Updated to include: Supplementary Angles, Complementary Angles, and Half Angle Formulas for Trigonometry

Remember, memorizing the formula is not enough. We need to know how to apply and use the formula! (The next level is to know how to derive the formulas, but that will not be tested in the exams. 🙂 )

Do you really really hate Math? Is it your most dreaded subject?

Why not learn to love Math as it is pretty much a compulsory subject until high school? Read this book, it may change your mindset about Math. From a well-known actress, math genius and popular contestant on “Dancing With The Stars”—a groundbreaking guide to mathematics for middle school girls, their parents, and educators

If you can read this clock, you are without a doubt a geek. Each hour is marked by a simple math problem. Solve it and solve the riddle of time. Matte black powder coated metal. Requires 1 AA battery (not included). 11-1/2″ Diameter.

It is a video of a girl who once did a math quiz and totally blanked out for the whole quiz. However, it turned out that her teacher did not actually ask for the quiz back, and gave her as much time as she wanted to complete the quiz. Under the relaxed circumstances, she completed the quiz and got a ‘C’. (big improvement from totally blank).

Then, she went to UCLA (very good school in US), and became a mathematics major, and wrote the book that is listed below the video!

Truly inspiring. For some kids, too much pressure may result in Math anxiety and totally blankout, while for other kids a little bit of pressure is needed to ensure that they do take studies seriously. Need to find the perfect balance for each child.

Anyone who has taken high school math is familiar with the constant .

Today we are going to prove that e is in fact irrational! We will go through Joseph Fourier‘s famous proof by contradiction. The maths background we need is to know the power series expansion: . The proof is slightly tricky so stay focussed!

Did you know the constant e is sometimes called Euler’s number?

Learn more about Euler in this wonderful book. Rated 4.9/5 stars, it is one of the highest rated books on the whole of Amazon.

Leonhard Euler was one of the most prolific mathematicians that have ever lived. This book examines the huge scope of mathematical areas explored and developed by Euler, which includes number theory, combinatorics, geometry, complex variables and many more. The information known to Euler over 300 years ago is discussed, and many of his advances are reconstructed. Readers will be left in no doubt about the brilliance and pervasive influence of Euler’s work.

Watch this video for another proof that e is irrational!

Is your child disinterested in Math? Looking for some fun and educational Math games?

Math Whiz plays like a video game and teaches like electronic flash cards. This portable ELA quizzes kids on addition, subtraction, multiplication and division, AND works as a full-function calculator at the press of a button. Problems are displayed on the LCD screen. Features eight skill levels, as well as lights and sounds for instant feedback. Two AAA batteries required (not included).

Chinese students typically outperform U.S. students on international comparisons of mathematics competency. Paradoxically, Chinese teachers receive far less education than U.S. teachers–11 to 12 years of schooling versus 16 to 18 years of schooling.

Studies of U.S. teacher knowledge often document insufficient subject matter knowledge in mathematics. But, they give few examples of the knowledge teachers need to support teaching, particularly the kind of teaching demanded by recent reforms in mathematics education.

This book describes the nature and development of the “profound understanding of fundamental mathematics” that elementary teachers need to become accomplished mathematics teachers, and suggests why such teaching knowledge is much more common in China than the United States, despite the fact that Chinese teachers have less formal education than their U.S. counterparts.

Using math in the fight against cancer WNYT
But a local college professor says many people don’t like math because they don’t see a connection to it. In this Friday’s STEM 13 report, learn how the …

In search of education The News International
Unless we start investing massively in education, science, technology and innovation, as was done by Singapore, Korea, Malaysia, China and others, …

NAFA inspires The Hindu
The safe and comfortable cosmopolitan environment Nanyang Academy of Fine Arts, Singapore makes it the perfect destination for education abroad.

This book is written by John Conway, one of the mathematicians who worked on the Monster Group. Rated highly on Amazon.

Start with a single shape. Repeat it in some way—translation, reflection over a line, rotation around a point—and you have created symmetry.

Symmetry is a fundamental phenomenon in art, science, and nature that has been captured, described, and analyzed using mathematical concepts for a long time. Inspired by the geometric intuition of Bill Thurston and empowered by his own analytical skills, John Conway, with his coauthors, has developed a comprehensive mathematical theory of symmetry that allows the description and classification of symmetries in numerous geometric environments.

This richly and compellingly illustrated book addresses the phenomenological, analytical, and mathematical aspects of symmetry on three levels that build on one another and will speak to interested lay people, artists, working mathematicians, and researchers.

A rational number is a number that can be expressed in a fraction with integers as numerators and denominators.

Some examples of rational numbers are 1/3, 0, -1/2, etc. Now, we know that .

Is the square root of 2 rational? Or is it irrational (the opposite of rational)? How do we prove it? It turns out we can prove that the square root of two is irrational using a technique called proof by contradiction. (One of the earlier posts on this blog also used proof by contradiction to show that there are infinitely many prime numbers.)

First, we suppose that , where is a fraction in its lowest terms.

Next, we square both sides to get .

Hence, . We can conclude that is even since it is a multiple of 2. Thus, itself is also even. (the square of an odd number is odd).

Thus, we can write for some integer k. Substituting this back into , we get , which can be simplified to .

Hence, is also even, and hence is also even!

But if both and are even, then is not in the lowest terms! (we could divide them by two). This contradicts our initial hypothesis!

Thus, the only possible conclusion is that the square root of two is not a rational number to begin with!

Who says math can’t be funny? In Math Jokes 4 Mathy Folks, Patrick Vennebush dispels the myth of the humorless mathematician. His quick wit comes through in this incredible compilation of jokes and stories. Intended for all math types, Math Jokes 4 Mathy Folks provides a comprehensive collection of math humor, containing over 400 jokes.

Check out this post by MIT almost perfect-scorer, on how to study. His secret is to study the material in advance, before the lessons even start! This is really a useful strategy, if implemented correctly. Imagine being in Primary 3 and already knowing the Primary 4 syllabus! Primary 3 Math will be a breeze then. This is one of the reasons why China students are so good at Math – they have already studied it back in China, where the Math syllabus is more advanced!

Do try out this strategy if you are really motivated to improve in your studies. The prime time to do this is during the June and December holidays – take some time to read ahead what is going to be learnt during the next semester.

This is an excerpt of the thread:

I graduated from MIT with a GPA of 4.8 (out of 5.0) in mathematics. I had two non-As, both of which were non-math classes.

That doesn’t imply that I have good study methods, but anyway, here’s how I studied at MIT. My main study method as an undergraduate, for math classes, was knowing a sizable chunk of the material in advance.

This isn’t a method that will work for everybody. I did a lot of mathematics outside of the classroom both in high school and at MIT, and I often saw a substantial portion of the material in a given class before I took it. I can’t emphasize enough how much easier this makes a class, and not just for the reasons you might expect: one of the most valuable things you get out of knowing a lot of the material already is just not being intimidated by it. (And you can get this benefit even if you’ve only seen some of the material before and possibly forgotten some of it too.) You’re much more relaxed, and that makes it easier to process the part of the material that you don’t know.

What that translates to in terms of practical advice is this:

cultivate a sense of curiosity,

don’t restrict your learning to the classroom,

only take classes that actually seem really interesting to you, and

try to learn something related to those classes the semester before.

None of this is advice for studying for a class you’re taking now, but it’s advice for reducing the extent to which you will need to study for classes you’ll take in the future.

Is it safe to log in through well known sites such as Facebook and Google? Think again, for Wang Jing, a PhD student in mathematics at the Nanyang Technological University in Singapore, has detected critical security vulnerabilities in the OAuth, OpenID security protocols. (Source: http://phys.org/news/2014-05-math-student-oauth-openid-vulnerability.html) [Second article in the list below]

Forward this information to your friends via the Tweet button below to warn them of the potential danger!

Tuition and divorce The Independent Singapore News
In September 2013, The Independent Singapore reported on Senior Minister of State for Education Ms Indranee Rajah’s observation on the perceived …

Why Indonesian education is in crisis Jakarta Post
Does anyone seriously believe “education” in Indonesia is on par with the west, or even Asian countries like Japan, Korea or Singapore? Ask the …

Recently, I saw on Arxiv (an online Math journal) that a professor from South-China Normal University, Mingchun Xu, has proved the notoriously difficult Riemann Hypothesis.

Quote: “By using a theorem of Hurwitz for the analytic functions and a theorem due to T.J.Stieltjes and I. Schur, the Riemann Hypothesis has been proved considering the alternating Riemann zeta function. “

In 1859, Bernhard Riemann, a little-known thirty-two year old mathematician, made a hypothesis while presenting a paper to the Berlin Academy titled “On the Number of Prime Numbers Less Than a Given Quantity.” Today, after 150 years of careful research and exhaustive study, the Riemann Hyphothesis remains unsolved, with a one-million-dollar prize earmarked for the first person to conquer it.

Most students will encounter the Maclaurin Series (also known as the Taylor’s Series centered at zero) when they are studying JC H2 or College Maths. The formula looks pretty intimidating at the start:

How on earth does one come up with that formula?

However, it turns out it is not that hard to prove the Maclaurin Series informally, or at least to derive the above formula. (The hard part is related to rigorous proof of convergence, etc.)

The idea is to approximate a function by a power series (a kind of infinite polynomial) and then find out what are the coefficients.

So, we assume we can write the function as such:

, where are the coefficients of the polynomial (to be determined).

We also assume that the above equation holds for all .

Then, letting , we get . We have just found the first coefficient!

Next, we differentiate the equation to get:

Letting again, we get: .

Now, differentiating the above equation one more time gives us:

Math wrath in Pincher Creek? Pincher Creek Echo
Protesters gather during a rally to support a petition calling for math curriculum reform at the Alberta Legislature Building in Edmonton, Alta., …

Math is logical, functional and just … awesome. Mathemagician Arthur Benjamin explores hidden properties of that weird and wonderful set of numbers, the Fibonacci series. (And reminds you that mathematics can be inspiring, too!)

Dr James Grime on the Pisano Period – a seemingly strange property of the Fibonacci Sequence.

Past Year Paper is an established Singapore Home Tuition Agency that brings to you quality private tutors and free exam papers for primary, secondary school to junior college level students.

This post is all about finding Vertical and Horizontal asymptotes of graphs.

Vertical Asymptotes

Usually, vertical asymptotes come about when there is a rational function with a numerator and a denominator, for instance, . When the denominator is 0, the function is undefined, and hence there is a vertical asymptote there.

Hence, to find the asymptote, let the denominator be 0. E.g. , so .

Another way vertical asymptotes can come about is via logarithmic graphs, e.g. .

is undefined, so when or , there will be a vertical asymptote at .

Horizontal Asymptote

Horizontal asymptotes usually come about when one of the terms approaches zero as approaches infinity.

To find the Horizontal Asymptote, find the value of y when x approaches infinity (i.e. when x becomes a very big number).

For example, . When x is a very big number, say x=10000, y will be close to 1 since 1/10000 is almost zero. Hence, the horizontal asymptote is .

Another time where Horizontal Asymptotes appear is for Exponential Graphs. For instance, . When x is very large, will be very small, and hence approaches 1. This means that the Horizontal Asymptote will be .

There are many free or affordable Kindle Math Books online for download/purchase. Other than Math books, the Kindle can also be used for reading other books, and also for playing games and using apps. It is a decent alternative to the Ipad, if you are not a fan of Apple.

The object of Math BINGO is to practice math facts while playing BINGO!

Features:
-Choose from 5 games: Addition, Subtraction, Multiplication, Division and Mixed
-Choose from 3 different levels of difficulty: Easy, Medium and Hard
-Create up to 5 player profiles
-Choose from 8 different fun cartoon avatars
-Keep track of number of games played by player profile
-The Scoreboard keeps track of scores for each game and level
-Collect and play with BINGO Bugs when you earn a high score!
-Fun bonus game: BINGO Bug Bungee

“I used to think that the iPad was the king of the tablets, but not any more. It’s time to whip the crown away from Apple and give it to its rightful holder – Amazon’s Kindle Fire HDX.”

Want to watch movies about Mathematics? There is a nice website with Free Movies involving Math. It is not the full movie, but the portion of the movie that involves math.

This is a collection of movie clips in which Mathematics appears. The site is now in HTML5 video and should be accessible by all devices. If not, chose the direct video links. To include a clip into a presentation, chose the quicktime version.

If you are really interested in Math in the Movies, check out this book: Math Goes to the Movies

Mel Gibson teaching Euclidean geometry, Meg Ryan and Tim Robbins acting out Zeno’s paradox, Michael Jackson proving in three different ways that 7 x 13 = 28. These are just a few of the intriguing mathematical snippets that occur in hundreds of movies. Burkard Polster and Marty Ross pored through the cinematic calculus to create this thorough and entertaining survey of the quirky, fun, and beautiful mathematics to be found on the big screen.

Math Goes to the Movies is based on the authors’ own collection of more than 700 mathematical movies and their many years using movie clips to inject moments of fun into their courses. With more than 200 illustrations, many of them screenshots from the movies themselves, this book provides an inviting way to explore math, featuring such movies as:

• Good Will Hunting• A Beautiful Mind• Stand and Deliver• Pi• Die Hard• The Mirror Has Two Faces

The authors use these iconic movies to introduce and explain important and famous mathematical ideas: higher dimensions, the golden ratio, infinity, and much more. Not all math in movies makes sense, however, and Polster and Ross talk about Hollywood’s most absurd blunders and outrageous mathematical scenes. Interviews with mathematical consultants to movies round out this engaging journey into the realm of cinematic mathematics.

This fascinating behind-the-scenes look at movie math shows how fun and illuminating equations can be.

Look no further! In this post I will recommend the Top 5 Math Games for kids, on Amazon.com. Amazon is one of the biggest companies in the world, and is a highly trusted and respected online retailer. Sometimes, it is hard to find Math Games in the local area, the internet provides a convenient and hassle-free way to buy Fun Math Games.

Without further ado, these are the Top 5 Math Games for children:

2048 is a highly popular Math/Puzzle game that can be played on the computer or on mobile devices. The game is about adding two tiles together until you get the number 2048.

Like most fun games, the concept of 2048 is deceptively simple, even a 5 year old kid could play it. However, it is hard to master it, and getting the coveted “2048” could prove quite tricky. Do not despair, for after reading this strategy guide, you have a much higher chance of winning the game!

Strategy Guide / Walkthrough / FAQ

The 3 Top Priorities for 2048 game:

1) Keep your highest tile in the top left corner of the grid. This is your top priority.

2) Do not let low tiles, especially 2’s or 4’s, clog up the upper two rows. This is your second priority.

3) Keep your top row in the following order, from left to right, . An example would be, 512, 256, 128, 64.

The reason for Priority 1 is that this immensely increases your chances of successful merges of two higher numbers into 1. It synergises with Priority 3 to create a chain-effect. For example, imagine you have 512, 256, 128, 64 on the top row. After merging another 64 with the 64 on the top row, you will have 512, 256, 128, 128. The two 128’s can merge together, making 512, 256, 256. The two 256’s can merge together, making 512, 512. And then, we have a 1024!

The reason for Priority 2 is that letting 2 or 4’s clog up the top rows is very bad. It greatly reduces your mobility (the top 2 rows clogged up with even a single ‘2’ is hard to move). The 2 or 4’s up there have little to no chance to get merged since most of the numbers at the top are high numbers.

Top 3 Guidelines for 2048 game:

1) Press up and left arrows only. Only press right when the upper row is full. Press down only when you have utterly no other choice.

2) Keep the top row filled up, as far as possible.

3) Your general aim is to target the lowest tile on the upper row, to set up the chain effect described above.

Reason for Guideline 1: Pressing right when the upper row is not full has the chance of introducing a new tile on the upper left corner, so now your highest tile is no longer on the upper left corner. This is not good. (Violates Priority 1)

Reason for Guideline 2: Keeping the top row filled up enables you to press “right” without fear of introducing a new tile on the upper left corner.

Reason for Guideline 3: After reaching the late game, we need to think a few steps in advance, and think of which is the best move in accordance to the Top 3 Priorities, and also can target the lowest tile on the upper row to set up a chain effect.

Top 3 Time Saving Quick and Fast Tips for 2048 game

1) The first few steps do not require thinking. Just spam up and left until you get a moderately high number like 128 or 256. There is no harm done about this as the board is uncluttered and there is little chance of losing. You only need to start thinking deeper during the later part of the game, when your highest tile is 512 or more.

2) If Priority 1 is violated, i.e. your highest tile is no longer in the top left corner of the grid, try a few steps to see if you can salvage the situation and get it back to the top left corner. If no, it is better to quit and start a new game to save time. Same for Priority 2, if there is a 2 or 4 clogging the upper row, try a few more steps to see if you can salvage the situation, by merging to make a higher number. If no, we can restart to save time. Priority 3 is less crucial, if the numbers in the top row do not form , no need to restart. But keep it in mind and keep trying your best to achieve the ideal order.

3) When there is only one possible move, make that move without thinking to save time. (No other choice anyway)

This is the best video on youtube about 2048 Strategy. (Note: They put the highest tile on the bottom right instead. Should be no difference due to the symmetry of the board)

Note: Even the expert maker of this video only has a 30% winning rate! 2048 has some element of luck (the tiles arrive randomly). Personally, I took quite some time to beat the game too.

This is the list of Top Weekly Math News around the world. To view more Math News, you can scroll down to the bottom of this website, and there will be more Free Math News for your viewing.

Math Academy expands program in county Shelby Star
The summer program that serves Cleveland County Schools students to enhance their math skills is planning to have a permanent location in Kings …

Math isn’t hard. Love is.
Currently in its eighteenth printing in Japan, this best-selling novel is available in English at last. Combining mathematical rigor with light romance, Math Girls is a unique introduction to advanced mathematics, delivered through the eyes of three students as they learn to deal with problems seldom found in textbooks. Math Girls has something for everyone, from advanced high school students to math majors and educators.

Praise for Math Girls!

“…the type of book that might inspire teens to realize how much interesting mathematics there is in the world—not just the material that is forced upon them for some standardized test.” “Recommended”
—CHOICE: Current Reviews for Academic Libraries

“Imagine the improbable: high-school students getting together on their own — not in a Math Club or Math Circle, not in preparation for any Math Olympiad or “regular” test, not on the advice of any of their teachers, not as part of any organized program — to talk about pure math, math more interesting than the math found in their textbooks. The three students in this book do that for the sheer love of it. That to me is the beauty and fascination of this novel for young people, mostly young people interested in math.”
—Marion Cohen, Arcadia University, MAA Reviews

“Sometimes the math goes over your head—or at least my head. But that hardly matters. The focus here is the joy of learning, which the book conveys with aplomb.”
—Daniel Pink, NYT and WSJ best-selling author of Drive and A Whole New Mind

“if you have a…teenager who’s really into math, this is a really interesting choice”
—Carol Zall, Public Radio International, The World

“Math Girls provides a fun and engaging way to learn and review mathematical concepts…the characters’ joy as they explore and discover new and old ideas is infectious.” —review, “Experiments in Manga” blog

Reviews from amazon.co.jp

“As a physics major, math has always been a painful tool to use and nothing more. But Math Girls changed the way I look at mathematics. Now I actually find it interesting!”
— “Au”

“Math Girls is a fun read, but I was surprised to find that it’s also a serious math book chock full of careful explanations. I hope that people who think they don’t like math will read it. Even when the formulas go over your head, just following the story gives you a great feel for how fun math can be.”
— “Nyanta”

“I got hooked on this book during summer vacation, and had a great time reading it by the pool. It was so good that I read it twice, the second time while working out the problems on the hotel stationary.”
— “Kei0210”

“Advanced math, explained in a playful way. But it’s not just a textbook, with dry solutions to problems. It’s a bittersweet story, with mathematics telling part of the tale. A brilliant comparison between the uncertainties of youth and the absolute proofs of symbols and numbers.”
— Shiori Oguchi

Paul Erdos’ Proof that there are Infinite Primes (with Examples)

Every integer can be uniquely written as , where is square-free (not divisible by any square numbers). For instance, 6 is square-free but 18 is not, since 18 can be divided by .

We can do this by letting to be the largest square number that divides , and then let . For instance, if , is the largest square number that divides 108, so we let .

Now, suppose to the contrary that a finite number k of prime numbers exists. We fix a positive integer , and try to over-estimate the number of integers between 1 and . Using our previous argument, each of these numbers can be written as , where is square-free and and are both less than .

By the fundamental theorem of arithmetic, there are only square free numbers. (The number of subsets of a set with k elements is 2^{k}) Since , we have .

Hence, the number of integers less than N is at most . ( choices for and choices for )

i.e. , for all N.

This inequality does not hold for sufficiently large. For instance, we can let , then .

Hence, this is a contradiction, and there are infinitely many primes!

An example of how the above argument works: Suppose the only prime numbers are 2, 3, 5. (k=3)

Then, there are only square-free numbers, namely, 1, 2, 3, 5, 2×3=6, 2×5=10, 3×5=15, 2x3x5=30.

Sometimes, E Maths can seem more difficult than A Maths!

Nevertheless, exposure and practising E Maths questions will make it easier as the student gets used to the problem solving techniques required to solve “tricky” E Maths questions.

Can you solve the following E Maths Problem? 🙂

It takes 8 workers to complete a certain job. The job can be completed 2 days earlier if 8 more workers are added. How many workers are needed if the house has to be completed 3 days earlier?

Once upon a time, there lived a Mathematician named Euclid.

Euclid came up with an ingenious method of proving that there are infinitely many prime numbers. Prime numbers are whole numbers that only have two factors, one and itself. For instance, 7 is a prime number while 6=2×3 is not.

The proof is by contradiction. Suppose that there are only a finite number of prime numbers, .

Now, we consider the number .

P is either a prime number, or it is a composite number.

If P is a prime number, then we have just contradicted the assumption that there are only a finite number of prime numbers !

If P is a composite number, then, it has a factor (smaller than P). However, none of can be a factor since P divided by those primes will leave a remainder of 1! Hence, P has another factor that is not in the original list, contradicting the initial assumption once again.

(Strictly speaking, Euclid’s proof is not by contradiction, instead he used a similar argument to show that given any finite list of primes, there is at least one other prime.)

This is the link to the Straits Times Education Forum, for parents who are interested to attend. The keynote speaker is Mr Heng Swee Keat, Minister of Education.

If you are looking for tuition for other subjects, be it English Tuition, Social Studies Tuition, Geography Tuition, Physics Tuition, Chemistry Tuition, Biology Tuition, Chinese Tuition, Economics Tuition, GP Tuition or Piano/Violin Lessons …

Come for E Maths Tuition at Bishan! Conveniently located at Block 230 Bishan Street 23 #B1-35 S(570230), near Catholic High School.

Schedule:

Monday 7pm-9pm (E Maths)

(Perfect for students who have CCA in the afternoon, or students who want to keep their weekends free.)

Do not underestimate E Maths!

Those who truly understand the Maths Syllabus at Secondary Level will know that E Maths is in some ways, more difficult than A Maths. Not to mention the infamous E Maths Bell Curve, which is definitely well above 75 marks to get a chance of A1. (Some sources say the Bell Curve for E Maths is 90 marks. See http://forums.sgclub.com/singapore/questions_about_o_378705.html )

“Amaths is good, like ditzy said, I tell you, if you practise amaths everyday and do past years papers, you can see amaths quite okay one. During my secondary school years i felt that emaths more difficult than amaths.”

“You can ask many people out there, which one is more tedious and require thinking more, they’ll say emaths.
Reason simple very simple:
Emaths:
They throw an entire paragraph at you which hidden clues here and there, slight errors is almost unavoidable.
Amaths:
Short and simple, they throw question at you. Just solve. End of story.”

How to use Math to calculate Chinese Zodiac and Impress your Friends

The Shengxiao (Chinese: 生肖, literally “birth likeness”), also known in English as the Chinese zodiac (“zodiac” derives from the similar concept in Western Astrology and means “circle of animals”), is a scheme and systematic plan of future action, that relates each year to an animal and its reputed attributes, according to a 12-year cycle. It remains popular in several East Asian countries, such as China, Vietnam, Korea and Japan. (Wikipedia: Chinese Zodiac)

Simple Mental Calculation: Years after 1900

Subtract 1900 from the year you are finding. E.g. 1988-1900=88

Divide your answer by 12 and find the remainder. E.g. 88/12=7R4

Add 1 to your answer. E.g. 4+1=5

The answer represents the Chinese Zodiac of that year! (1-Rat 鼠, 2-Ox 牛, 3-Tiger 虎, 4-Rabbit 兔, 5-Dragon 龙, 6-Snake 蛇, 7-Horse 马, 8-Goat 羊, 9-Monkey 猴, 10-Rooster 鸡, 11-Dog 狗, 12-Pig 猪) Hence, 1988 is the Year of the Dragon, since 5 represents Dragon.

Calculation for Years after 2000

Very similar to the above, with some slight changes.

Subtract 2000 from the year you are finding. E.g. 2015-2000=15

Divide your answer by 12 and find the remainder. E.g. 15/12=1R3

Add 5 to your answer. E.g. 3+5=8 (Note: If your answer is greater than 12, subtract 12 from it)

The answer represents the Chinese Zodiac of that year! (1-Rat, 2-Ox, 3-Tiger, 4-Rabbit, 5-Dragon, 6-Snake, 7-Horse, 8-Goat, 9-Monkey, 10-Rooster, 11-Dog, 12-Pig) Hence, 2015 is the Year of the Goat, since 8 represents Goat.

Year 1900: Rat
Year 1901: Ox
Year 1902: Tiger
Year 1903: Rabbit
Year 1904: Dragon
Year 1905: Snake
Year 1906: Horse
Year 1907: Goat
Year 1908: Monkey
Year 1909: Rooster
Year 1910: Dog
Year 1911: Pig
Year 1912: Rat
Year 1913: Ox
Year 1914: Tiger
Year 1915: Rabbit
Year 1916: Dragon
Year 1917: Snake
Year 1918: Horse
Year 1919: Goat
Year 1920: Monkey
Year 1921: Rooster
Year 1922: Dog
Year 1923: Pig
Year 1924: Rat
Year 1925: Ox
Year 1926: Tiger
Year 1927: Rabbit
Year 1928: Dragon
Year 1929: Snake
Year 1930: Horse
Year 1931: Goat
Year 1932: Monkey
Year 1933: Rooster
Year 1934: Dog
Year 1935: Pig
Year 1936: Rat
Year 1937: Ox
Year 1938: Tiger
Year 1939: Rabbit
Year 1940: Dragon
Year 1941: Snake
Year 1942: Horse
Year 1943: Goat
Year 1944: Monkey
Year 1945: Rooster
Year 1946: Dog
Year 1947: Pig
Year 1948: Rat
Year 1949: Ox
Year 1950: Tiger
Year 1951: Rabbit
Year 1952: Dragon
Year 1953: Snake
Year 1954: Horse
Year 1955: Goat
Year 1956: Monkey
Year 1957: Rooster
Year 1958: Dog
Year 1959: Pig
Year 1960: Rat
Year 1961: Ox
Year 1962: Tiger
Year 1963: Rabbit
Year 1964: Dragon
Year 1965: Snake
Year 1966: Horse
Year 1967: Goat
Year 1968: Monkey
Year 1969: Rooster
Year 1970: Dog
Year 1971: Pig
Year 1972: Rat
Year 1973: Ox
Year 1974: Tiger
Year 1975: Rabbit
Year 1976: Dragon
Year 1977: Snake
Year 1978: Horse
Year 1979: Goat
Year 1980: Monkey
Year 1981: Rooster
Year 1982: Dog
Year 1983: Pig
Year 1984: Rat
Year 1985: Ox
Year 1986: Tiger
Year 1987: Rabbit
Year 1988: Dragon
Year 1989: Snake
Year 1990: Horse
Year 1991: Goat
Year 1992: Monkey
Year 1993: Rooster
Year 1994: Dog
Year 1995: Pig
Year 1996: Rat
Year 1997: Ox
Year 1998: Tiger
Year 1999: Rabbit
Year 2000: Dragon
Year 2001: Snake
Year 2002: Horse
Year 2003: Goat
Year 2004: Monkey
Year 2005: Rooster
Year 2006: Dog
Year 2007: Pig
Year 2008: Rat
Year 2009: Ox
Year 2010: Tiger
Year 2011: Rabbit
Year 2012: Dragon
Year 2013: Snake
Year 2014: Horse
Year 2015: Goat
Year 2016: Monkey
Year 2017: Rooster
Year 2018: Dog
Year 2019: Pig
Year 2020: Rat
Year 2021: Ox
Year 2022: Tiger
Year 2023: Rabbit
Year 2024: Dragon
Year 2025: Snake
Year 2026: Horse
Year 2027: Goat
Year 2028: Monkey
Year 2029: Rooster
Year 2030: Dog
Year 2031: Pig
Year 2032: Rat
Year 2033: Ox
Year 2034: Tiger
Year 2035: Rabbit
Year 2036: Dragon
Year 2037: Snake
Year 2038: Horse
Year 2039: Goat
Year 2040: Monkey
Year 2041: Rooster
Year 2042: Dog
Year 2043: Pig
Year 2044: Rat
Year 2045: Ox
Year 2046: Tiger
Year 2047: Rabbit
Year 2048: Dragon
Year 2049: Snake
Year 2050: Horse
Year 2051: Goat
Year 2052: Monkey
Year 2053: Rooster
Year 2054: Dog
Year 2055: Pig
Year 2056: Rat
Year 2057: Ox
Year 2058: Tiger
Year 2059: Rabbit
Year 2060: Dragon
Year 2061: Snake
Year 2062: Horse
Year 2063: Goat
Year 2064: Monkey
Year 2065: Rooster
Year 2066: Dog
Year 2067: Pig
Year 2068: Rat
Year 2069: Ox
Year 2070: Tiger
Year 2071: Rabbit
Year 2072: Dragon
Year 2073: Snake
Year 2074: Horse
Year 2075: Goat
Year 2076: Monkey
Year 2077: Rooster
Year 2078: Dog
Year 2079: Pig
Year 2080: Rat
Year 2081: Ox
Year 2082: Tiger
Year 2083: Rabbit
Year 2084: Dragon
Year 2085: Snake
Year 2086: Horse
Year 2087: Goat
Year 2088: Monkey
Year 2089: Rooster
Year 2090: Dog
Year 2091: Pig
Year 2092: Rat
Year 2093: Ox
Year 2094: Tiger
Year 2095: Rabbit
Year 2096: Dragon
Year 2097: Snake
Year 2098: Horse
Year 2099: Goat
Year 2100: Monkey

Contrary to popular opinion, 1 divided by 0 is not infinity! Wikipedia states that “the expression has no meaning, as there is no number which, multiplied by 0, gives a (assuming a≠0), and so division by zero is undefined“.

How to show that division by zero is undefined

The limit of 1/x as x approaches zero from the right is positive infinity.

However,

The limit of 1/x as x approaches zero from the left is negative infinity.

Since the left limit and right limit are different, the limit of 1/x as x approaches infinity does not exist!

Note: There are mathematical structures in which a/0 is defined for some a (see Riemann sphere, real projective line, and section 4 for examples); however, such structures cannot satisfy every ordinary rule of arithmetic (the field axioms).

There are also more advanced methods of proving 0.999…=1, listed here on Wikipedia. (http://en.wikipedia.org/wiki/0.999…) Some of the techniques include Infinite series and sequences, Dedekind cuts, and Cauchy sequences.

Most people educated in the past would simply use the usual subtraction working to arrive at the answer of 111.

But lately, in the USA, the Common Core Math approach gets very complicated and teaches an approach that even the father who has a “Bachelor of Science Degree in Electronics Engineering, which included extensive study in differential equations and other higher math applications” cannot explain it, nor get the answer correct.

See also the Common Core method of adding 26+17:

Is it really necessary to use “number bonds” to “skip-count by seven” to add 7+7?

I agree with the father above that in Maths, “simplification is valued over complication”. This is why I always teach the easiest to understand method and the shortest method to solve questions to my students. No point using a complicated method when a shorter and simpler method can give the same answer!

As an international student, I assume your lodgings are taken care of, hence the bulk of your money will go to food and entertainment.

1. Eat at the university as much as possible

2. Eat at kopitiams rather than restaurants- and gives you a sense of the ‘authentic Singapore’, as it were.

3. Get this card. It gives you a 10% discount at every Kopitiam (the chain, not the general type of coffee shop). You can get back the deposit later.

4. Get second-hand textbooks. Sometimes, they have sales in the canteen. I got my marketing textbook for $5, and my biochemistry one for $20. Campbell and Reece went for $21. Genetics was $20, too, down from the original prices of $45-80.

5. Or don’t get textbooks at all. Go to the library.

Some people doubt that dogs are capable of even the most rudimentary form of quantitative thinking. The most basic form of analyzing the world in a quantitative way involves the judgment of size, namely, answering the question of whether one thing larger than another. Early researchers would put out two balls of hamburger, one large and the other small, and when they found that dogs were as likely to choose the small one as the large, they concluded that dogs could not estimate size. However there is a flaw in this test. Dogs think in an opportunistic manner, a sort of “A bird in the hand is worth two in the bush” mentality. If the two plates were a different distances, the dog would always grab the closest. However if they were at equal distances, the dog would show that he understood the notion of size by going after the larger one. Norton Milgram, at the University of Toronto confirmed that dogs can judge size well using a tray which contains two objects of different sizes. If the dog pushes the correct object, then underneath it he will find a food treat. Dogs can be taught always to pick the larger (or smaller) of two objects, regardless of the shape or identity of the objects, and they learn this fairly easily.

Watch the above video, about Maggie the Jack Russell that can do Arithmetic!

Maggie the Jack Russell is a maths genius (for a dog!), watch her compete with a class of 7 year olds.

However, there has been the case of Clever Hans, a horse that seemingly can do Mathematics.

However, it was “not actually performing these mental tasks, but was watching the reaction of his human observers”. (Wikipedia) Could Maggie be observing the subconscious signals of her owner? The owner might not even be aware of that, like in the case of Clever Hans.

Is Maths Dog Maggie a Fraud? Robert De Franco devises a series of tests to see if owner Jessie is somehow prompting Maggie the right answers.

Wikipedia:

Clever Hans (in German, der Kluge Hans) was an Orlov Trotter horse that was claimed to have been able to perform arithmetic and other intellectual tasks.

After a formal investigation in 1907, psychologist Oskar Pfungst demonstrated that the horse was not actually performing these mental tasks, but was watching the reaction of his human observers. Pfungst discovered this artifact in the research methodology, wherein the horse was responding directly to involuntary cues in the body language of the human trainer, who had the faculties to solve each problem. The trainer was entirely unaware that he was providing such cues.^{[1]} In honour of Pfungst’s study, the anomalous artifact has since been referred to as the Clever Hans effect and has continued to be important knowledge in the observer-expectancy effect and later studies in animal cognition. Hans was studied by the famous German philosopher and psychologist Carl Stumpf in the early 20th century. Stumpf was observing the sensational phenomena of the horse, which also added to his impact on phenomenology.