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I'm a children's book writer and illustrator, and I want to to create a book for young readers that exposes the beauty of mathematics. I recently read Paul Lockhart's essay "The Mathematician's Lament," and found that I, too, lament the uninspiring quality of my elementary math education.

I want to make a book that discredits the notion that math is merely a series of calculations, and inspires a sense of awe and genuine curiosity in young readers.

However, I myself am mathematically unsophisticated.

What was the first bit of mathematics that made you realize that math is beautiful?

For the purposes of this children's book, accessible answers would be appreciated.

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For me Euclid's proof of the infinitude of primes was the first thing that made me realize the beauty of mathematics. – Manjil P. Saikia Mar 7 '13 at 7:02
Wow. Just last night I had a fierce argument with one of the bartenders of my usual watering hole who is a mechanical engineering student. He insisted that he has a better idea than me of what is mathematics. I am so going to print him a copy of Lockhart's text. Thank you for that link! – Asaf Karagila Mar 7 '13 at 7:59
I can’t remember a time when I didn’t think that mathematics was beautiful and fascinating. – Brian M. Scott Mar 7 '13 at 15:06
Although I don't know if it's what you are looking for, try looking up "vihart" on youtube--Even if it's not helpful, I guarantee you will appreciate it. – Bill K Mar 8 '13 at 2:57
I think it's a shame that this question was voted closed... – Will Mar 10 '13 at 19:50

160 Answers 160

I don't remember what the first beautiful piece of math I encountered was, but here are a couple of candidates:

  • Proof that the square root of 2 is irrational

  • Euclid's proof that there are infinitely many prime numbers

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+1: I must confess that I have showed those two proofs to many people, hoping that I will be able to recreate in them the awe and fascination that those proofs ignited in me. Unfortunately, although some did find it a neat curiosity, I have failed to kindle in them a permanent interest for mathematics. – user5501 Mar 7 '13 at 10:07
When I first saw the proof for infinitely many primes it blew my mind as well! – sdm350 Mar 7 '13 at 17:59
@SteveChambers I hope you find it even more beautiful to learn that Euclid lived over 2000 years before 1782! – Erick Wong Mar 9 '13 at 21:37
@ErickWong If he lived over 2000 years he probably wasn't very beautiful toward the end :-J – Trevor Wilson Mar 9 '13 at 21:45

This wasn't the first, but it's definitely awesome:

A Proof of the Pythagorean Theorem (without words)

This is a proof of the Pythagorean theorem, and it uses no words!

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@11684: This picture is a proof of the Pythagorean Theorem, which says that $a^2 + b^2 = c^2$ in right triangles. – Jesse Madnick Mar 7 '13 at 12:06
Wow... You're right this is amazing. Perhaps it's an idea to add that to your answer? (Accessibility was a requirement.) – 11684 Mar 7 '13 at 12:16
@double_squeeze, the idea is that the area of both big squares is the same, and the area of the two sets of four triangles is the same. In the first picture, we see that the remainder of the area after taking away the four triangles is $a^2 + b^2$. In the second picture, the very same remainder is just $c^2$. So, we have $a^2 + b^2 = c^2$. It should be clear that one can draw a similar picture for any right triangle. QED. – Will Mar 8 '13 at 5:30
This is not a proof, this is THE proof of the Pythagorean Theorem. If you traveled back in time, and told Pythagoras himself that $a^2 + b^2 = c^2$, he either wouldn't understand you, or (after you defined what the square root means) wouldn't believe you. – vsz Mar 8 '13 at 7:11
@AKE, the point of this picture is that it doesn't require 'a little algebra'. – jwg Mar 12 '13 at 13:36
  • Like Trevor Wilson, I was awed by human ingenuity where just by looking at "few" given numbers, one can deduce that there are infinitely many primes from reason alone and doing basic operations. (Here is more on Euclid's proof.)

  • As freshman undergrad in college, always loved Theoni Pappas' Joy of Mathematics before being introduced to Raymond Smullyan, Charles Seife (Biography of Zero), Rudy Rucker and Hoftstadter's books.

  • As far as Math.SE's question is concerned, this was an interesting brain teaser and simplicity at best.

===================EDITED THE FOLLOWING BELOW==================== I just realized although the above have been influential, but earliest memory of the workings of mathematics came in the manner of following magic trick aged six or seven:

Effect: Performer asks someone to write down a random long number 4567829872367783456753745673456347567346534756 and he writes the another line of the matching digit, so let's say she writes 1263347567346534756378567563434543534543534545 and after that performer asks another audience member to approach the blackboard on dimly lit stage. Let's say the next random line 8636652432653465243621432436565456465456465454 and random number and as he writes 5555555555555555555555555555555555555999999990 the performer quickly writes below 4444444444444444444444444444444444444000000009 and then pauses. Then he continues his patter: Now I could not have possibly known what digits you would have chosen, right ladies and gentlemen? Well, let me gather my thoughts for a while and clear my I attempt to add this rather cumbersome mess in just the time of writing it down. Then he approaches the blackboard and without hesitation calculates the answer:


which of course proves to be correct.

Method: There is of course no telepathy and the trick is entirely mathematical in nature. If the reader wants the audience to choose the first line make sure the last digit does not end in 0 or 1. So what the practitioner would do is matching the digits of audience write the complement of the number adding to 9. Say the line is of a 10-digit sequence of 5s then the performer should write a 10-digit sequence of 4s. To add the whole block one simply copies down the first line with 2 in front of it and subtracting 2 from the last digit. Hence the need for no 0 or 1 in the first line.

Tips: To make it realistic, make sure it is a cumbersome mess that is not too big of a block. Because say one smart aleck chooses all digits of 0000... and then when performer writes 999999... it may be a give away. Strike a balance between how big the mammoth block should be to appeal awe from students and the reality of randomness in the numbers. The rest is, of course, all up to the showmanship of mentalist.

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+1 for the first bullet point! I had the same enthralling reaction to the Euclid's proof that there are infinitely many primes. It was the first time in my life that I realized there was more to mathematics than mindless calculation, and it fundamentally changed the course of my life. – user5501 Mar 7 '13 at 9:58

I remember being very pleased at an early age, perhaps five or six, by the following bits of calculator tinkering, among others:

  • 12345679 × $n$ × 9 = nnnnnnnnn.
  • The cyclic behavior of the decimal expansions of $\frac n7$. For example, $4\times 0.142857\ldots = 0.571428\ldots$.
  • The reciprocity of digit patterns in numbers and their reciprocals. For example, $\frac12 = 0.5$ and $\frac15 = 0.2$; $\frac14 = 0.25$ and $\frac 1{2.5} = 0.4$. This is the earliest pattern I can remember observing completely on my own. Similarly, I enjoyed that the decimal expansions of $\frac1{2^n}$ (0.5, 0.25, 0.125…) look like powers of 5.
  • The attraction of the map $x\mapsto \sqrt x$ to 1, regardless of the (positive) starting point. I liked that numbers greater than 1 were attracted downwards, and numbers less than 1 were attracted upwards. Later on I noticed, from looking at the calculator, that $\sqrt{1+x} \approx 1+\frac x2$ when $x$ is small; for example $\sqrt{1.0005} \approx 1.0002499$, and similarly when $x$ is negative. When this useful fact recurred later in calculus and real analysis classes, I was already familiar with it.

When I got a little older, I loved that I could find an $n$th-degree polynomial to pass through $n+1$ arbitrarily chosen points, and that if I made up the points knowing the polynomial ahead of time, the method would magically produce the polynomial I had used in the first place. I spent hours doing this.

I also spent hours graphing functions, and observing the way the shapes changed as I varied the parameters. I accumulated a looseleaf binder full of these graphs, which I still have.

As a teenager, I was thrilled to observe that although the number "2 in a pentagon" in the Steinhaus–Moser notation is far too enormous to calculate, it is a trivial matter to observe that its decimal expansion ends with a 6.

I realize that your book wants to discredit the notion that math is merely a series of calculations, but I have always been fascinated by calculation, and I sometimes think, as the authors of Concrete Mathematics say in the introduction, that we do not always give enough attention to matters of technique. Calculation is interesting, for theoretical and practical reasons, and a lot of very deep mathematics arises from the desire to calculate.

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Related: $\displaystyle \frac{1}{7}=0.(142857)(142857)(142857)(142857)(142857)\ldots$ and $$7\times 142857=999999$$ $\displaystyle \frac{1}{13}=0.(076923)(076923)(076923)(076923)(076923)\ldots$ and $$13\times 076923=999999$$ $$\vdots$$ – Git Gud Mar 7 '13 at 7:22
@DheeB Perhaps it sounds like that, but I was not. – MJD Mar 7 '13 at 20:05
Wow, you sound like me. Also hitting the cosine button on the calculator converges 0.739..., like $\sqrt{x}$. Rational approximations of $\pi$. But 1/7 was THE thing for me when I was nine. I looked at all the $x/n$ up to $n = 25$. All the questions I had about $1/7$ weren't answered until I took Abstract Algebra in college, a decade of curiosity! About that time, I switched to a math major. Even now I don't have a good answer to why the multiples of 3 are missing in the decimal expansion, other than the expansion has to be 6 digits long, which means some digits are missing. – Michael E2 Mar 11 '13 at 15:01
12345679 x 8 = 98765432 – jwg Mar 12 '13 at 13:39
@JoeZeng Thanks. I have a similar one: The fractions occurring at equal intervals, you have to cast out four digits equally spaced, starting with 0. It happens that every third one works perfectly, which seems accidental. Is there a reason that connects divisibility by 3 to the prime 7? That's the question I had as a child. I think the question is a bit childish, in that I dreamt a certain sort of answer might exist, but our explanations are probably better and closer to the "real reason." Not all dreams come true, but they propel us into searching for answers. – Michael E2 Mar 12 '13 at 15:19

These amazed me quite a lot when I first saw them:

$1.$ Prove that $|(a,b)| =|\Bbb R|$, $\forall a,b\in\Bbb R$ and $a<b$.

$2.$ Both $\Bbb Q$ and $\Bbb R\setminus \Bbb Q$ are dense in $\Bbb R$, but $\Bbb Q$ is countable set while $\Bbb R\setminus \Bbb Q$ is uncountable.

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At what age?${}$ – mrf Mar 7 '13 at 7:42
My brother used to tell me these kind of things from an early age. – Aang Mar 7 '13 at 11:15
You learned about dense, countable and uncountable sets at that age?? Its hard to believe...and its very strange that you have to get so far to see something you considered beautiful. – Integral Mar 7 '13 at 15:23
@tttppp … I read $(a,b)$ as an tuple and completely forgot it could also denote an open interval (which I write $(a..b)$ now) – even after searching for different meanings! It’s times like this that I wonder whether I have some serious brain condition. – k.stm Mar 15 '13 at 15:51

The first "math thing" that just blew my mind was the identity $$ e^{i\pi} = -1 $$ Namely the fact that the two independently discovered transcendent numbers and imaginary one so simply and elegantly bound.

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Didn't the title say childrens book? If you knew the concept behind $e^{i\pi} = -1$ as a child then you must be a brillant prodigy. – Q.matin Mar 7 '13 at 7:30
I was about 15 :), but my bad overlooked that part. – Kaster Mar 7 '13 at 7:33
I've always thought that it's kinda cheating to teach this to someone who doesn't fully understand complex numbers. I remember I first heard about this when I was about 16, and I thought it was some miraculous numerical coincidence, when in reality this exponentiation doesn't work like the one you know, so this identity does not mean what you think it means at that age. – Javier Mar 7 '13 at 12:04
@DheeB oh really? Complex exponential:Proofs – Kaster Mar 7 '13 at 20:37
@Dylan: many arguments involving $\pi$ are circular :) [couldn't resist bad pun, sorry..] – Marek Mar 8 '13 at 21:20

It is really difficult to remember my days as an elementary student. I just remember how beautiful I found math to be: the connections I saw between everything I was learning, the beauty of the patterns, the sense-making, the sheer marvel of it all. I cannot pinpoint one "fact" I learned, or one particular "ah-ha!" moment (there were so many), but I attribute my love for math as much to the freedom I was given to actively inquire and explore mathematics, as much as to the many marvels I discovered in this way.

I recall being encouraged (by remarkable teachers) to explore, ask questions, and try to find answers to those questions. I was given a lot a lee-way, apart from classroom lessons, to pursue the connections and patterns I saw, to conjecture, and confirm conjectures, or find counterexamples. Given this encouragement and flexibility, I found mathematics to be akin to solving mysteries. I wondered about what I was learning, and was able to anticipate what this would lead to, before receiving formal instruction. And this was terribly satisfying: the wonder, the pursuit, the discovery, and even "invention" (for myself) of things I would later find to be true.

So in a sense, I discovered as much about math as I learned through formal instruction, and didn't get trapped into the mechanistic learn-a-rule/apply-the-rule/produce-an-answer mode which so many students come to define as "doing math."

So it wasn't so much a matter of the facts I learned that drew me to, and keeps me enamored by, math: it was/is the activities of mathematics: the process of questioning why certain relationships hold, conjecturing, exploring, testing, discovering and chasing down implications, constructing an understanding, and defending or rejecting my hypotheses, and on and on...

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1) Modular arithmetic fascinated me. I could not believe that with just a few tools, I could find the remainder left when $3^{100}$ is divided by 8. ($3^2\equiv 1$ mod $8$ and hence the result.)

2) Euclid's proof of the infinitude of primes. (Let the number of primes be finite. Let them be $P=\{p_1,p_2,\dots,p_r\}$. Take $k=p_1p_2\dots p_r+1$. None of the primes in $P$ divides $k$, hence $k$ is a prime or divisible by a prime not in $P$, and so we have a contradiction.)

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I remember in geometry using direct reasoning once and another by the absurd, and I manage to show that lines are parallel, intersecting at a point, a triangle is isosceles, it is inscribed in a circle ..... I was fascinated by geometry.

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The fact that you can always divide something by two. That is an amazing discovery my dad tells me I made as a toddler.

I think that ever since I remember abstract mathematics was a fascination of mine, even before I knew what it was (because it was obvious school mathematics wasn't that).

Another fact I stumbled upon as a teenager and fascinated me was that if you hold a magnifying glass over a ruled paper the parallel lines bend, and eventually meet at the edge of the glass. That, in a nutshell, is a non-Euclidean geometry.

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That is a really amazing discovery. It reminds me of my own happy discovery that iterating $x\mapsto x/2$ produces numerals that look like powers of 5. Did you do your division with paper or with some sort of calculator? – MJD Mar 7 '13 at 7:32
@Asaf It reminds me of my discovery that $a+b\geq 2\sqrt{ab}$ by trial and error when I was in 8th grade. – Ishan Banerjee Mar 7 '13 at 7:41

This isn't what did it for me, but it's fairly simple and quite nice:

$$0.9999999999\ldots =1$$

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+1: This one, or actually the binary version $0.\overline{1} = 1$, was a source of early fascination for me. – copper.hat Mar 7 '13 at 7:34
I remember thinking "ahhhh I broke maths!!!" when I first stumbled over this. – Ben Mar 7 '13 at 10:56
@infact We are performing $10x-x=9.99999..... -0.999999......=9 $ – Ben Mar 7 '13 at 12:31
An "argument" that I find to convince a surprisingly high number of people goes like this: do you agree that $1/3 = 0.3333...$? And that $2/3 = 0.6666....$? Well, how about $3/3 = 0.9999....$? – Jesse Madnick Mar 8 '13 at 9:25
A YouTube video titled "9.999... reasons that .999... = 1": – Kevin Mar 10 '13 at 4:46

For me, I suppose it was Pascal's triangle. I was first formally introduced to it in one of my high school math classes, where my teacher explained Pascal's triangle, and challenged us to find as many patterns as we could in it. We spent a decent chunk of time doing so, and I was amazed by how a simple rule to generate a simple pattern of numbers could yield so many interesting patterns and properties.

I also found it cool how Pascal's triangle could be used to solve a variety of patterns from binomial distribution to the problem where you try and find the total number of paths on a grid assuming you can only travel in two directions, and demonstrated to me how mathematics is a lot more interconnected then I thought.

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I found it completely amazing that the angles in a triangle always added up to 180 degrees. No matter how you drew a triangle, you could measure the angles with a protractor and they always add up to about 180 degrees, like magic. Even more amazing when I realized it wasn't some rule of thumb or approximation, but true in some deeper sense for the ideal, platonic triangle.

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When I came home and told my father, he drew a triangle on the skin of an orange. All angles were 90°. I was deeply disturbed. – gerrit Mar 7 '13 at 11:04
The sum of the angles of a triangle is 180 degrees only in an Euclidean geometry (where the shortest distance between any two points is a straight line). On an orange, the shortest distance between two points is a curve. – utnapistim Mar 7 '13 at 11:54
your father is awesome =) – Glougloubarbaki Mar 7 '13 at 12:28
@Buksy: Imagine the earth (as a sphere). Draw a line from the north pole to the equator, then a quarter of the way around the equator, then back north to the pole. All angles are 90°. – Christian Mann Mar 7 '13 at 16:25
This is indeed a very pretty result, but lost on children. The reason it's lost is because they've been told this rule in school already, it's been "spoiled". I was taught this in middle school, and my attitude was "what's this? just another rule for doing calculations? okay.". It was only years later I understood how pretty it is. – Jack M Mar 8 '13 at 20:40

The game of Nim and its solution are pretty cool. The proof might be a bit difficult, but I think kids would love to learn a game like that and how to beat their parents at it.

There's a lot of other fun mathematical games like that too. But I think the first thing I learned that turned me towards mathematics was the existence of multiple infinities, and things like Hilbert's infinite hotel.

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+1, Hilbert's hotel could be great for a children's book! See's_paradox_of_the_Grand_Hotel. – Will Mar 7 '13 at 16:25

For me it was the Times Table of $9$.

We are usually forced to memorize the multiplication tables in school. I remember looking at the table for $9$, and seeing that the digit in ten's place increased by one, while the digit in the one's place decreased by one.

$$ \begin{array}{r|r} \times & 9 \\ \hline 1 & 9 \\ 2 & 18 \\ 3 & 27 \\ 4 & 36 \\ 5 & 45 \\ 6 & 54 \\ 7 & 63 \\ 8 & 72 \\ 9 & 81 \\ 10 & 90 \end{array} $$

After this, I realized that I could always add $10$ and subtract $1$ to get the next result. For a $7$ year old, this was the greatest discovery ever made.

And that your hands could give you the answer immediately: $7 \times 9$ = hold down your $7$th finger, leaves $6$ fingers on left of held down finger, and $3$ on right: $63$.. works all the way up to $9\times10$, beautiful.

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Fo me, it was the "flipping" of the digits along 9's times table. (e.g. $9 \times 9 = 81$ and $9 \times 2 = 18$ or $9 \times 3 = 27$ and $9 \times 8 = 72$. – Evan Teitelman Mar 7 '13 at 11:50
Reminds me of when I realized that "skip counting" (like $3, 6, 9, 12, \ldots$) is the distributive law. – Jesse Madnick Mar 7 '13 at 12:05
Add 10 and subtract 1 to get the result. I like that :) – Arch Mar 7 '13 at 16:22
Also, both digits of the result, sum 9. (18, 27, 36...) – Francisco R Mar 7 '13 at 17:13
I love this example, because it shows how the abstract side — finding patterns, understanding the reasons for them — arises directly out of the concrete side. Too often (as in the Lockhart) the two sides are presented as in opposition to each other — really, they’re intimately connected the whole way. – Peter LeFanu Lumsdaine Mar 7 '13 at 19:53

Euclidean geometry was the first thing that got me (about grade 9 or 10). That's where I first found out that

1) There is such a thing as mathematical proof (rather than just calculation).

2) Mathematics is not a closed subject: new and interesting results can still be found.

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The one that I was particularly intrigued in my late years was the execution of the proof of Gambler's Ruin. However, it might be too deep for small children.

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I believe it was when I was in 5th grade. I used to enjoy adding the digits of the plate numbers of vehicles until it resulted in a single digit result. I was excited to realize that all I had to do was eliminate nines from the number. Example 9468 is 9 (removing 9,6+3, 8+1), 3454 is 7 (what remains after removing 5+4). It's simple but it sure made travelling fun for me.

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That (including the removing of nines) was almost a compulsion for me... well,. it still is (i'm 46) – leonbloy Mar 7 '13 at 12:08

For me, it was the discovery that the sum of the digits in all multiples of three are themselves multiples of three, and you can recursively sum them to get to 3, 6, or 9 (i.e. an 'easy' multiple of three)


The sum of the digits in $13845$ is $21$,

The sum of the digits in $21$ is $3$

Edit: Should probably add that what made this useful to me was that numbers that are not multiples of three do not have this pattern.

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Or, more generally, that the ultimate result of this digital summing is the same as the remainder when dividing by 9. – cobaltduck Mar 7 '13 at 17:11
This generalises via arithmetic modulo 10 to produce tests for divisibility by other numbers based on similar properties of their decimal digits. – user3490 Mar 7 '13 at 20:21

Here are some things that I found interesting back when I was in junior high school. I hope they are not too advanced for young children:

  • Archimedes' method for computing areas and volumes (which is really cool).
  • The "limit" card magic. Take 27 cards from an ordinary deck of playing cards. Invite your audience to pick one of them, without telling the choice. Deal the 27 cards into three stacks, say $A, B$ and $C$, each containing 9 cards. The deal order is $A\to B\to C\to A\to B\to\cdots\to C$. Ask the audience which stack contains the chosen card. Collect the three stacks into one deck, where the stack containing the chosen card is placed in the middle. Repeat this deal-and-ask procedure twice more (so, thrice in total). Now the chosen card is the middle one in the stack as told by the audience.
  • The remainder of a whole number, when divided by $3$, is the remainder of the sum of its digits when divided by $3$.
  • The cyclic decimal expansion you get when a whole number is divided by $7$.
  • $1+2+\ldots+n=\frac{n(n+1)}2$. $$ n\left\{ \begin{array}{ccccc} \bullet&\color{red}\bullet&\color{red}\bullet&\color{red}\bullet&\color{red}\bullet\\ \bullet&\bullet&\color{red}\bullet&\color{red}\bullet&\color{red}\bullet\\ \bullet&\bullet&\bullet&\color{red}\bullet&\color{red}\bullet\\ \bullet&\bullet&\bullet&\bullet&\color{red}\bullet\\ \end{array}\right. $$ (Actually $1^2+2^2+\ldots+n^2=\frac{n(n+1)(2n+1)}6$ is even more interesting, but its proof is certainly too advanced for most young children.)
  • The (slanted) cross section of a cone has a symmetric shape (an ellipse). (Provided that the cross section does not cut into the base of the cone, of course.) This is rather inobvious to me because I thought the slant will break symmetry.
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+1 for visual 1 + 2 + ... + n = n*(n+1)/2. – Rauni Mar 7 '13 at 11:34

Pythagorean theorem

If $(a, b, c)$ is a Pythagorean triple, then so is $(ka, kb, kc)$ for any positive integer $k$.

$3^2 + 4^2 = 5^2$

Pythagorean triples

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I always thought cycles in decimal fractions were magic, until I realized I can easily create whichever cycle I wanted:

  • ${1\over9} = 0.111...$
  • ${12\over99} = 0.12\ 12\ 12...$
  • ${1234\over9999} = 0.1234\ 1234...$

I failed a number theory exam because the professor did not know this trick and said I needed to prove it.

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You sound like you don't agree with the professor, but of course you need to prove it! – Kundor Mar 15 '13 at 20:24
Isn't it obvious by long division? I knew it since I was a kid.. – user21820 Apr 24 '14 at 0:41
One proof is very similar to the proof $0.999... = 1$. If we know ${1\over9999} = 0.00010001...$, then $1234{1\over9999} = 1234 * 0.00010001... $, ${1234\over9999} = 0.1234 1234...$ – Kobi Feb 22 '15 at 19:46

It's not the first one that made me love math (what made me love math isn't math itself at all, but rather someone pointing out to me that I was pretty good at math -- and then I proceeded to like math haha), but this is the most amazing discovery I made when I was 15.

$$ Pr(X = r) = \frac{(nCr)(x-1)^{n-r}}{x^{n}} $$

Which is really just a restatement of the binomial distribution:

$$ Pr(X = r) = (nCr)(p^{r})(p^{n-r}) $$

where $p = 1/x$, so it works only makes sense for integer values. For example, the chances of choosing one blue jar out of 10 differently colored ones would be $x=10$, but also $p=0.1$.

I discovered it after one week of exhaustively listing down all the permutations of the letters n, t, g, and b and figuring out what patterns they looked like when you took only 1, 2, 3, and 4 elements. Then I went ahead and added more and more letters until I arrived at that formula by inspection.

In my opinion, it isn't the math itself that makes kids dislike math. It's all the people around them who dislike math who make kids dislike math.

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Hilbert's infinite hotel, the realization that $\mathbb{Z}$ is equinumerous with $\mathbb{N}$, and the uncountability of the set of all functions $\mathbb{N} \rightarrow \{0,1\}$. Basically: if it involves infinity, it's interesting.

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The Golden ratio

It was presented to me like this: There exists a number that you can square, subtract itself, and you'll get 1. Or, you can inverse the number, add 1, and you'll get the number back. What a beautiful number, I thought. Of course, I later realized the number was just a solution to:

$$x^2 - x - 1 = 0.$$

However, I was really impressed when later I learned this number also shows up in nature in the patterns of plant growth. Wow! Who would have thought?

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The way I first saw it was like this: there is a number that, if you add one to it, becomes its square. Then they showed that you could easily calculate it from just that information with only the quadratic formula (which I had thought was boring and had never suspected of hiding anything beautiful), and that if you kept multiplying it by itself, you'd get a simple expression with fibonacci numbers as coefficients. That was a happy day. – Michael Shaw Mar 8 '13 at 5:17
Showing that number on a calculator really drives its magic home. Square it, and all the digits after the decimal point stay the same. Or hit the 1/x button, same thing. – Joubarc Mar 8 '13 at 8:51
I think it's more impressive that if you subtract one from that number, you get its reciprocial. – Zsbán Ambrus Mar 8 '13 at 11:51
3 Vi Hart explains why the "magic" is actually completely inevitable, and the connection with the Fibonacci numbers. – Ben Millwood Mar 9 '13 at 23:06

When I was a child, I spent the whole summer at a camping at the coast of Catalonia. There I was always around my grandfather. He himself had no proper education and never finished school. Nevertheless he liked to read books on his own, about many things, grammar, the French language, mechanics, mathematics...

I remember he taught me many things. He was the first to explain me, as I fell asleep in his arms, under the starry night, that the Earth was a ball, and that there were people underneath the ground where we stood, on the other side of the planet, who were standing upside down without falling, because we were all attracted to the center of the ball. I did not understand, at that moment, how was that possible. But I trusted him and knew that there were many things I did not understand about the world.

One particular thing related to mathematics that he told me and that got me thinking, making myself questions and reaching the boundaries of my mind, was that one frog could try to jump her way across a puddle (we also went together to catch frogs), jump first to the half of it, then to the half of the remaining half and so on, and that after an infinite number of jumps she would arrive at the other shore.

This was, I think, one of the first things that made me feel that the world or that reality itself was infinitely bigger, more complex and more beautiful that anything we could understand or even begin to grasp. I guess this sense of real magic is what makes me have a special love for mathematics.

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When I was a kid my parents explained basic arithmetic to me. After thinking for a while I told them that multiplying is difficult because you need to remember if $a \cdot b$ means $a+a+\ldots + a$ ($b$ times) or $b + b + \ldots + b$ ($a$ times). I was truly amazed by their answer.

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What was their answer? – User 17670 Mar 9 '13 at 17:56
@Dan Petersen It's bad manners to tell this story and not tell us what the answer was. – Git Gud Mar 10 '13 at 10:04
I assumed their answer was the trivial one: "They're the same, you don't have to remember it". – user5501 Mar 12 '13 at 15:28
Maybe it was "Uuuuh, bedtime!"? : - ) – k.stm Mar 12 '13 at 15:37
The answer should be obvious to anyone using this site, and it would spoil the humour of the story to spell it out explicitly. – Donkey_2009 Mar 17 '13 at 11:09

Whether this is 'simple' enough is debatable... the method to generate the Mandelbrot set is likely to be far too complicated for the book in question, but the mathematical expression that's at its heart couldn't be much simpler.

$z_{n+1} = {z_n}^2 + c$

After implementing the Mandelbrot set I learned about the Buddhabrot, which is basically a way of rendering the stages of the Mandelbrot algorithm, and after some considerable processing time I had a render:

Buddhabrot whole

I then tweaked my input parameters to 'zoom in' on a particular area, and when I saw the result my jaw hit the floor. This is when I saw the true beauty in mathematics beyond 'nice' results. Again, it's probably too advanced for your book because of the steps involved in creating the visual, but maybe it'd make for a nice final hurrah to inspire further exploration? It still boggles my mind to see such amazing results from something so simple.

enter image description here

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It was just a C program I wrote myself after reading about the Mandelbrot sait in one of A. K. Dewney's books. When I was a kid I was fascinated with the Mandelbrot set but never understood it, after reading a brief description of the algorithm in the book as an experienced developer it surprised me how simple it was. Took about half an hour to do the Mandelbrot set code, but getting nice renders of the Buddhabrot was harder - not least because it was taking several hours to perform a high-resolution render. – LaceySnr Mar 7 '13 at 11:05
Would you make the code open source? – Wouter Zeldenthuis Mar 7 '13 at 14:25
Hoooooooooly. That Buddhabrot looks like a nebula. – Joe Z. Mar 7 '13 at 15:00
I cant imagine a child reading about a recursive formula with complex numbers and be amazed. This is surely beautiful and in a sense is simple, but not for a child. – Integral Mar 7 '13 at 15:36
@Integral You would do it the other way around. Show the pretty pictures, maybe show an animation, or allow them to play with some fractal software that allows you to zoom in. Then mention that it’s just a short formula behind it. I’d consider most answers here too complicated for children but with fractals they have something they can immediately see and be amazed of. – poke Mar 7 '13 at 16:54
  1. 17 + 20 = 8
  2. 17 − 20 = 26
  3. 17 · 20 = 21
  4. 17^(−1) = 12 (inverse of 17)

I got really upset when I saw this. The professor explained, to do network communication you will need to understand this.

I found maths awesome after dealing with these. What we are normally learning can not always help (it's real numbers mathematics). But the best things deal with fields. Therefore, the below is the explanation of the above meaningless things.

(i) Addition: 17 + 20 = 8 since 37 mod 29 = 8

(ii) Subtraction: 17 − 20 = 26 since −3 mod 29 = 26

(iii) Multiplication: 17 · 20 = 21 since 340 mod 29 = 21

(iv) Inversion: 17^(−1) = 12 since 17 · 12 mod 29 = 1

The elements of F29 are {0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27 28}

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My son loved this when he was little - patterns everywhere:

enter image description here

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Except the white boxes, they doesn't have pattern! XD – ᴊ ᴀ s ᴏ ɴ Mar 8 '13 at 11:19
Why isn't 2 red, 3 yellow, 5 green etc? – AidanO Mar 8 '13 at 12:12
crayons can be provided... :) – cormullion Mar 8 '13 at 12:14
@OmarKooheji: yes, but the pattern would be more natural if the prime numbers themselves were coloured — as solid squares of whatever colour they’re given in subsequent composites. // The difficult question is: what colour, if any, should 1 be? – Peter LeFanu Lumsdaine Mar 8 '13 at 19:59
Pretty cool! People fascinated by this might also like prime factorization diagrams:… – Matt W-D Mar 9 '13 at 20:31

protected by Zev Chonoles Mar 7 '13 at 22:43

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