Memorize Trick For Ring:

1. Ring (R) is for Marriage with 2 operations: + (addition of kids), * (multiply asset).

2. (R , +) is Commutative Group
Analogy: your kids are also your wife’s kids, vice versa.

3. (R ,*) is Semi-Group (only closure & associative)
Analogy: your asset to multiply (*) is semi (50%) owned by your wife.

4. Fair distributive law for * with respect to +
=> distribute your asset (a) to your kids (k) & wife (w):
a*(k+w) = a*k + a*w

5. No division (/) operation => Ring can’t be broken.

A few weeks ago I wrote a post in shock at the fact that a number of schools in the locality had been teaching rounding wrongly. By this I don’t mean I took issue with their pedagogical approach, but rather that the method of rounding they were teaching was wrong and would result in the wrong answer in some instances. This post received the following comment from Old Andrew (@andrewolduk):

“Is anybody going to raise the obvious issue here that a lot of maths teachers don’t have maths degrees? In recent years the rise of “conversion courses” means that many have a background that isn’t in the least mathematical.”

This stirred up some quite strong feelings on both sides. Sophie Skinner (@miss_skinner) wrote this excellent piece stating her views, and Dave Gale (@reflectivemaths) wrote this piece reflecting his view. Twitter has been buzzing with conversation on the matter, and…

Description: *FREE app at time of post *  Sushi Monster is a ridiculously cute and engaging way to practice math fact fluency (addition and multiplication)!  Students must use their critical thinking and reasoning skills to strategically decide which sushi pieces (each has a number above it) to add/multiply together to feed the monster the total he desires.  Within each round, students see all the upcoming totals in the top left-hand corner so they can plan ahead.  Next to this bar is a Number Sentence bar that builds the addition/multiplication sentence as the student chooses sushi.  If it’s wrong, the monster throws a fit and tosses the sushi plates back!  (What a picky monster…)  Levels within this app are sequenced so that they become progressively more difficult as the student completes them.

Compatible with the following device(s): iPad, iPod Touch, and iPhone

Implementation in classroom:

– Activity idea:  This is a…

http://en.m.wikipedia.org/wiki/Paul_Erd%C5%91s

What is Erdős Number ?

Amazon Book: “My Brain Is Open”

Paul Erdős (er-dish) was one of the greatest mathematicians in 21st Century, a Jewish Hungarian, single and no home. He traveled around the world in one small suitcase containing his mathematical papers. He would knock at the door of his former students or math collaborator impromptu, started working and published the papers, then moved on to next destination the moment his overstay became unwelcome by the host’s wife. In this way he published 1,500 articles in his lifetime.

Example:

Einstein had the Erdős Number 2 , through his assistant who co-wrote a paper with Paul Erdős.

Ref:

http://tomcircle.wordpress.com/2013/03/30/paul-erdos/

Prof Benedict Gross is one of the best Algebra professors I have seen – he can explain so well the abstract concepts, without injecting fear and confusion to the students.

As Prof Gross had brilliantly said in the beginning of this Lecture 1:

Algebra is the language of Math.

Since Math is the language of science,
therefore any serious Science needs to speak in Algebra language.

Today, if you read a research paper on any math (or Computer Science, Mathematical Physics…) topic, hardly you can avoid these “basic” algebraic lingoes: Group, Ring, Field, Vector Spaces, Quotient Group, Ideal, …

I strongly recommend to anyone who likes to study Modern Algebra but afraid of the abstractness, this is the course (free) for you. I can guarantee you by the halfway (15th lecture) you will have a solid foundation, and by the last lecture you will be able to follow high-level…

$latex boxed{n!=int_{0}^{infty}x^{n}e^{-x}dx}$

For original post: < Click Here >

The romantic gallic Frenchmen like to joke and give pranks. We have already seen the Number 1 Mathematical ‘prank’ in Our Daily Story #1 (The Fermat’s Last Theorem), here is another 20th century Math prank “Nicolas Bourbaki” – the anonymous French mathematician who did not exist, but like Fermat, changed the scene of Modern Math after WW II.

http://en.m.wikipedia.org/wiki/Nicolas_Bourbaki

André Weil (not to confuse with Andrew Wiles of FLT in Story #2 ) and his university classmates from the Ecole Normale Supérieure (Évariste Galois‘s alma mater which expelled him for involvement in the French Revolution), wanting to do something on the outdated French university Math textbooks, formed an underground ‘clan’ in a Parisian Café near Jardin du Luxembourg. They met often to brainstorm and debate on the most advanced Math topics du jour. Finally they decided to totally re-write the foundation of Math based on Set…

Description: *FREE app*  Although Math vs Zombies is probably meant for children, I found this app to be really fun, especially if there were watching me play and screaming in anxiety!  Basically, cute/quirky zombies are coming to get you, and the only way to change them back into humans (and save yourself) is to solve math equations correctly.  There’s nothing like a little urgency to get those brain juices flowing! Students using this app can choose from four basic operations: addition, subtraction, multiplication, and division.  Within each Operation Level, there are six mini levels and a boss level to beat, and the equations become progressively more difficult between mini levels.  Also, the difficulty level can be set to easy, medium, or hard by clicking on the gear in the bottom left-hand corner of the main menu.

Compatible with the following device(s): iPad, iPod Touch, and iPhone

Implementation in classroom:

–…

The 3 valuable  takeaways from this 1- hour interview with Steve Ballmer, who co-founded Microsoft with the richest man in the world Bill Gates:

1. Have a great idea before startup.
– The financial,  talent, etc come later.

2. Great company does (at least) one trick well.
– Microsoft has 2.5 tricks: PC operating system (Windows); Microprocessors in data center (Winservers); the half trick is X box game console. 

– Apple also has 2 tricks: Mac; Mobile computing in iPhone/iPAD/Appstore

– Facebook (Social Network) and Google (Search Engine)  have 1 trick.

– Sony has 1 trick : Audio TV

– HP has 1 trick: Tester Equipment (now in Agilent)

– IBM has 1 trick: Entreprise Data Centre

– Samsung has 1 trick: hardware manufacturer (LCD flat panel, Smartphone…)

– Amazon has 1 trick: online bookshop

– Alibaba.com has 1 trick: eCommerce

Many great companies degenerate into smallness or extinction, because…

Remember that you can solve the puzzles from the League online on the Sudokucup Guest League page with a 24h delay (though I’m not certain whether this is too late to be put up today).

I’m sorry dearest readers, I think I’ll have to skip an explanation of why I’m inactive on my blog right now for later. There’s just too many plans right now, that all seem to end around May, and if it all goes as planned, you’ll see at least some of the reasons why I’ve been too busy to post here.

Anyway, here’s a Killer Sudoku that was used for the Polish Sudoku Championship qualifiers. These qualifiers were open to International solvers online, so I think many solvers would have seen this already. I still think it’s nice enough to put it up for those of you who haven’t.

Rules – Follow Classic Sudoku rules.
Additionally…

Here are some amazing spiritual math and science facts:

1 + 1 + 1 = 3 when it’s the Trinity. God is three in one! No, we can’t figure it out but God is not limited by the math He invented for us to make sense of our little world. I delight in worshiping a God who is great beyond my understanding .

2 wrongs don’t make 1 right. It’s not like multiplying two negative numbers which gives us a positive.
For example, 1 unintended pregnancy + 1 abortion does NOT equal one “right.” (No one has the “right” to kill a baby—including his/her mother!)

Gravity (physical law) always pulls us down. But glory lifts us upward. Our body is temporal; it has to stay on this earth whether we’re alive or dead. Our spirit can receive God’s never-ending life that give us supernatural power during this life and eternal…

Check out Bedtime Math!

Bedtime Math

Go do a fun Math problem daily with kids of any age! You can also have it sent to you daily. My daughter asks to do her Bedtime Math and so will your kids!!

Have a happy day★

Jamie★

Are You A Genius? Read These Signs And Find Out

From the previous O.D.S. stories (#3, #4) on Quintic equations (degree 5) by Galois and Abel in the 19th century, we now trace back to the first breakthrough in the 16th century of the Cubic (degree 3) & Quartic (degree 4) equations with radical solution, i.e. expressed by 4 operations (+ – × /) and radicalroots {$latex sqrt{x} , : sqrt [n]{x} $ }.

Example: Since Babylonian time, and in 220 AD China’s Three Kingdoms Period by 趙爽 Zhao Shuang of the state of Wu 吳, we knew the radical solution of Quadratic equations of degree 2 :
$latex ax^2 + bx + c = 0 $

can be expressed in radical form with the coefficients a, b, c:

$latex boxed{x= frac{-b pm sqrt{b^{2}-4ac} }{2a}}$

Are there radical solutions for Cubic equation (degree 3) and Quartic equations (degree 4) ? We had to wait till the European Renaissance…

Zhang is the typical demonstration of pure perseverance of traditional Chinese mathematicians: knock harder and harder until the truth is finally cracked.

His work is based on the prior half-way proof by 3 other mathematicians “GPY”:

Gap between Primes:

Let p1 and p2 be two adjacent primes separated by gaps of 2N:

p1 – p2 = 2 (twin primes)
eg. (3, 5), (5, 7)… (11, 13) and the highest twin primes found so far (the pair below: +1 and -1)

p1 – p2 = 4 (cousin primes)
eg. (7, 11)

p1 – p2 = 6 (sexy primes)
eg. (23, 29)
…
p1 – p2 = 2N

Euclid proved 2,500 years ago there are infinite many primes, but until today nobody knows are these primes bounded by a gap (2N) ?

Zhang, while working as a sandwich delivery man in a Subway shop…

Carl Friedrich Gauss is named the “Prince of Math” for his great contributions in almost every branch of Math.

As a child of a bricklayer father, Gauss used to follow his father to construction site to help counting the bricks. He learned how to stack the bricks in a pile of ten, add them up to obtain the total. If a pile has only 3, for example, he would top up 7 to make it 10 in a pile. Then 15 piles of 10 bricks would give a total of 150 bricks.

One day in school, his teacher wanted to occupy the 9-year-old children from talking in class, made them add the sum:
1 + 2 + 3+ ….+ 98 + 99 + 100 = ?

Gauss was the first child to submit the sum within few seconds = 5,050.

He used his brick piling technique: add

1 + 100…

To prove the FLT, Prof Andrews Wiles used all the math tools developed from the past centuries till today. One of the key tool is the Galois Group,  invented by a 19-year-old French boy in 19th century, Evariste Galois. His story is a tragedy – thanks to the 2 ‘incompetent’ examiners of the Ecole Polytechnique (a.k.a. “X”), the Math genius failed in the Concours (Entrance Exams) not only once, but twice in consecutive years.
Rejected by universities and the ugly French politics and academic world, Galois suffered set back one after another, finally ended his life in a ‘meaningless’ duel at 20.

He wrote down his Math findings the eve before he died – “Je n’ai pas le temps” (I have no more time) – begged his friend to send them to two foreigners (Gauss and Jacobi) for review of its importance. “Group Theory”…

The story of Ramanujian:

http://en.m.wikipedia.org/wiki/Srinivasa_Ramanujan

http://mathworld.wolfram.com/Hardy-RamanujanNumber.html

We have seen how two 19th century greatest mathematicians Cauchy and Gauss who were not helpful to two young unknown mathematicians Galois and Abel, now let’s see an opposite example — the discovery of an unknown math genius Ramanujian by the greatest Pure Mathematician in 20th century Prof G.H. Hardy.

References:
1.
http://tomcircle.wordpress.com/2013/11/27/163-and-ramanujan-constant/

2.
http://tomcircle.wordpress.com/2013/07/04/ramanujian/

In the previous story (#9) we mentioned Ramanujan having the luck of being spotted by Prof G.H. Hardy as the treasure of mathematics, another Chinese Hua Luogeng 华罗庚, 20 years younger than Ramanujan,  was also coached by Prof Hardy, although Prof Hardy did not realize Hua’s potential later to the modern mathematics in China.

Hua dropped out of secondary school due to poverty, he worked in his father’s little grocery shop as the shop assistant. His talent was spotted by the French-educated mathematician Prof Xiong Qinlai ( 熊庆来) in Tsinghua University 清华大学 from a paper the young boy published – on Quantic Equation Solvability error made by a Math Professor Su. Hua was admitted to Tsinghua University as assistant math lecturer on exception. Later he was sent to Cambridge on 庚子赔款 Boxer Indemnity scholarship.

When Prof Hardy met Hua, he let Hua choose between:
1) Work on a PhD…

This video explains how to perform stick multiplication and shows how it relates to the method of partial products.

http://en.m.wikipedia.org/wiki/Augustin-Louis_Cauchy

We mentioned Augustin Louis Cauchy in the tragic stories of Galois and Abel. Had Cauchy been more generous and kind enough to submit the two young mathematicians’ papers to the French Academy of Sciences, their fates would have been different and they would not have died so young.

Cauchy was excellent in language. He was the 2nd most prolific writer (of Math papers) after Euler in history. When he was a math prodigy, his neighbor — the great French mathematician and scientist Pierre-Simon Laplace — advised Cauchy’s father to focus the boy on language before touching mathematics. (Teachers / Parents take note of the importance of language in Math education.)

Cauchy’s language education made him very rigorous in micro-details. This was the man who developed the most rigorous epsilon-delta Advanced Calculus (called Analysis) after Newton / Lebniz had invented the non-rigorous Calculus (why?).

Rigorous epsilon-delta…

We all know how terrible homework is and how long it takes and how you could be doing SOOO many other things that are not homework. Like nothing productive. That kind of stuff. And once we actually get started, it’s so difficult to keep on track. Sometimes we hit a brick wall and so we just stop and do something else. And then there are those moments where you have so much work to do that you take a nap. Or the infamous “due tomorrow do tomorrow” mentality. When teachers assign a certain portion of reading you think, “Sweet! No homework!” just because you don’t get credit for actually reading it. This affects almost every single person and extends beyond schooling and even into your real life work. What should you do? Well I can’t tell you, but I can confide in the type of mentality I have and what…

Arcademic Skill Builders :

Free, engaging educational games for students in K-8 to practice skills in math, language arts, geography, and other subjects

Features multiplayer mode so students can practice while competing against classmates in real time

via tumblr published on March 02, 2014 at 05:30PM

I hope that most of you have either asked yourselves this question explicitly, or at least felt a vague sense of unease about how the definitions I gave in lectures, namely

$latex displaystyle cos x = 1 – frac{x^2}{2!}+frac{x^4}{4!}-dots$

and

$latex displaystyle sin x = x – frac{x^3}{3!}+frac{x^5}{5!}-dots,$

relate to things like the opposite, adjacent and hypotenuse. Using the power-series definitions, we proved several facts about trigonometric functions, such as the addition formulae, their derivatives, and the fact that they are periodic. But we didn’t quite get to the stage of proving that if $latex x^2+y^2=1$ and $latex theta$ is the angle that the line from $latex (0,0)$ to $latex (x,y)$ makes with the line from $latex (0,0)$ to $latex (1,0)$, then $latex x=costheta$ and $latex y=sintheta$. So how does one establish that? How does one even define the angle? In this post, I will give one possible answer to…

I remember my secondary teacher told me that 1 cannot be divided by 0.

Why 0 cannot be divisor?

“Because it is just not permitted in arithmetic”, he said.

Well. It is not permitted because it is not permitted. A kid shouldn’t ask much…

Throughout my N years of learning and teaching, I know that 0 cannot be a divisor because if it does, something will go wrong in mathematics. But how things can go wrong other than the calculator showing some error messages?

Here is a simple example.

Let say

  1 / 0 = SOMETHING,

is valid.

If so, I can multiply both left hand and right hand sides by 0.  And, it becomes

1 = SOMETHING * 0.
1 = 0.

Oops, something is wrong with the arithmetic. That’s why we can’t have 0 as the divisor.

Well. Perhaps you are not yet convinced and ask “how about…

Galois and Abel had many things in common: both worked on the Quintic equation (of degree 5). Abel first proved there was NO radical solution; Galois, who was 9 years younger, went one step further to explain WHY no solution (with Group theory).

Both were young Math genius not recognized by the world of mathematics. Their fates were ruined by the same French mathematician Cauchy, who hid their Math papers from the recognition of the French Academy of Science.

Both died young: Abel at 26,  Galois 20.
Abel was poor and weak in health. His dream job of professorship came 2 days (too late) after his death.

Ironically, today the top Math award in monetary term (US$ 1 million) for the world’s top mathematician is named after this extremely poor mathematician – the Abel Prize.

http://scienceworld.wolfram.com/biography/Abel.html

(Go to YouTube “Niels Henrik Abel” to read the English sub-title)

 

If you want to feel comfortable with math, you have to understand that the world exists through a series of complex patterns. Admittedly, we can’t say this for sure. The universe is so infinitely large (and by the time you finish reading this blog post it will have already grown by an immeasurable amount), and our grasp on math is so tentative as humans that it’s impossible to notice or even understand the pattern at work on a large scale. However, the belief in patterns can be thought of more as a philosophy rather than an explanation for the world.

 

One of the most fundamental patterns in Algebra and number theory is the Fibonacci sequence. Now, this can get pretty dense, so bear with me.

 

The Fibonacci sequence was first introduced in 1202 by an (you guessed it) Italian mathematician Leonardo Fibonacci. Though, as we’ll soon see…

Follow up with the story #1 on FLT (Fermat’s Last Theorem),  it was finally cracked 358 years later in 1994 by a British mathematician Professor Andrew Wiles in Cambridge.
The proof of FLT is itself another exciting story, a 7-year lonely task on the attic top of his Cambridge house, nobody in the world knew anything about it, until the very day when Prof Wiles gave a seemingly unrelated lecture which ended with his announcement: FLT is finally proved. The whole world was shocked!

http://en.m.wikipedia.org/wiki/Wiles%27_proof_of_Fermat%27s_Last_Theorem

Part 1/5 Andrew Wiles and FLT Proof:

(Part 2 – 5 to follow from YouTube)

Speech at IMO by Andrew Wiles: