematics.
At the end of each of the three terms in the academic year there were examinations, and for the first of these I was the best student in mathematics, scoring over 90 marks.
I was good at mathematics and the sciences and had a solid grounding in the English language.

These are the quotes from the book that let us know more about the mathematical side of Lee Kuan Yew.

Hence, it is indeed apt and suitable that there is a prestigious award named “Lee Kuan Yew Award for Mathematics and Science” to honor students who have done well in Mathematics and Science subjects.

I hope you are inspired to learn more about Math, since even a famous person like LKY liked Math so much as to major in it. You can read more about Recommended Books for University Math where we list down all the best books for learning University Math.

GEP Exam Paper / GEP Questions

Firstly, do check out our GEP Recommended Books if you are interested in books that will help boost your chances of entering GEP.

A typical sample GEP Exam Paper Question (Math) is the Chicken and Rabbit Question. This test is highly popular in the GEP Screening / GEP Selection Test, Round 1 or even extremely difficult Round 2.

A sample question would go like this:

A farmer has 36 chickens and rabbits in total.
He counted 124 legs altogether.
How many chickens and how many rabbits are there?

(By the way, this question is generated from my Chicken and Rabbit Question Generator!)

This is a highly typical GEP Exam paper question that may come out in the GEP test.

How do we solve it? One way is the trial and error or Guess and Check method. However, this method may not work for high numbers. What if the farmer had 10000 sheep?

The GEP Exam Paper has limited time, hence, we would need to solve it in an efficient way.

There is one method called the Assumption method, where students can remember the acronym ASSD!

Steps of ASSD to solve GEP Exam Paper “Chicken and Rabbit” Question:

Step 1 (A): Assume) Let’s assume all the 36 animals are chickens.

Then, there would be 36×2=72 legs in total.

Step 2 (S): Subtract) Clearly, 72 legs is too few.

In reality, there are 124-72=52 legs more.

Step 3 (S): Subtract) 4-2=2

Each rabbit has 2 more legs than a chicken.

Step 4 (D): Divide)

The extra 52 legs must be due to the rabbits, and each rabbit contributes 2 more legs.

Hence, there are 52/2=26 rabbits!

There must be 36-26=10 chickens then.

Check

During the GEP Exam Paper, checking is essential to avoid careless mistakes. 26×4+10×2=124, which tallies! Hence, we are right!

To practice more Chicken and Rabbit questions, which is highly likely to come out in the GEP Exam Papers, check out our Chicken and Rabbit Worksheet Generator.

Lastly, do check out our GEP Recommended Books which may be the most useful books on the market (not found in Singapore since most major bookstores like Borders have closed down).

Scientists have proven that IQ can be increased, and hence reading a book like Match Wits With Mensa: The Complete Quiz Book would increase your score in the GEP Exam Paper (Logic) section during the GEP Screening / Selection Test.

Coursera Cryptography I Review

I have just completed the Coursera Cryptography I course by Dan Boneh successfully, and received the statement of accomplishment!

Review of the Course

Difficulty: 4.9/5

This course is really difficult for those with no computer science background. Although there is a section on number theory, most of the sections are new to me as my background is mostly undergraduate mathematics. (Though I did take a course IT1002 (from NUS) called Introduction to Programming, which is mostly on Java Programming.)

Especially the programming exercises are very tough for people with limited programming knowledge! However, note that the programming assignments are entirely optional.

Course Content

This course covers the theory and practice of cryptographic systems. Topics included symmetric encryption, data integrity, public-key encryption, and key exchange. The course emphasized the correct use of these primitives.

Interesting Things about this Course

It is interesting to note how complex the field of cryptography is, and how smart hackers have become. It is possible to do a timing attack where even the time taken to respond to say a login, can be used by hackers to guess your password. Every logical operation in a computer takes time to execute, and the time can differ based on the input; with precise measurements of the time for each operation, an attacker can work backwards to the input. – Wikipedia

Needless to say, as our world becomes increasingly digital, cryptography becomes increasingly important.

I wrote two JavaScript applications to help solve some of the programming challenges in this course:

JavaScript XOR Hexadecimal App is a simple JavaScript application that can XOR Hexadecimals (Hex), 2 characters at a time. The output is in 2 digits, meaning that “0” is written as “00”.
Javascript Application to Split Text Every Few Characters

Sadly, WordPress doesn’t support JavaScript, so I have to write them on my sister blog: http://www.mathtuition88.blogspot.com

If you are interesting in programming, particularly app programming, why not check out this book Learning iOS Game Programming: A Hands-On Guide to Building Your First iPhone Game. You may be the creator of the next “Flappy Bird” which reportedly earned its creator $50,000 a day! Wow!

Read this to be the next Flappy Bird creator! Michael Daley walks you through every step as you build a killer 2D game for the iPhone.

Solution to HP A4 Printer Paper Mysterious Question

A while ago, I posted the HP A4 Paper Mysterious Question which goes like this:

Problem of the Week

Suppose $f$ is a function from positive integers to positive integers satisfying $f(1)=1$ , $f(2n)=f(n)$ , and $f(2n+1)=f(2n)+1$ , for all positive integers $n$ .

Find the maximum of $f(n)$ when $n$ is greater than or equal to 1 and less than or equal to 1994.

So far no one seems to have solved the question on the internet yet!

I have given it a try, and will post the solution below!

If you are interested in Math Olympiad, it is a good idea to invest in a good book to learn more tips and tricks about Math Olympiad. One excellent Math Olympiad author is Titu Andreescu, trainer of the USA IMO team. His book 104 Number Theory Problems: From the Training of the USA IMO Team is highly recommended for training specifically on Number Theory Olympiad questions, one of the most arcane and mysterious fields of mathematics. He does write on other Math Olympiad subjects too, like Combinatorics, so do check it out by clicking the link above, and looking at the Amazon suggested books.

Now, to the solution of the Mysterious HP A4 Paper Question:

We will solve the problem in a few steps.

Step 1

First, we will prove that $\boxed{f(2^n-1)=n}$ . We will do this by induction. When $n=1$ , $f(2^1-1)=f(1)=1$ . Suppose $f(2^k-1)=k$ . Then,

$\begin{aligned} f(2^{k+1}-1)&=f(2(2^k)-1)\\ &=f(2(2^k-1)+1)\\ &=f(2(2^k-1))+1\\ &=f(2^k-1)+1\\ &=k+1 \end{aligned}$

Thus, we have proved that $f(2^n-1)=n$ for all integers n.

Step 2

Next, we will prove a little lemma. Let $g(x)=2x+1$ . We will prove, again by induction, that $\boxed{g^n (1)=2^{n+1}-1}$ . Note that $g^n(x)$ means the composition of the function g with itself n times.

Firstly, for the base case, $g^1(1)=2+1=3=2^2-1$ is true. Suppose $g^k (1)=2^{k+1}-1$ is true. Then, $g^{k+1}(1)=2(2^{k+1}-1)+1=2^{k+2}-1$ . Thus, the statement is true.

Step 3

Next, we will prove that if $y<2^n-1$ , then $f(y)<n$ . We will write $y=2^{\alpha_1}x_1$ , where $x_1$ is odd. We have that $x_1<2^{n-\alpha_1}$ .

$\begin{aligned} f(y)&=f(2^{\alpha_1} x_1)\\ &=f(x_1) \end{aligned}$

Since $x_1$ is odd, we have $x_1=2k_1+1$ , where $k_1<2^{n-\alpha_1-1}$ .

Continuing, we have

$\begin{aligned} f(x_1)&=f(2k_1+1)\\ &=f(2k_1)+1\\ &=f(k_1)+1 \end{aligned}$

We will write $k_1=2^{\alpha_2}x_2$ , where $x_2$ is odd. We have $x_2<2^{n-\alpha_1-\alpha_2-1}$ .

$\begin{aligned} f(k_1)+1&=f(2^{\alpha_2}x_2)+1\\ &=f(x_2)+1 \end{aligned}$

where $x_2=2k_2+1$ , and $k_2<2^{n-\alpha_1-\alpha_2-2}$ .

$\begin{aligned} f(x_2)+1&=f(2k_2)+1+1\\ &=f(k_2)+2\\ &=\cdots\\ &=f(k_j)+j \end{aligned}$

where $k_j=1$ , $1=k_j<2^{n-\alpha_1-\alpha_2-\cdots-\alpha_j-j}$ .

Case 1: All the $\alpha_i$ are 0, then $y=2(\cdots 2(k_j)+1=g^j(1)=2^{j+1}-1$ . Then, $j+1<n$ , i.e. $j<n-1$ .

Thus, $f(y)=f(k_j)+j<1+n-1=n$ .

Case 2: Not all the $\alpha_1$ are 0, then, $1=k_j<2^{n-\alpha_1-\alpha_2-\cdots-\alpha_j-j}\leq 2^{n-j-1}$ . We have $2^0=1<2^{n-j-1}$ , thus, $0<n-j-1$ , which means that $j<n-1$ . Thus, $f(y)=f(k_j)+j<1+n-1=n$ .

Step 4 (Conclusion)

Using Step 1, we have $f(1023)=f(2^{10}-1)=10$ , $f(2047)=f(2^{11}-1)=11$ . Using Step 3, we guarantee that if $y<2047$ , then $f(y)<11$ . Thus, the maximum value of f(n) is 10.

Ans: 10

Singaporean’s Plan for the Long Weekend (Aug 7-Aug 10)

As every Singaporean should know by now, August 7 is going to be a public holiday this year, due to SG 50, i.e. Singapore’s 50th anniversary as a nation.

A quick check using the mental calculation of dates (Doomsday Algorithm), we can know that August 8 is Saturday, hence August 7 is a Friday. Thus, August 7 to August 10 is indeed a long weekend!

What are Singaporeans doing during the long weekend? Hint: It has something to do with studying. This is a really funny cartoon by Singaporean cartoonist “Chew on it”

HP Printer Paper Mysterious Math Problem

On the cover of HP Printer A4 Paper, there is a mysterious Math Problem, which is strangely not solved by anyone on the internet yet! At least, when I gave it a search on Google, no one seems to have solved it…

Click on the images above to zoom in!

I will also type out the question in case the image is not clear:

Problem of the Week

Suppose $f$ is a function from positive integers to positive integers satisfying $f(1)=1$ , $f(2n)=f(n)$ , and $f(2n+1)=f(2n)+1$ , for all positive integers $n$ .

Find the maximum of $f(n)$ when $n$ is greater than or equal to 1 and less than or equal to 1994.

I will give it a try when I have free time this week.

Anyone who is interested to try please leave your answer in the comments below! I will help to verify if it is correct. If it is, you may be the first person on the internet to solve the HP A4 Paper Math Problem! 🙂

You may also forward this problem to your friends to try!

Update: Solution can be found here (Solution to HP A4 Printer Paper Mysterious Question)

Problems of Number Theory in Mathematical Competitions (Mathematical Olympiad Series) is a Math Olympiad book written by 2 Chinese authors. As everyone may know, China is one of the big guns of International Math Olympiad, winning the most medals out of all the countries almost each time, due to the huge talent pool of kids, and also their specialized Chinese Math training. So, to discover the Chinese Math secrets of Number Theory, this book cannot be missed. (By the way, this book is written in English, in case of any misunderstanding.)

Math and Motivation

Like many human endeavors, Math is one subject that requires motivation to excel.

There is this inspirational story that I found on https://schoolbag.sg/story/from-rock-bottom-to-top-of-the-class

With no interest in studying and thoroughly convinced he will never do well, M Thirukkumaran was on a downward spiral in secondary school and was almost retained in Secondary One and Two.

Fast forward nearly 10 years, Thiru, 23, is now studying Business Analytics at the National University of Singapore (NUS) under NEA and PUB’s National Environment and Water (NEW) Scholarship.

What turned Thiru a full 180 degrees around? It was the first taste of success, through the efforts of a teacher, Mr Tan Thiam Boon.

In Secondary Three at Monfort Secondary School, Thiru felt that he had hit rock bottom. During a rudimentary algebra test, he scored one mark out of a total score of 50. Instead of shaking his head in disbelief and despair, his mathematics teacher, Mr Tan, went out of his way to coach Thiru after school. But his efforts were in vain.

“Mentally, I had already accepted that I would not be able to do it,” said Thiru.

Undeterred, Mr Tan encouraged Thiru to pay full attention for the next topic, Trigonometry. Dejected and with little left to lose, Thiru came early to sit at the front of the class and followed the lesson attentively. About a week later, Mr Tan distributed the results of a test starting from the lowest to the highest scorers.

Thiru recalled the incident with great clarity.

“Naturally, I had expected my name to be the first to be called. But it was not and I was afraid my paper was lost. But instead, it was the last name called! I still remember the smile on Mr Tan’s face, and the confusion on everyone else’s that day. I sat dumbfounded. Mr Tan had managed to do what nobody else had. In one fell swoop, he eliminated the negative labels that society and I had placed on myself, and reinstated my confidence. More importantly, he showed me that I was capable, that I wasn’t a “delinquent” or a failure. It was akin to recovering from blindness.”

That initial spark ignited a passion, drive and desire to test his potential and accomplish what he did not even dare to imagine before. Thiru topped his class in mathematics and did well at the N-level and O-level examinations. At Tampines Junior College, his teachers gave him the opportunity to take H2 physics and mathematics, even though he did not take the prescribed secondary school subjects. Thiru’s hard work and determination paid off, often in the top 5% of the cohort for the regular examinations, and emerging as one of the college’s top students at the A-level examinations.

As a tutor, I know this is not easy. I have coached students from Fail to A grade. However, 1/50 is a really bad fail, and to achieve A in a matter of months requires extreme effort short of a miracle. For Thiru to overcome his challenges and extremely weak Math foundation to achieve his amazing accomplishments, required a motivational figure in the form of his Math teacher.

Such motivational teachers are rare, and I am glad that Thiru has found his mentor.

Helen Exley — ‘Books can be dangerous. The best ones should be labeled This could change your life.’ Books are another source of motivation. For those who have not yet found their motivational figure, do not wait as true motivational teachers are few and far between. To encounter one like Thiru requires luck and good fortune. However, good motivational books are there and available if you look for it. I have compiled a list of Motivational Books for the Student, which is available by clicking on the link.

In Math, you must always believe that you can solve the answer, in order to solve it. Just like Thiru, if you believe in yourself, you can do it! A nice book to read about Math and Motivation is Math Doesn’t Suck: How to Survive Middle School Math Without Losing Your Mind or Breaking a Nail. Ideal for teenagers, this book features an award winning actress who struggled with math, but later overcame her fears to be one of the top in UCLA Math faculty. Even Terence Tao praised her.

Book by Truly Gifted Kid (GEP Book)

This is the book written by a truly gifted kid, the book of all gifted books. If you are looking for GEP books, this is the GEP book to rule all GEP books, written by the gifted kid himself.

The book is titled: We Can Do

I was reading the online news, and discovered this story about Moshe Kai Cavalin.

The one thing 14-year-old Moshe Kai Cavalin dislikes is being called a genius.

All he did, after all, was enroll in college at age 8 and earn his first of two Associate of Arts degrees from East Los Angeles Community College in 2009 at age 11, graduating with a perfect 4.0 grade point average.

Now, at 14, he’s poised to graduate from UCLA this year. He’s also just published an English edition of his first book, “We Can Do.”

Not only is he focused on academics (he researches on Wormhole Theory), he is also proficient in Chinese martial arts, scuba diving, and also writing books. He is also a math major!

Personally, I think the title “We Can Do” is a clever word play on Jeet Kune Do, a style of martial arts invented by Bruce Lee!

If you want to read the Chinese version, it is also available: （Go to College At 8-year-old!）八歲進大學

More books on giftedness, GEP at: GEP Books Compendium

Also, thanks to one of my readers who bought this book via my website: The Stanford Mathematics Problem Book: With Hints and Solutions (Dover Books on Mathematics)
This volume features a complete set of problems, hints, and solutions based on Stanford University’s well-known competitive examination in mathematics.

Video on Moshe Kai Cavalin, of half Jewish, half Chinese descent:

Poll: Is Math easier or more difficult than other subjects?

Hi everyone, please vote on this poll, and forward it to your friends! You may click on the buttons below this post to post to Facebook or Twitter!

Recently, I was reading this article online (Science and maths exams are harder than arts subjects, say researchers) by a reputed online magazine Guardian. In it, Durham researchers have found that, “At A-level, science, maths and technology subjects are not just more difficult than the non-sciences, they are without exception among the hardest of all A-levels. At GCSE, the sciences are a little more difficult than the non-sciences.“.

However, there are many rebuttals, for instance: Ian McNeilly, director of the National Association for the Teaching of English, said: “It seems scientists, on the one hand, decry the quality of their intake at universities and, on the other, say that their exams are so very hard. Any view of English as a ‘soft option’ is absolute nonsense. If the scientists tried to do it, they would find it wasn’t such a breeze.”

My view is that on one hand, yes Math is hard especially if the student doesn’t know what he/she is doing. It is very possible to get zero marks if the student makes a careless mistake in the very first step, or goes completely off tangent in approaching the question. However, in humanities, the only possible way to get zero marks is to hand up a blank script of paper. As long as you write something, most likely you will get some marks.

However, if the student knows what he/she is doing, and has sufficient practice, Math is actually easier. (We are not talking about the Riemann Hypothesis here, just high school and perhaps up to college mathematics.) In Math, we often hear of students scoring full marks, which they fully deserve if they know how to do every question. On the other hand, it is impossible to score full marks in English Literature, especially if there is an essay component. No essay is perfect!

Of course, there are ways to be good at both the Science and the Arts. For instance, From STEM to STEAM: Using Brain-Compatible Strategies to Integrate the Arts is a highly regarded book on how to integrate arts curriculum into STEM (Science, Tech, Engineering, Mathematics) courses.

Personally, I will not dismiss arts as “easy”, as arts do need time and effort to achieve mastery. Not everyone can write essays like Shakespeare, nor can everyone write calligraphy like the famous Wang Xizhi of China.

However, in the context of examinations, based on my personal experience, and readings on the internet, Math / Hard Science exams are indeed harder and more difficult than arts exams. This has a very negative effect of students dropping Math / Science subjects to major in “easier” subjects. Another point of concern is grade inflation in other subjects.

Most importantly, what do you think? Is Math easier or harder than other subjects?

Please vote, and forward this to your friends for voting! I will be really interested to know the results.

Indian bride walks out of wedding when groom fails math test

Source: https://sg.news.yahoo.com/groom-fails-math-test-indian-bride-walks-wedding-065433753.html

Here’s one more reason to learn math!

NEW DELHI (AP) — An Indian bride walked out of her wedding ceremony after the groom failed to solve a simple math problem, police said Friday.

The bride tested the groom on his math skills and when he got the sum wrong, she walked out.

The question she asked: How much is 15 plus six?

His reply: 17.

The news may seem like a joke, but on a more serious note, the usage of calculators in Primary 5 & Primary 6 has led to a drop in the mental arithmetic standards of children in Singapore! As a tutor, I have clearly observed the difference before and after the calculator usage was introduced. Many a times, students need a calculator to calculate what is 18+15, for example. Also, 8 times 8 (mental calculation) would pose a challenge nowadays to some students ages 9-12, whereas in the past students would know it is 64 in less than a second.

What is happening is that students are getting a bit too reliant on the calculator, and hence their human calculator (a.k.a brain) has lack of training in this aspect. How to remedy it is to practice and train in more multiplication questions.

The Multiplication Worksheet Generator listed on this page, would generate endless amount of multiplication questions for practice. Students really need to have their multiplication tables internalized for patterns questions. It makes it easier to spot patterns, if one knows their multiplication well.
Secrets of Mental Math: The Mathemagician’s Guide to Lightning Calculation and Amazing Math Tricks is the book to refer to, to learn more mental calculation tips and tricks.

Happy Pi Day!

Tomorrow (March 14) is an important day for math lovers! It is the famous Pi Day! Pi is approximately 3.14 which corresponds to the date March 14.

Check out our post last year on Pi Day:

Did you know, Pi day is also Einstein’s Birthday?
Was Einstein destined to be a Math genius? He was born on Pi Day!

Here are some other previous posts on Pi:

How to get Pi on Calculator – Without pressing the Pi Button
You may want to think about it before looking at the answer. How on earth do we get Pi on the scientific calculator without pressing it? Hint: It is possible, in at least 5 different ways!

The Mystery of e^Pi-Pi (Very Mysterious Number)
Without giving out the spoiler, just note that $e^\pi-\pi$ is a very mysterious number. How mysterious? Take a look at the post to find out!

If you are looking for a Pi Day T Shirt, Pi Day Once in a Century March 14, 2015 T-Shirt Large Black is definitely what you are looking for. By the way, this year’s Pi Day is a league above the other Pi Days, as you can see from the image below. (click the image for a larger picture) This year’s Pi Day date culminates in the lengthy decimal expansion of Pi: 3.141592653! The year 2015 (i.e this year) is needed for the “15” in the decimal.

6 Common Misconceptions About Mathematics Degrees

Some interesting info on studying mathematics in university.

ACEI-Global

March 12th, 2015

[Note: This blog, written by Samantha Woodcock, was originally posted on http://www.topuniversities.com, and reposted here on Academic Exchange by permission from the author.]

Considering studying mathematics at university but not sure you fit the right mold? Think it’ll be too difficult, too nerdy, or won’t provide enough career options? Get ready to re-think your idea of the “typical” math student, and what’s involved in a mathematics degree…

1. Maths students are giant geeks.

Now there are the students out there that would remind you of the Sheldon Coopers of the world, but for the most part maths students are just normal people who have a passion for numbers. Not all of them wear glasses; they all don’t carry a calculator everywhere and they also don’t insist on wearing white shirts and plaid. Mathematics is also easy to combine with another subject, including art, all sciences…

View original post 497 more words

101 Mathematical Trivia

Very interesting Math Trivia

Buy Math Notes with Exam Papers

We are selling Math Short Summarized Notes, and giving away Free Exam Papers bundled together! These two complement each other, the Short Condensed notes will help when doing the Exam Papers, and practice makes perfect for Maths.

Check out the Highly Condensed Math Notes at: https://mathtuition88.com/math-notes-worksheets-sale/

Currently, we have 4 products:

Highly Condensed E Math Notes (Bundled with E Maths Prelim Papers)
Highly Condensed A Math Notes (Bundled with A Maths Prelim Papers)
Chicken and Rabbit Worksheet Generator (Bundled with PSLE P6 Prelim Papers)
Multiplication Worksheet Generator (Bundled with Primary 4 Exam Papers)

All of them can be accessed from this page: https://mathtuition88.com/math-notes-worksheets-sale/

Many times, it is possible to be stuck on a Math Problem. I have personally been stuck on Math Problems many numerous times, for instance when doing the Math Problem of the Month! When you finally discover the solution by yourself, the feeling of accomplishment is like pure nirvana. There are some books that specifically address How to Solve Math Problems, one of them is How to Solve It: Modern Heuristics. Very highly rated on Amazon, this book describes modern methods to solve heuristics and optimization problems (suitable for students interested in Computer Science). Another well written book by Polya is How to Solve It: A New Aspect of Mathematical Method (Princeton Science Library). This book is suitable for students who like challenging math problems to train their mathematical skills!

Page keywords: Buy Exam Papers Singapore, Buy Test Papers, Buy Prelim Papers

20% Discount on the following Math Resources!

I have recently revamped my Math Resources for Sale page, currently there are 6 products for sale! From the wacky Chicken and Rabbit Question Generator, to the very serious Highly Condensed Notes series for students preparing for exams!

For a limited time only, there is currently a 20% discount promotion on all the products on the page!

10% Discount Code: ilovemath

(This Discount Code is only known to my website viewers! :))

10% Social Discount: Tweet or Share on Facebook to get another 10% Discount

Do check out the Math Resources for Sale page!

Is there a set of 2015 consecutive positive integers containing exactly 15 prime numbers?

Check out this intriguing Math Olympiad Number Theory question: Is there a set of 2015 consecutive positive integers containing exactly 15 prime numbers?

For instance, the number of primes in the set {1,2,3,…,2014,2015} is 305. This can be computed by entering $\pi (2015)$ in WolframAlpha.

The solution to this problem can be obtained at: http://www.fen.bilkent.edu.tr/~cvmath/Problem/problem.htm (February 2015 Problem of the Month)

To learn more about techniques for Math Olympiad style questions (including Number Theory and more), check out this book Mathematical Olympiad Treasures by noted author Titu Andreescu. Truly a treasure trove of useful tips and techniques.

GR0568 Q2 Solution (Tangent to Graph)

This is indeed a GRE Math question that even high school students can solve!

Q2) Which of the following is an equation of the line tangent to the graph $y=x+e^x$ at $x=0$ ?

(A) $y=x$
(B) $y=x+1$
(C) $y=x+2$
(D) $y=2x$
(E) $y=2x+1$

Solution:

Firstly, at $x=0$ , $y=0+e^0=1$ .

$\frac{dy}{dx}=1+e^x=1+e^0=2$

Substituting these values into $y=mx+c$ , we get

$1=2(0)+c$ , hence $c=1$ .

Thus, the equation is $y=2x+1$ .

Answer: (E)

By the way, recently someone purchased a set of Copic Markers 36-Piece Sketch Basic Set from my website. Thanks! I was curious why a set of Copic Markers cost a whopping $150 USD, and thus went online to research. It seems that Copic Markers are one of the best markers on the market. If you are into animation / illustration, or just love drawing, Copic Markers will be the perfect choice. The difference between Copic Markers and ordinary markers is that Copic Markers are alchohol based, while usual markers are water based. Alchohol based markers mix better and can create rich blends, unlike water based markers which tend to oversoak the paper while blending.

GR0568 GRE Math Test Solutions

I have decided to post some solutions to the GRE Mathematics Test Sample Practice Past Year Papers that are available online. Hope it helps!

The GRE (Graduate Record Examinations) Mathematics Test is a standardized test by the company ETS (Educational Testing Service), and meant for students considering PhD in Math in the United States.

Question 1) In the xy-plane, the curve with parametric equations $x=\cos t$ and $y=\sin t$ , $0\leq t\leq\pi$ , has length

(A) 3
(B) $\pi$
(C) $3\pi$
(D) $\frac{3}{2}$
(E) $\frac{\pi}{2}$

Solution:

Using the integral formula for arc length, $\displaystyle s=\int_0^\pi \sqrt{(\frac{dx}{dt})^2+(\frac{dy}{dt})^2}\ dt$ , we obtain

$\displaystyle \begin{aligned}\int_0^\pi \sqrt{(-\sin t)^2+(\cos t)^2}\ dt &=\int_0^\pi 1\ dt\\ &= \pi\end{aligned}$

Thus, the answer is option (B).
Do check out this book Cracking the GRE Mathematics Subject Test, 4th Edition for more information about the GRE Math Subject Test, and how to crack it! It contains valuable and concise information on the topics tested in the GRE Mathematics Test.

Exciting New Math Course at Probability, and Book by Eccentric Mathematician

Recently, I just finished my Coursera course on Cryptography, and I have started a brand new course on Probability. This is a rather introductory course on Probability, probably at the junior undergraduate level. However, looking at the course syllabus, later on in the course some deeper aspects of probability will be taught.

Welcome message from the Probability team:

Thank you for signing up for Probability! This is a beguiling subject with a fascinating historical tradition and vibrant modern applications. We are delighted that you are joining us in this journey of exploration. And you are entitled to bask in the warm light of the realisation that you are one of the pioneers in a novel form of online learning!

Class has already begun, so please join us. We are looking forward immensely to our journey together!

Best wishes,
The Probability Course Team

The course is based on the book by Professor Venkatesh: The Theory of Probability: Explorations and Applications. Rated 5/5 on Amazon, this book is a good read, and covers many aspects of probability theory.

Another new Math book on the international scene is Birth of a Theorem: A Mathematical Adventure by the eccentric French Mathematician Cédric Villani who received the Fields’ Medal in 2010. It contains the story of his life, his favorite songs, his love of manga, and the imaginative stories he tells his children.

The Useless Number

In this exciting video, the “Useless Number” refers to complex numbers, which were first thought to be useless when first discovered. However, nowadays everyone knows that complex numbers are very useful in engineering, physics and science!

Also, check out these related posts on complex numbers:

An interesting book about complex numbers is An Imaginary Tale: The Story of [the Square Root of Minus One] (Princeton Science Library). Styled like a storybook, this book tells the history of the imaginary numbers, which was discovered as early as during ancient Egypt. However, people didn’t realize the immense usefulness of complex numbers until much later. Click the image below to read more!

Interesting but Hard to Understand Math Comic

Check out this interesting comic riddle from Spikedmath.com! The sky is the limit when it comes to Math!

Do give it a try, before scrolling down to view the answer.

Featured Amazon Product:

Buy yourself a gift from Amazon! Amazon Gift Card – Email is an excellent way to send a gift for any memorable event.

Answer: sky

First, we use L’Hopital’s Rule since the initial limit is a 0/0 indeterminate form.

$\displaystyle \lim_{x\to k} \frac{sx^2y\sin (k-x)}{k^2-kx}=\lim_{x\to k}\frac{2xsy\sin (k-x)-sx^2y\cos (k-x)}{-k}$ (We have also used the product rule and chain rule in this step)

Substituting x=k into the expression, we get $\frac{-sk^2y}{-k}$ which is equivalent to $sky$ .

Education/Math/Science Niche Website Directory

I have recently created a Education/Math/Science Niche Website Directory at: https://mathtuition88.com/education-website-directory/.

Currently, it is free to submit to this directory! (Just need to provide a link back to our website.) Benefits of submitting to this link directory include a valuable dofollow backlink from our Page Rank 4 website, mathtuition88.com. Basically, Page Rank determines how high your website is ranked on search engines like Google. The more quality backlinks a site has, the higher the website’s Page Rank, and the higher it will rank on Google. Ranking on the first page of Google is an incredible way of getting organic and highly targeted traffic from the search engines.

The blog directory will be only accepting Education/Math/Science websites hence it is a niche blog directory. Evidence have shown that blog directories which are reviewed and curated, and from a niche topic is highly beneficial to SEO (Search Engine Optimization). Furthermore, Page Rank from our blog will add to the links submitted, and raise the Page Rank of the websites listed on the Blog Directory, and also vice versa.

Hence, do submit to this directory if you are interested to boost your search engine rankings, and have more people find your blog or website. 🙂

Also, do check out our latest blog posts:

Finally, if you are using Android or Amazon Kindle, do check out this free Math app MathOpen Cool Math Kids Game. Preschool to 5th Grade. It has cool Math games and exercises for children aged preschool to 5th Grade.

Also, wishing all my readers from Singapore, China, and Hong Kong (and all other Chinese readers) a very happy Chinese New Year!

Driving Schools

Today we will introduce two driving schools for the benefit of our readers who live in Canada. Canadians are one of the top 10 countries who read our blog!

One of them is Driving school in Mississauga. Located in the city of Mississauga, one of Canada’s rising cities, this driving school has patient and friendly instructors. Learning driving is just like learning any subject, it requires patient and friendly teachers to make the whole process more enjoyable.

The next school is a Driving school in Kitchener. Also nicely located in Canada, this driving school teaches how to drive safely under winter conditions. Just like in Math, students need to know how to prevent careless mistakes, as road accidents can be deadly.

In case you need to know the address about where to find those Driving schools, I have provided the addresses below. Hope it helps and have fun learning driving!

Ultimate Drivers – Driving School Mississauga
3173 Erindale Station Rd #101
Mississauga, ON
(905) 848-6565

Ultimate Drivers – Driving School Kitchener
301 King St E #319, Kitchener, ON N2G 2L3
(519) 954-2266

Free Audiobooks by Amazon (Audible)

Are you a auditory learner? (one who learns best through hearing) Do you wish you could listen to some books on your long journey to work or school? Try Audible and Get Two Free Audiobooks! Audible is a company owned by Amazon, and a provider of audio books.

Some other interesting blog posts you may want to check out are:

I have also set up a page to promote Singapore Math Books hopefully to a global audience!

How community colleges will help fill one million jobs in the US

President Obama is making community college free in the US!

The Numerology Of The Dewey Decimal

Very interesting discussion on the special properties of certain numbers like 153!

The Mystery of the Sine of Nines, and a Math Olympiad question whose answer is 2015

Here are some of the latest updates back on http://mathtuition88.blogspot.com.

These two are highly recommended to view: The Mystery of the Sine of Nines, and a Math Olympiad question whose answer is 2015.

By the way, if you are looking for a good iPhone 6 case, do consider the iPhone 6 Case, Spigen Slim Armor Case for iPhone 6 (4.7-Inch) – Metal Slate (SGP11169). I just purchased it from Amazon, and it is working well. Looks quite good quality too. By the way, buying from Amazon is much faster than buying from the Spigen Singapore website. (see the reviews of the Spigen SG Website at Hardwarezone) Amazon delivery is always fast and reliable, even when shipping from a faraway place like the US.

iPhone 6 Case, Spigen Slim Armor Case for iPhone 6 (4.7-Inch) – Metal Slate (SGP11169)

Building a strong career in Logo Design

This is for student inclined in arts and graphic designing. Hope it is helpful! 🙂

Building a strong career in Logo Design

Before you seek to make a career in logo designing, you should determine whether your interests and skills are aligned to those required for a logo designer. The following are the basic interests and inclinations that a logo designer should have:

Superior ability with the use of Photoshop, Adobe Illustrator and other graphic design software suites
Ability to concentrate on work that needs visual precision
Interest in working with computers and employing your creative abilities in the process
Ability to hold concentration on a single task for long periods
Excellent communication skills to gain understanding of client requirements and project needs
Interest in a career that gives the flexibility of formal or self-employment

Skills necessary for a career in logo design

There is a set of skills that will ensure that you perform the best in your career should you choose to pursue logo design. The core point of having these skills is to be equipped to work competently, combining natural abilities with school knowledge and in-field experience.

Design skills – Logo designers will tend to have an artistic inclination, with the flare for design in particular. Aside from that, they should be equipped with operative knowledge of graphic design software such as Photoshop, and Adobe Illustrator, among others.

Communication skills – it is important to have the skills required to gain a comprehensive understanding of client or employer requirements, and for that you’ll need superior communication skills. You also should be able to explain clearly why each design element was chosen over other options. If you’re a freelancer, communication skills will be invaluable when you are building networks and selling your business to potential clients.

Business skills – logo design is a very business oriented field. As a freelance logo designer, you should have the skill to build various aspects of your business, including marketing yourself to new clients, maintaining relationships with existing clients and having the time management and planning skills to coordinate multiple projects within the time and budgetary allocations for each project. You should be able to plan your finances also so that you can grow your business and knowledge.

Practical/physiological abilities – A logo designer will probably spend a good portion of their life staring at a computer screen. You should be physically and mentally prepared to put in the long work hours and have a long concentration span. Your eyesight should also be very good, since logo design demands a high level of precision.

Personality traits – There are certain demands of the Logo Design profession that should be ingrained in your character to help you succeed in your career and live through the various challenges presented on the job:

Innate creativity and passion for design
Willingness to keep yourself informed on the changes in the industry, and learn new skills that are required to remain relevant
Precise and unerring attention to detail, so that you can present work that you and your client are proud to own and display
Discipline to plan and deliver projects on time and within budget, especially when you have the flexibility of being self-employed or working from home

Author Bio

John Lewis has 5 years’ experience in logo design at http://bigdropinc.com/. For more information and guidance on logo design and to view and receive top reviews for design work, contact us below

Career Tips for Shaping the Career as a Digital Signage Reseller

Lately we have a lot of posts on Career Tips specially for students who have just graduated and are looking at career options. Hope these posts will help!

Career Tips for Shaping the Career as a Digital Signage Reseller

Digital signage can be a great advertising and promotional tool. You can choose your profession as a retail digital signage reseller and can earn handsome salary. Here are some tips for shaping a career as a retail digital signage reseller.

Know The Benefits: You need to know all the advantages of the digital signage before thinking about the career in this field. With low printing costs, the digital signage has gained a lot over the traditional print advertisement. With the large multinationals, the small as well as medium enterprises are also entering into the digital signage world. If you are aware of all the advantages, you will be able to promote your products and re-sell them in different organizations.

Consumer Behavior: The buying behavior of the consumers is important to know if you have decided to choose this profession. The market including the restaurants, food joints and other take-away counters can simply convert their print media into the digital mode of advertisement. Therefore, the operation activities require proper skill and personnel. The messages should be compelling and attractive to the consumers. So, the behavior of the consumers is required to be categorized in the right direction. If the impulse can be triggered in the right direction, the promotional messages can be targeted to the right section of the audience.

Increase Profile: The company profile and their dignity can easily be increased if the digital signage can be used to promote the advertisements. The signage cost needs to be managed properly. We at dopublicity.com provide the best digital signage cost and provide high quality digital signage.

Reach More Customers in Less Time: Local business ads can be displayed digitally through the local businesses. The big brands may not be interested in displaying their ads in any retail shop but the local brands can advertise their products locally.

Spend Smartly: If you are looking forward to a career in the digital signage, you should know the process of spending. As the digital signage allows saving paper and the print costs, it is actually a smart expenditure for a business owner. If you are acting as a reseller and promoting the factor of the smart expenditure, the business owners will easily get convinced.

Creative Displays: The advertising displays are creative as well as unique in nature. If you can use the software in an innovative way, different ideas like, using the media players and the images will easily come into your mind. Most of the screens are required to be legible 24/7 so that it creates a good impact on the audience’s mind.

Operations and Control: The operational and controlling activities can be easily managed by the skilled personnel who have studied about the profession and have chosen it as their career. Some of the companies may start the digital campaigns with their products and services and can opt for the reselling activities by hiring the knowledgeable personnel. So, when you are choosing the digital signage partners, you should be aware that they are operating and controlling in a smart way.

Knowledgeable Partner: When you have chosen your profession as a digital signage reseller, you may choose a knowledgeable partner who can provide sufficient time for your training and development.

Therefore, in every step you will have to be aware with the skills of selling the digital signage products to other companies. Before re-selling, you should have proper product knowledge with good marketing skills.

Author Bio: We at dopublicity.com provide the best digital signage cost with proper training and development to all the aspiring professionals who are interested to build their career in digital signage.

Latest Updates on Mathtuition88.blogspot.com

Just to update on the latest blog posts at http://mathtuition88.blogspot.com. I am trying to bring up the page rank for my Blogspot site, hence blogging there for the time being.

Topics taught in JC 1 / JC 2 H2 Maths Many schools are now functioning more or less independently, and as a result teach various JC topics in different orders. Here is a suggested order followed by HCI, and to some extent RJC.
H2 Maths Syllabus and Formula Sheet can be found here.
Some amazing statistics about the top scorers of the IB program. Did you know the tiny Singapore island with only a few schools (notably ACS(I) and SJI) taking the IB syllabus actually produces more top scorers than the entire Australian continent? Asia power?
Noah Ark is round or square? Check out this link which debates whether the Noah’s Ark is round or square, based on mathematics.
O Level Top Scorer 2015
Winnie The Pooh Maths Olympiad question
The latest trend: Parents going for Maths Tuition

This book Why Before How: Singapore Math Computation Strategies, Grades 1-6 is surprisingly very popular. As an Amazon Affiliate, one of the most popular book I promote on my site is actually this book. It is a Singapore Math book that focuses on the “Why” before the “How”, which is very important as nowadays questions are designed to be solved using thinking, not just mechanical procedures.

Building a strong career in Internet Marketing

Hope this post is useful for students considering studying marketing in the future!

Building a strong career in Internet Marketing

If you want to break into the internet marketing field, there’s no time like the present. According to research, the amounts dedicated to digital marketing in enterprises of all sizes will steadily increase over the course of the next few years.

The internet marketing field is rather wide, and you’ll have to choose one or two areas you’re most comfortable with, but for all of these fields, the requirements for you in order to build a strong career and hence prove your worth to prospective clients and employers will mostly be the same.

The following is a list of five things you can do to distinguish yourself as an Internet marketing consultant, given by key players in the industry:

Marketing experience

A diploma or degree in a marketing related field will only go so far in honing your skill as a marketer; the more valuable education is the more experience you’ll gain working different cases out in the field. Most clients and employers look for candidates who have noteworthy marketing experience, so start building your expertise as soon as possible.

While in college, take up jobs and internships that will allow you to apply your classroom education and be involved hands-on in any marketing campaigns that come around you, no matter how small. Search for internships to occupy your holiday times while you are in school, so that you create for yourself an advantage over other recent graduates once you’re churned out into the job market.

Understand the field

Read everything you can in the field of Internet marketing that interests you: trends and statistics, new technologies, news, terminologies, metrics etc. You should also have a good working knowledge of all the metrics and analytical tools that are applied to the measurement of the efficacy of your campaign.

There are many methods of measuring effectiveness, so you should know as many as you can, and where each can be applied in order to add value to your client’s projects and campaigns.

Have an online presence

Create for yourself a vibrant online presence because all your clients will look into that before taking you on. Employers are also likely to choose the candidate with the stronger online presence when they have matching qualifications. Build your personal online brand as this may also be effective if you want to launch out and begin your own Internet marketing gig.

Jack of all trades, master in one

It’s good to know a little something about every field of internet marketing, but it’s better to have one or two fields that you know inside out. Distinguish yourself in areas of greatest interest, but know where and how you can find assistance in any other field if the need for it arises in the course of handling your employer’s or client’s businesses.

Begin by dipping your foot into the practical and basic aspects of everything, and then narrow down to the one you’re performing the best at.

Peer reviews and networks

Take advantage of all opportunities to build your knowledge and networks. Engage with peers in career fairs, conferences and other meet-up forums, and exchange ideas and experience. Also take advantage of such gathering to evaluate your work and gain tips on how they can be improved. Jealously nurture your offline relationships as you do the online relationships.

Author Bio

John Lewis has 5 years’ experience in internet marketing. For more information and guidance on internet marketing and to view and receive top web reviews for design work, contact us below

Tips to becoming an Oracle DBA

Here are some tips on becoming an Oracle database administrator. It will be useful for students interested in computer science and IT. 🙂

Tips to becoming an Oracle DBA

The Oracle database is the most sophisticated and complex database today. Knowing enough to become a database administrator is not an easy task. Oracle dataset administration is only recommended for information technology, information systems and computer science professionals who have undergone the relevant Oracle training.

The Oracle DBA’s position and salary attracts many young graduates, but few know exactly what it takes before they can earn this hallowed title in any company. An Oracle DBA is typically a senior-level manager in the smaller companies, mid-level in much bigger corporations, and their annual salaries can go as high as $ 100,000-250,000.

But the money does come with its load of responsibility – data management for the whole company, with the risk of grounding all operations with just a slight glitch. The first thing any aspiring DBA will have to remember is that database administration is just like any other profession; there’s no shortcut leading to the top.

Preparation, preparation and more preparation!

You can’t wake up one morning and decide to enroll for a DBA course, more so an Oracle DBA course. Majority of the successful DBAs in Oracle began to learn after obtaining their Bachelor’s and even Master’s degree in an IT related field. This is a career move that will require thought and plenty of preparation beforehand, and a lot of dedication after.

Oracle DBA requires 24/7 unfailing commitment, with long hours (since most of your work begins after everyone has checked out for the day) and on-call duty during holidays and other important occasions. That’s not all, as a DBA, you must keep yourself updated with every new change that comes in the Oracle and computer systems technology or you risk obsolescence.

You can now see that the Oracle DBA does earn every one of the dollars he’s paid, right?

Below is a list of qualities you should have in order to succeed in the pursuit of a career in Oracle Database Administration:

Communication skills

An Oracle DBA will constantly be in communication with end-users and managers to determine their data needs and act accordingly. College-level communication skills will be a must, with outstanding abilities to communicate through any medium and interpret information provided.

Business degree

It helps a lot if you have a college degree in business administration from a proper graduate training institution. This equips you with the foundational understanding of business systems and processes, which is required in order to best perform your job as an Oracle Applications DBA.

Foundational DBA skills

Prior to becoming an Oracle DBA, which is more demanding and complex, it’s recommended that you practice database administration at lower levels. Oracle has designed levels of learning for beginners, and practicing as you pursue OCA and OCM certification will be very helpful. This is because an Oracle DBA should understand other related database technologies as well e.g. Oracle Application Server and Java – J2EE, JDeveloper, Apache.

College classes

In preparation for a career in DBA, Operations Research is a class you want to take in your CS or IT course. This involves the development of complex decision rules and their application to real-life datasets. The back-end data store is usually Oracle, and it provides a great platform to prepare for the job setting.

Author Bio

Tom Foster is an expert in database administration. For more information on WordPress, SQL, Oracle and other database management and administration, contact the database professionals by visiting their website.

Free Career Analysis (With Prizes to be Won)

Free Career Analysis (With Prizes to be Won):
Are you a Careerist, Entrepreneur, Harmoniser, Idealist, Hunter, Internationalist, or Leader?

Find out by doing this Free Career Analysis by renowned survey company Universum:
http://uledge.co/ambWWCY

Thanks a lot for your help! You will also benefit by finding out your Career Personality, which is released at the end of the survey!

1st prize: Win a week-long trip to Singapore, Stockholm, or New York.

Website: http://uledge.co/ambWWCY

Latest Posts on Mathtuition88.blogspot.com

Recently, I have been blogging at http://mathtuition88.blogspot.com. Here are the latest posts that you may want to check out!

JavaScript XOR App (Free)

This is a JavaScript app that I wrote to perform the XOR function on two hexadecimal strings. This is to help solve one of the cryptography assignements on Coursera.

SMU Tuition for Calculus Math 001

I am providing tuition for SMU Calculus module: Math 001. A previous student of mine passed the module, despite not having any H2 Maths background. The module is highly challenging, and includes multivariable calculus.

Dragon Curve and Jurassic Park
This is to introduce two videos on the mysterious and fascinating fractal: The Dragon Curve. Not many people have heard about it!

Featured Book:

The Fractal Geometry of Nature

Review

“A rarity: a picture book of sophisticated contemporary research ideas in mathematics.”–Douglas Hofstadter, author of Godel, Escher, Bach

Free JavaScript App to split text every few characters

This is a simple JavaScript application to split characters every few characters, separated by a comma. Useful to make long strings more easy to read.

Site: http://mathtuition88.blogspot.sg/2015/01/javascript-application-to-split-text.html

This is a simple JavaScript application that can split text every “n” characters, and add a separator of your choice in between. (E.g. separated by comma, space, semicolon, etc.) For instance, given the input:”abcde12345″, separating using the default settings will output:”ab,cd,e1,23,45″. Hope it helps!

Screenshot:

Important Guidelines For Cracking Interview In Finance Company

Hope this article will be useful for students wishing to enter into the highly popular but also competitive Finance Industry! Finance jobs are open to students who major in diverse disciplines, including Engineering.

Important Guidelines For Cracking Interview In Finance Company

Cracking an interview at times seems to be a daunting task. If you are facing the jitters because of the upcoming interview in the finance company, then the best way to do so is to be prepared for the interview. Though there is no on-stop solution for the same, but remembering few important points can help you to overcome the fear related to the interview. During the interview in the financial sector, the interviewers assess whether the candidate is appropriate for the job, both in the professional and personal level. It is important for the job-seekers to communicate their ideas clearly during the interview process so that the prospective employer can gage how the candidate will perform under pressure.

Few tips to remember

Some amount of research, planning and preparation is required to crack the interview. Adequateknowledge regarding the subject is the key to crack the interview. Following few simple tips can help you to master the interview.

Research about the company as the services and products that are catered by the company. You can go through the website of the company and have an insight of the same. Recent news, press releases and recent accomplishments can help you to stay abreast about the financial company.
Identify your character traits as the strength and weaknesses and assess whether you are suitable for the job. It is imperative to say that the recruiters emphasize on the skill set of the candidate while they describe a job responsibility. Prior to the interview, make sure that you go through the job responsibility so that you can gage the qualities that the recruiters are looking for. This will also give you the hint of the questions that they might ask you during the interview. So after you have identified your behavioral traits, you can determine the ways through which you can add value to the company.
Interview gives the applicants the platform to showcase their skills and knowledge. So, make sure that you carry a detailed portfolio. This will include two copies of the resume, and along with the different certificates that will reflect your credentials. Make sure that you have these documents in one folder so that you can produce them with ease without going through piles of paper. Prepare the documents in accordance to the job description and the requirement of the job and the company. This also reflects your professionalism. If you have prior experience of working in any financial company, it is wise to showcase that.
Your appearance and presentation are also importance for creating an impression on the recruiters. Your attire affects the psyche of the recruiters, so pay heed to what you wear for your interview. If you are wearing any perfume, make sure that it is not too strong.
It is important to be polite during the interview. Make sure that you communicate clearly with the interviewer. Remember the way you present yourself will create a negative or positive impression about you. Convey your dedication towards your work and also your pride. Refrain from interrupting your recruiters as this might irk them.
While responding to the questions maintain a decent way. Your posture and gait also createsa positive impression on the recruiter.
Reach the venue of the interview ahead of the scheduled interview in formal attire.Great the receptionist or the concerned person on meeting them.
You can prepare a list of questions that are likely asked in the interviews and practice the same.

Knowing these tricks will help you to answer the questions asked during the interview seamlessly. It is important to remember that the key to cracking the interview is to stay positive and remain cool all through.

Author Bio: Ralph Spencer is a career counselor, and he also writes blogs on the tips for a successful career. You can click here to know the tips of cracking an interview.

New Year Maths Movie to Watch: The Imitation Game

Reblogged from: http://mathtuition88.blogspot.sg/2015/01/math-movie-imitation-game.html

Featured Book:

Alan Turing: The Enigma: The Book That Inspired the Film “The Imitation Game”

This book is currently ranked Top 100 book in the whole world! (#95 according to Amazon.com)

It is only a slight exaggeration to say that the British mathematician Alan Turing (1912-1954) saved the Allies from the Nazis, invented the computer and artificial intelligence, and anticipated gay liberation by decades–all before his suicide at age forty-one. This acclaimed biography of the founder of computer science, with a new preface by the author that addresses Turing’s royal pardon in 2013, is the definitive account of an extraordinary mind and life.

Capturing both the inner and outer drama of Turing’s life, Andrew Hodges tells how Turing’s revolutionary idea of 1936–the concept of a universal machine–laid the foundation for the modern computer and how Turing brought the idea to practical realization in 1945 with his electronic design. The book also tells how this work was directly related to Turing’s leading role in breaking the German Enigma ciphers during World War II, a scientific triumph that was critical to Allied victory in the Atlantic. At the same time, this is the tragic account of a man who, despite his wartime service, was eventually arrested, stripped of his security clearance, and forced to undergo a humiliating treatment program–all for trying to live honestly in a society that defined homosexuality as a crime.

The inspiration for a major motion picture starring Benedict Cumberbatch and Keira Knightley,Alan Turing: The Enigma is a gripping story of mathematics, computers, cryptography, and homosexual persecution.

Free Psychometric Career Matching Service

If you know or are a stressed teen, returning veteran, or a high school counselor then this free online service may be of great service to you. MyVerse is a free online service dedicated to psychometrically matching individuals to 1-5 careers via a color test. MyVerse will be a great first step towards a career path, and create a foundation for a person to build upon.

Right now 92% of high school seniors are not sure what career path they want to take and there is a 52% drop out rate among first year college students. MyVerse will help guide individuals along the correct path and give people a goal to achieve which will drastically reduce the dropout rates across the nation.

Right now MyVerse has a campaign on RocketHub found here . There you can contribute to the MyVerse campaign so we can see this awesome new online career service that has so much potential come to life.

New Year, New You from Amazon

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Shop Amazon – New Year, New You Event

Start Date: Dec 26, 2014
End Date: Jan 20, 2015
Type: Seasonal
Categories: Miscellaneous

Check out this New Year offer from Amazon, which ends Jan 20, 2015!

Buy a gift for yourself!
Shop Amazon – New Year, New You Event

The Significance of 8888: MH17+MH370+QZ8501

Although the QZ8501 plane is now confirmed to be in the Java Sea, we still hope that there may be some survivors, who may have possibly swum to a nearby island. Hopefully, the rescuers may be able to rescue some of the passengers.

This is a followup on the previous post on the Probability of adding up to 8888.

Other than the QZ 8501 incident, there are many other incidents that have relation to “8888” or “8” in general. Some of them are the “8888” Uprising in Myanmar, the May 12 Sichuan earthquake, the Blizzard of January 25, and the Tibetan Riots of March 14.

One special fact about 8, is that it is the only natural perfect power that is one less than another perfect power. (2^3=3^2-1) This is known as Catalan’s conjecture or Mihăilescu’s theorem.

Lastly, you may want to watch this video on Chinese Lucky Numbers by Numberphile.

Featured Book:

Biblical Numerology: A Basic Study of the Use of Numbers in the Bible

A sane explanation of biblical numerology. Davis explains the conventional, rhetorical, symbolic, and mystical use of numbers in this fascinating study of the structure and syntax of biblical numbers.

Numbers are present throughout the Bible, and do have some meanings. Why did God create the world in seven days? (rest on the seventh day) Why did Jesus have 12 disciples, not more or not less? Read this book to find out!

What is a Tensor?

Official Definition by Wikipedia

Derivation

H2 Maths Condensed Notes and Prelim Papers

What is the JC Ranking Point System?

How to Repair the TI-84 (Cannot even switch on)

GEP Screening Test Sample Math Question (The Tap Question):

Steps of ASSD to solve GEP Exam Paper “Chicken and Rabbit” Question:

Check

Review of the Course

Course Content

Interesting Things about this Course

Step 1

Step 2

Step 3

Step 4 (Conclusion)

More books on giftedness, GEP at: GEP Books Compendium

Free Career Analysis (With Prizes to be Won): Are you a Careerist, Entrepreneur, Harmoniser, Idealist, Hunter, Internationalist, or Leader?

Review

Free Career Analysis (With Prizes to be Won):
Are you a Careerist, Entrepreneur, Harmoniser, Idealist, Hunter, Internationalist, or Leader?