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
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    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
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    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
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    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
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    I think it's a shame that this question was voted closed... – Will Mar 10 '13 at 19:50

164 Answers 164

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
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    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
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    @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
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    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
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    @AKE, the point of this picture is that it doesn't require 'a little algebra'. – jwg Mar 12 '13 at 13:36

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 $9 \times 8 = 72$. – Smith John Mar 7 '13 at 11:50
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    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
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    Add 10 and subtract 1 to get the result. I like that :) – Archy Wilhes 魏何 Mar 7 '13 at 16:22
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    Also, both digits of the result, sum 9. (18, 27, 36...) – Francisco R Mar 7 '13 at 17:13
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    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

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. – Matt Lacey Mar 7 '13 at 11:05
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    Would you make the code open source? – Wouter Zeldenthuis Mar 7 '13 at 14:25
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    Hoooooooooly. That Buddhabrot looks like a nebula. – Joe Z. Mar 7 '13 at 15:00
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    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
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    @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

I used to love naughty $37$.

$37 \times 3 = 111;$

$37 \times 6 = 222;$

$37 \times 9 = 333;$

$37 \times 12 = 444;$

$37 \times 15 = 555;$

$37 \times 18 = 666;$

$37 \times 21 = 777;$

$37 \times 24 = 888;$

$37 \times 27 = 999;$

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    That is naughty! – Loki Clock Mar 7 '13 at 12:57
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    But, there isn't a naught in any of that... – Gary S. Weaver Mar 7 '13 at 14:05
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    Wow - 37 is my new favourite number! – Steve Chambers Mar 8 '13 at 11:21
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    If you like 37, you'll love double-37: 37037*3=111111, 37037*6=222222, 37037*9=33333... – MiniQuark Mar 8 '13 at 14:00
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    And in general any repetition of 037. For the infinite case, take 1/27. – Joe Z. Mar 8 '13 at 20:05

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
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    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
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    your father is awesome =) – Glougloubarbaki Mar 7 '13 at 12:28
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    @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
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    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 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.

In the marginally rearranged form $$ e^{iπ}+1=0 $$ it uses absolutely nothing but nine essential concepts in mathematics:

  • five of the most essential numbers, $\{0,1,i,e,π\}$,
  • three essential operations, { addition, multiplication and exponentiation }, and
  • the essential relation of equality.
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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
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    I was about 15 :), but my bad overlooked that part. – Kaster Mar 7 '13 at 7:33
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    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
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    @DheeB oh really? Complex exponential:Proofs – Kaster Mar 7 '13 at 20:37
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    @Dylan: many arguments involving $\pi$ are circular :) [couldn't resist bad pun, sorry..] – Marek Mar 8 '13 at 21:20

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
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    @DheeB Perhaps it sounds like that, but I was not. – MJD Mar 7 '13 at 20:05
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    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
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    12345679 x 8 = 98765432 – jwg Mar 12 '13 at 13:39
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    @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

Adding to LaceySnr’s answer, I’d like to mention fractals in general. While fractals will probably count as a higher application of maths, they are very often very visually beautiful. So you could easily show a picture of a fractal and explain that there is just a simple formula behind it all.

Mandelbrot Astrophyton Darwinium 3D fractel 3D fractal 2

Some more examples:

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    holy crap the last 3.... – im so confused Mar 7 '13 at 15:43
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    All the fractals shown in this and other answers are too complex for little kids, but the basic concept is easy for them to understand. Start with a tree with each branch being another tree. It doubles as a introduction to recursion for those more inclined towards programming than pure math. – Izkata Mar 7 '13 at 17:12
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    @Izkata But they are visually appealing and that will spark the interest in it. As a kid I wouldn’t care how to prove that 2 is irrational, probably because I wouldn’t know what irrational is to begin with, or that there is some equality with some weird constants I don’t even know ($e^{i\pi} = -1$). It’s things like fractals, or other natural things like Fibonacci flowers that makes math interesting beyond just numbers. – poke Mar 7 '13 at 17:57
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    This is fantastic. – josh Mar 8 '13 at 0:58
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    @Izkata Perhaps fractal theory is too complicated for kids, but using fractal generation software might be within their reach. And if they start playing with that, they'll be learning a whole ton of math in a way that is incredibly fun. – Kevin Mar 10 '13 at 4:42

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
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    I remember thinking "ahhhh I broke maths!!!" when I first stumbled over this. – Ben Mar 7 '13 at 10:56
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    @infact We are performing $10x-x=9.99999..... -0.999999......=9 $ – Ben Mar 7 '13 at 12:31
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    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
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    A YouTube video titled "9.999... reasons that .999... = 1": youtube.com/watch?v=TINfzxSnnIE – Kevin Mar 10 '13 at 4:46

Here is my favorite classic illustrating the power and beauty of mathematical argument. Consider the question:

Question: Can an irrational number raised to an irrational number be rational?

Answer: One of the classic answer goes as follows. Consider the number $x=\sqrt{2}^\sqrt{2}$. If $x$ is rational, we are done. If $x$ is irrational, then consider $x^{\sqrt2}$, which is $2$ and now we are done.

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    I love those "anyway" proofs. – Joe Z. Mar 8 '13 at 1:00
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    The really amusing thing is that after the proof is given, we still don't know whether $\sqrt{2}^{\sqrt{2}}$ is rational or not! – Jesse Madnick Mar 8 '13 at 9:29
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    @ted $(a^{b})^c = a^{bc}$. Hence, $(\sqrt2^{\sqrt2})^{\sqrt2} = \sqrt{2}^{\sqrt2 \cdot \sqrt2} = \sqrt2^2 = 2$. – user17762 Mar 8 '13 at 20:49
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    How about $e^{\ln 2} = 2$? – Jas Ter Feb 27 '14 at 21:24
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    @SimenK. It is significantly more difficult to prove that either $e$ or $\log 2$ is irrational than it is to prove that $\sqrt 2$ is. – guest196883 Mar 21 '14 at 4:52

enter image description here

The number of pennies stacked in a triangle $(1,3,6,10,\cdots)$ is along one diagonal line of Pascal's Triangle. The number of spheres stacked in a tetrahedron $(1,4,10,20,\cdots)$ is the line next to it. The next line is the number of hyperspheres in a pentachoron.

enter image description here

I was about $10$ and living in a hotel and home sick from school, stacking up pennies and "red hots" in pyramids, etc. I made a table of these numbers. Noticing the simple addition rule in the table, I extrapolated to the $4$th, $5$th, dimensions. Later when I learned of Pascal's triangle that moment was probably my biggest joy of mathematics, realizing I'd run into that years before.

enter image description here

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    Back in School i noticed that each row of pascals triangle is a multiple of 11: 11^0, 11^1, 11^2 etc. – Nils Werner Mar 8 '13 at 12:14
  • @NilsWerner I'm not sure that's entirely accurate. How then do you handle numbers beyond single-digits, or are you perhaps working in some 'arbitrary base'? – Sean Allred Mar 8 '13 at 21:05
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    @vermiculus: I think what Nils Werner is saying is that the $n$th row consists of the numbers $\binom{n}{k}$, and that $\sum \binom{n}{k} 10^k = (10 + 1)^n = 11^n$. Of course some of the $\binom{n}{k}$ will have more than one digit. – Jesse Madnick Mar 9 '13 at 22:25
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    Mark odds and evens in Pascal's triangle and you get something nice :) – Manishearth Mar 10 '13 at 7:27
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    @vermiculus Shift multiples of 10 one position to the left, so 1 5 10 10 5 1 becomes 1 6 1 0 5 1 – Nils Werner Mar 11 '13 at 9:01

As a child, the Fibonacci numbers $$1,\; 1,\; 2,\; 3,\; 5,\; 8,\; 13,\; 21,\; 34,\; 55,\;\ldots$$ were very fascinating to me. They are named after the the Italian mathematician Fibonacci, who described these numbers in his 1202 book Liber abaci modeling a growing rabbit population:

enter image description here

Formally, the Fibonacci numbers $F_n$ are defined recursively by $$F_1 = 1, \quad F_2 = 1, \quad F_{n+2} = F_{n+1} + F_n$$ It was a lot of fun to compute them, one after the other, and to collect the results in ever-growing tables: $$F_3 = F_2 + F_1 = 1 + 1 = \mathbf{2}\\F_4 = F_3 + F_2 = 2 + 1 = \mathbf{3}\\F_5 = F_4 + F_3 = 3 + 2 = \mathbf{5}\\F_6 = F_5 + F_4 = 5 + 3 = \mathbf{8}\\F_7 = F_6 + F_5 = 8 + 5 = \mathbf{13}\\\vdots$$

At some point, I asked myself the question: To compute $F_{10}$, do I really have to compute all the Fibonacci numbers up to $F_9$ beforehand? So I tried to figure out some formula where you can plug in $n$, do some basic arithmetics, and get $F_n$ as a result. I've spent a lot of time on this. However no matter how hard I tried, I didn't succeed.

After a while I found the closed form $$F_n = \frac{1}{\sqrt{5}} \left(\left(\frac{1 + \sqrt{5}}2\right)^{\!n} - \left(\frac{1 - \sqrt{5}}{2}\right)^{\!n}\right) $$ in some book. I was paralyzed.

How can it happen that such an easy recurrence formula needs to be described by such a complex expression? Where do the square roots come from, and why does the expression still always evaluate to an integer in the end? And, most importantly: How on earth can one find such a formula??

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    I first discovered the Fibonacci sequence by playing with an 8-digit calculator. There's a bug in most standard 8-digit calculators (the kind you get as promotional gifts, for example) so that if you press "1 + = + = + = ...", it will give you the Fibonacci sequence. – Joe Z. Mar 7 '13 at 21:16
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    In order to find this formula, you usually will use linear algebra, and the property that linear recurrences can always be represented in matrix form. In particular, the two exponential terms correspond to the eigenvalues of the matrix $\left [ \begin{matrix}1 & 1 \\ 1 & 0\end{matrix} \right ]$, which, as you could probably guess, has a characteristic polynomial of $x^2 - x - 1$. – Joe Z. Mar 7 '13 at 21:23
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    JoeZeng: Wow, that's a nice bug-abuse :) Nowadays I know, of course, how to derive the formula. Besides the linear algebra method you indicated (my favorite), there is also a standard way to do it by generating functions. – azimut Mar 7 '13 at 21:32
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    There's a way to derive this using only plain algebra, which would be nicer for a children's book. Let a = (1+sqrt(5))/2 and b = (1-sqrt(5))/2. Show that a^2 = a + 1 (and similarly for b). Multiply by a and simplify to get a^3 = 2a + 1, and show that you can repeat this to get a^n = F(n)*a + F(n-1) (and similarly for b). Take a^n - b^n, simplify and rearrange and you're done. – Michael Shaw Mar 8 '13 at 5:55
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    A bug? I thought it was a feature? – Raskolnikov Mar 15 '13 at 20:56

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 – A. Chu Mar 8 '13 at 11:19
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    Why isn't 2 red, 3 yellow, 5 green etc? – AidanO Mar 8 '13 at 12:12
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    crayons can be provided... :) – cormullion Mar 8 '13 at 12:14
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    @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
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    Pretty cool! People fascinated by this might also like prime factorization diagrams: datapointed.net/visualizations/math/factorization/… – Matt W-D Mar 9 '13 at 20:31

(Copy from http://mathforum.org/library/drmath/view/57919.html)

There is a well known story about Karl Friedrich Gauss when he was in elementary school. His teacher got mad at the class and told them to add the numbers 1 to 100 and give him the answer by the end of the class. About 30 seconds later Gauss gave him the answer.

The other kids were adding the numbers like this:

$$ 1 + 2 + 3 + ... + 99 + 100 = ? $$

But Gauss rearranged the numbers to add them like this:

$$ (1 + 100) + (2 + 99) + (3 + 98) + ... + (50 + 51) = ? $$

If you notice every pair of numbers adds up to 101. There are 50 pairs of numbers, so the answer is $$ 50 * 101 = 5050 $$ Of course Gauss came up with the answer about 20 times faster than the other kids.

In general to find the sum of all the numbers from 1 to n:

$$ 1 + 2 + 3 + 4 + ... + n = (1 + n) * \bigg(\frac{n}2\bigg) $$ That is "1 plus n quantity times n divided by 2".

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    This was also one of my favorites, perhaps because of the story. I'll suggest a slight improvement in the last statement. It's easier to imagine it as number of items times average value, i.e. $n \frac{n+1}{2}.$ Then it becomes common sense and no need to remember any formula. – karakfa Mar 15 '13 at 21:41

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
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    @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
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    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
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    Maybe it was "Uuuuh, bedtime!"? : - ) – k.stm Mar 12 '13 at 15:37
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    The answer should be obvious to anyone using this site, and it would spoil the humour of the story to spell it out explicitly. – John Gowers Mar 17 '13 at 11:09

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.

  • It's just as much fun in binary... Take 1/10. Then, 1 / 5 / 2 = 3 / 15 / 2 = (binary) 0.0 0011 0011 0011 .... where 15 in binary is 1111 so any binary value repeated every 4 digits is evenly divisible by 15. – ErikE Mar 8 '13 at 7:27
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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
  • You can use the result from this question to prove that you can create any cycle in any base. – Joe Z. Aug 27 '13 at 4:09
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    Isn't it obvious by long division? I knew it since I was a kid.. – user21820 Apr 24 '14 at 0:41
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    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

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
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    When I first saw the proof for infinitely many primes it blew my mind as well! – sdm350 Mar 7 '13 at 17:59
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    @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
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    @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
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    Exactly these two examples are provided in 'Mathematician's apology' – Alex Feb 11 '15 at 22:16

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
  • @MJD: Dividing by two? I have no idea. That's a story my dad keeps telling (the age varies from 3 to 5, and I am inclined to believe 5 is the correct age). He says I knew that if you have a floor tile, you can break it in half, and again and again and again. I don't think I came across many calculators at that age. – Asaf Karagila Mar 7 '13 at 7:34
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    @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
  • @Ishan: That's pretty nice! – Asaf Karagila Mar 7 '13 at 7:53

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.

  • I always found it fascinating! – Integral Mar 7 '13 at 15:29
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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
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    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
  • this is the one I was going to submit, and it's easy for a 7 year old to grasp, even if the proof is difficult at that stage. – Alnitak Mar 8 '13 at 13:06
  • I discovered a special case of this in third (?) grade: 3 times 37 is 111. It caught my attention -- how could THAT happen? – SixWingedSeraph Apr 29 '13 at 13:28

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.

Who remembers magic squares? Those sparked my interest in mathematics.

enter image description here

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    Althought this isn't suitable for children. My interest in magic squares was only sparked much later in life, during my undergrad, where I realised that the set of $n\times n$ magic squares (over a field $k$) form a subring of the matrix ring $M_n(k)$. I also spent many hours trying to work out the dimension of the ring as a $k$-vector space. – Dan Rust Mar 7 '13 at 13:21
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    How isn't this suitable for children? Finishing one out that has already been started (but has some missing elements) can be a fun task for someone just learning addition... – apnorton Mar 8 '13 at 1:27
  • I was first introduced this in 4th grade. We were given a blank square (with the exception of 5 in the middle) and the goal was to make the sum of each row/column equivalent. It was a great way to challenge ourselves in math. – krikara Mar 9 '13 at 2:31
  • What I loved about the magic square was that the numbers form a symmetric pattern. To remember the order of the numbers, you just need to remember the pattern. – Paddy Landau Mar 9 '13 at 19:00
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    @anorton: Daniel's comment's opening clause "Although this isn't suitable for children..." referred to the rest of his comment, not to the answer. ("Although the following isn't suitable for children,...") – ShreevatsaR Mar 24 '13 at 10:33

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.
  • +1 for $n(n+1)/2$. I "proved" this in the 5th grade using the same method :). – Jacob Mar 7 '13 at 21:27
  • Actually, the one that's interesting is the sum of the first $n$ odd integers. – asmeurer Mar 22 '13 at 3:13
  • @asmeurer Yes, that's more interesting! – user1551 Mar 22 '13 at 12:05
  • You can prove the formula for the sum of squares using the same wort of picture: stack a 1x1x1 on a 2x2x1 on a 3x3x1, on a ... on a nxnx1 block. Six such things form a nx(n+1)x(2n+1) block. If you actually made the shapes, I think this would be just as accessible to young children as your proof of the sum of the first n integers. – Steven Gubkin Jan 13 '14 at 16:55
  • I've just learnt that the card trick I mentioned in the above is called Gergonne’s Card Trick, named after Joseph Diaz Gergonne, who first published an analysis of the trick circa 1813 in Annales de Mathématiques. – user1551 Dec 13 '16 at 8:44

A few things come to mind:

Here's a beautiful JavaScript demo of these graphs being generated: http://www.datapointed.net/visualizations/math/factorization/animated-diagrams/

  • Even as an adult, I think continued fractions and generalized continued fractions are amazing. One of the simplest is the golden ratio: $$\varphi = 1 + \cfrac{1}{1+\cfrac{1}{1+\cfrac{1}{1+\ddots}}}$$ And this identity is downright incredible:

    $$ \frac{\pi}{2} = 1+\cfrac{1}{1+\cfrac{1}{1/2+\cfrac{1}{1/3+\cfrac{1}{1/4+\ddots}}}}$$

I should stop myself now... But math is really filled with astounding phenomena like I've mentioned above...

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    I've never seen that one for $\pi$ before! It's much better than the one with integers, which has no pattern at all. – Ryan Reich Jun 12 '14 at 16:45
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    I wonder if there exist an intuitive argument explaining the pattern that appears in the continued fraction of $\tfrac\pi2.$ – Hakim Jun 28 '14 at 23:08
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    The Galton device makes a really important (philosophical?) point that kids can understand: through math, you can be completely certain that a pattern will emerge from the aggregate behavior of a large number of individuals, even if those individuals don't communicate with each other or organize, and even if they act completely randomly! There was (is?) one at the Museum of Science and Industry in LA when I was a kid, and I remember trying it over and over again -- how on earth did all those balls know where to line up? – Tad May 3 '15 at 12:59

I always had a peripheral understanding that there was something more to maths than working out how your change or divvying up sweets with your siblings. But the day I really, truly understood was when I learned about $\pi$.

$\pi$ was magical to me. For one thing, it's a funny-looking Greek letter with a funny-sounding name. But, more captivatingly, it introduced me to an epiphany: that somewhere, on some level, the fundamental structure of reality itself could be understood through mathematics.

Let's assume your childen understand what a circle is, and how to measure things with a measuring tape. Introduce them to circumference and diameter. Give them a table with three columns—circumference, diameter and "the secret of circles"—and a big tape measure. Tell them to go out and measure as many circles as they can find: plates, car tires, stop signs, plant pots, lines on a basketball court… anything so long as it's circular. Let them loose.

Later, once they're done measuring everything in the neighbourhood, hand them a calculator and tell them to go through each of their circles and divide the circumference by the diameter, and write the number they get in the mystery third column. Tell them that a pattern will start to appear, and they need to see if they can spot it.

Once they're done, you can explain to them that the reason the first couple of numbers is the same is because there's a number, a magical number, that tells us a secret about every circle in the universe—from rings we wear on our fingers to the sun and moon in the sky and the whole planet Earth. No matter how big or how small, how grand or how humble, every single circle is a bit more than three times bigger around than it is from one side to the other. This number is so special that it has its own name, pi, and its own special letter, $\pi$. It's not three and it's not four—it's somewhere just after three, and we can't write down exactly where because it goes on forever. Luckily, we only really need to know the first few numbers most of the time, so we can use this magical number whenever we need it.

The sense of revelation that came from knowing that every circle in the universe is connected by this strange, special number stayed with me for a long time, and is at least partly responsible for my love of mathematics in later life.

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    That's an awesome idea for something to do with your kids. +1 – Will Mar 7 '13 at 16:43

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 en.wikipedia.org/wiki/Hilbert's_paradox_of_the_Grand_Hotel. – Will Mar 7 '13 at 16:25
  • If one knows base two, the optimal strategy for Nim is pretty simple. I didn't see this until much later, but this would have been a favorite of mine if I'd seen it earlier. – robjohn Dec 18 '16 at 17:51

Everybody loves fractals. I think this one - The Dragon Curve - is particularly easy to explain, and it is very surprising and aesthetically pleasing:

enter image description here

Here's a video I've seen which explains how it comes about: The Dragon Curve from Numberphile

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    I have a friend who is disgusted by pictures of fractals. Might relate to trypophobia. – Nikolaj-K Mar 7 '13 at 20:31
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    I seem to recall this fractal making an appearance in Jurassic Park (the book). – icurays1 Mar 8 '13 at 18:23
  • Check out the video I attached at the bottom of the picture, it states that appearance in Jurassic Park! – Adar Hefer Mar 9 '13 at 13:42
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    I found by myself this fractal by folding and reopening a long strip of paper. The strip was actually the edge (with holes) of the continuous paper used in old pin printers. – Emanuele Paolini Feb 17 '14 at 14:15


Disney, as it was long time ago.

  • Oh yes. Ooohhhhh yes. – Sean Allred Mar 8 '13 at 21:06
  • Ooohhhhh yes indeed! I can still remember the theme music now, and picture the pentagrams arms expanding into golden rectangles. – Lucas Mar 15 '13 at 19:07
  • You can watch it again online: youtube.com/watch?v=YRD4gb0p5RM – Lucas Mar 15 '13 at 19:10
  • @Lucas removed for copyright infringement. hurray for copyright law, making the world a better place for everyone. – Alexander Gruber May 6 '13 at 7:03
  • @AlexanderGruber Weird: "multiple third-party notifications of copyright infringement" – Lucas May 7 '13 at 13:11

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
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    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
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    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
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    youtube.com/watch?v=ahXIMUkSXX0 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

Realising why zero is not nothing, and understanding numbers

I first understood the difference between zero and none when thinking about thermometer readings. If you had a ton of thermometers scattered around the world, and you collected their readings periodically and put them in a database, what would you do if any thermometer was broken? If you just put a zero reading, you'll screw up your averages, but if you put a null value, you can handle broken thermometers easily.

That made me realise what a number is.

I was in elementary school, drawing 3D shapes in class while bored. I drew cubes by drawing two overlapping squares and connecting the vertices, like the top row in this image:

enter image description here

Then I thought, what if I did the same procedure, but to a cube? So I drew four squares and connected the vertices, like this:

enter image description here

I was struck by the beauty of the resulting image, with its intricate structure of star-like patterns. Here's a static version:

enter image description here

It was years later that I discovered, to much fascination, that this was in fact the four-dimensional analogue of the cube: the hypercube. Hence my username.

Edit: Another thing I remember thinking about when I was younger was that I could not always draw a straight line through three points, but was surprised to find that it would always work for two points.

  • IIRC Emmy Noether also used to draw such pictures to visualise higher dimensions as a child. – Marc van Dongen Oct 17 '13 at 10:33

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

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