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What was the first bit of mathematics that made you realize that math is beautiful? (For children's book)

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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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Original Q&A
107

There are 107 best solutions below

11
On

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

19
On

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!

1
On

  • 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 mind....as 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:

24567829872367783456753745673456347567346534754

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.

13
On

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.

10
On

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.

20
On

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.
1
On

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...

2
On

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.)

0
On

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.

4
On

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.

23
On

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

$$0.9999999999\ldots =1$$

0
On

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.

15
On

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.

2
On

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.

20
On

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.

0
On

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.

0
On

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.

5
On

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.

7
On

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)

E.g.

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.

5
On

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.
0
On

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

6
On

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.

1
On

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.

1
On

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.

4
On

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?

0
On

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.

12
On

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.

31
On

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

0
On
  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}

15
On

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

enter image description here

7
On
  1. Take your age, and reverse it.
  2. Subtract smaller number from bigger.
  3. Add the digits of subtraction.
  4. You get 9.
2
On

Complex numbers was awesome to me 

        i² = - 1
0
On

I think the first thing that amazed me in this way was $\pi$. An irrationnal number, which means it has an infinite number of digits, which involves humans can't manage it, we can't know it on the whole, but already the Greeks discovered it. They knew it has something to do in the circumference or the area of a circle, that is, they could manipulate it, and I find this unbelievable.

9
On

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??

8
On

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:

2
On

That the roots of $z^n-1 = 0$ start to form a circle as $n$ increases. It starts out with the simple solution, the quadratic which you've already seen, then the complex plane comes in for $z^3$ and all of a sudden it's like "Hey! Those form circle!"

0
On

I didn't so much discover mathematics was beautiful as I discovered everything I found beautiful was mathematics. I would have said the dodecahedron was my favorite elementary shape when I was little, but as a teenager I was exposed to the fourth dimension. I became obsessed with symmetry and analogy-based objects, and with the Johnson solids, which gave me a then-ineffable feeling of filling out the quality of symmetry that it always creates good shapes, that there are rules you can make that name exactly the set of good shapes. That "looks right" and "looks wrong" can be made precise, and hence the feeling can be explained, you can learn what it is you're noticing about those shapes that you wouldn't feel when you look at a 3D mesh of a face or a blanket.

3
On

Arithmetic series might be interesting: straightforward to explain and amenable to pictorial representation ...and the child might love the fact that they've learnt how to do huge sums that might stump many (non-mathematical) adults.

You could show how $1 + 2 + 3 + \cdots + 100$ could be worked out by pairing numbers from opposite ends of the sum together $(1 + 100) + (2 + 99) + \cdots + (50 + 51) = \underbrace{(101 + 101 + \cdots )}_{\text{50 terms}} = 5050$.

or by adding the series to itself with terms running in ascending and descending order $1 + 2 + \cdots + 99 + 100$

$100 + 99 + \cdots + 2 + 1$

to get $101 + 101 + ... = 101 \times 100$ which is twice the sum.

2
On

When I was young I found a riddle:

  • Think about a number
  • Multiply it by 3
  • Add 1
  • Multiply it by 3
  • Add the number you thought at the beginning
  • Tell me the result:)

The number you thought about is your result without the digit 3 at the end, so i.e. if your result is 53, then you thought about 5.

1
On

It was probably not the first thing that made me realize that math is beautiful, but it was something that amazed me the most and still does to this day: The fact that the Mandelbrot set is not only infinite - in a way that eg. the Koch snowflake is infinite - but that it is infinitely complex, the complexity never ends, you can zoom it forever and you will never find exactly the same patterns, the information that is contained in it is infinite and yet it is described by such a simple formula.

It made me wonder whether math was discovered or created, whether things like the Mandelbrot set existed independently from their discovery or not, whether the infinitely complex pictures existed if they were never seen etc.

I remember the sleepless nights in elementary school when I was writing programs to explore the Mandelbrot set, to find nice looking colors, to animate it - impossible to do live at that time so I had to learn how to script some animation program that I had, wait an hour to realize that I had the colors wrong, change one number, wait another hour, rinse and repeat.

I didn't know about complex numbers at that time. I only knew that I was looking at something most amazing in the world and just couldn't stop exploring. Fractals became my obsession and were probably one of the reasons why I started programming more seriously.

I was fascinated by the fact that I could zoom it so much that it was like finding some proton on the face of Earth and zooming it to the size of a planet, and then looking at that planet-sized proton with an electron microscope. I could print what I found and I knew that no one in the Universe has ever seen it before me and no one will ever be able to find it even after looking on my printout - the scale was so amazing.

I remember how I got scared when I eventually saw large pixels in my Mandelbrot set! Finally I realized that I hit the limits of the floating point number precision on my 386 but I knew that the complexity of the Mandelbrot set was there, somewhere, even if I couldn't see it with my computer at that time.

Those are some of my favorite pictures that I posted to Wikipedia:

Cool Mandelbrot:

Cool Mandelbrot

Calm Mandelbrot:

Calm Mandelbrot

Hot Mandelbrot:

Hot Mandelbrot

You can download them from Wikipedia.

One of those pictures was magnified 248,034,982,258 times - probably the Cool Mandelbrot but I'm not sure because strangely all of them have the same description on Wikimedia Commons (something had to go wrong when they were copied from Wikipedia to Wikimedia Commons).

I would be honored if you'd like to use those pictures in your book. If you need higher resolution pictures or more information about them then I might be able to find something in some very very old backups.

Good luck with the book!

5
On

http://en.wikipedia.org/wiki/Donald_in_Mathmagic_Land

Disney, as it was long time ago.

13
On

I used to love naughty $37$.

$37 \times 3 = 111;$

$37 \times 6 = 222;$

$37 \times 9 = 333;$

$37 \times 12 = 444;$

$37 \times 15 = 555;$

$37 \times 18 = 666;$

$37 \times 21 = 777;$

$37 \times 24 = 888;$

$37 \times 27 = 999;$

7
On

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

enter image description here

2
On

To me, it was probably an old animated book "Och ta geometria" (eng. "Oh that geometry") written and illustrated by Zlatko Šporer, Nedejko Dragić. In the form of funny comix (check the link for samples), this book introduces basics of geometry from points, segments (not sure if this is the correct name), lines, flat figures and their area to cartesian coordinate system. This was probably the catalyst for my interest in math.

0
On

My mother repeatedly tells this story about me.

In German television there is a series called Telekolleg (not Kellog you silly, more like in college) which is broadcated for remote learning. One series deals with Math.

I was about 5 or 6 years old, when I sat in front of the TV watching this Telekolleg Mathematik series, turning to my mother and insisting: 'This is a good programme, you have to watch this'.

I don't remember what the exact topic was, perhaps quadratic function graphs.

0
On

This was my favourite equation. I was 16 or so, when my father showed it to me. I was amazed, and I programmed an application which drew this:

The equation of love.

The interval should be <-6;6> maybe. I made it a looong time ago after all ;)

orig http://imageshack.us/scaled/thumb/221/heartwl.png white http://imageshack.us/scaled/thumb/23/heart01q.jpg black http://imageshack.us/scaled/thumb/268/heart21.bmp

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The first interesting mathematics problem I remember in my limited memory is 1 + 1/2 + 1/4 + 1/8 + 1/16 + ... It never totals to TWO :-)

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This was probably the very first mathematic riddle which absolutely got me. It is called Algebrogram in my language, but I couldn't find a reference in English.
I was attending mathematic group after normal school (at age 11-14) and then I made few of my own for my classmates. I loved it ^^

You use characters instead of numbers and you construct some words. You then let others solve it.

F O R T Y
    T E N
    T E N
---------
S I X T Y

Solution:

2 9 7 8 6
    8 5 0
    8 5 0
---------
3 1 4 8 6

It was common to construct sentences as well, but it is kind of hard. This is only an example, which is unsolvable ;)

You could specify if there were some other operations or you could let your solvers find it out by themselves.

        O U R
    H O U S E
        H A S
      - T E N
-------------
W I N D O W S
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My favorite was when I was asked:

"If you were to save 1 penny on day one, and double your money for a month every day after that, how much money would you have?"

One I realized the answer was $10,737,418.24 I was flabbergasted. That was when I was able to understand that there is a mathematical model/equation for just about everything in this world; now that's beautiful.

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I'm not sure there was a first bit; realizing the beauty of mathematics was a gradual process for me, turning it from a fun little thing I was doing into a full-fledged appreciation.

One of the more recent things, I suppose, is some of the patterns that appear in modular arithmetic. The concepts of continued fractions and aliasing in signal processing are closely related. When continuously adding 9 to a number, the ones digit appears to decrease by 1 constantly. If you mark all the multiples of 3 on a 10-by-10 grid, they form diagonal stripes down the page. Things like that, which actually have quite significant uses in real life, are things that make math beautiful (and tricky!) to me.

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If you've ever heard of $3,529,411,764,705,882$ being multiplied by $3/2$ to give $5,294,117,647,058,823$ (which is the same as the 3 being shifted to the back), you might consider including that in the book.

There are lots of other examples, like $285,714$ turning into $428,571$ (moving the 4 from back to front) when multiplied by $3/2$, or the front digit of $842,105,263,157,894,736$ moving to the back four times in a row when you divide it by $2$. (There's a leading zero in front of the last term, though.)

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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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Pi has always fascinated me. The notion that perimeter of every possible circle imaginable divided by its diameter always results Pi is astonishing.

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Two instances where I thought math was amazing:

  1. In like 4th grade or whenever you learn areas of rectangles, one of the exercises in my book was to estimate the area of some squiggly shape overlaid on a rectangular grid. I thought this was pretty cool, and reasoned that if you could make the grid "smaller" (higher resolution), you could be more accurate. I mentioned this to my mom, who proceeded to tell me that was basically how Calculus 2 worked. :) That was very fun for me.
  2. Deriving the quadratic formula in Algebra 1. That was fun--it showed that some totally un-intuitive formula could be easily found using other previously found results.
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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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I was pretty good at math from an early age, but what was the clincher for me was the existence of non-Euclidean geometry. In grade 6 my math professor gave me a book on axiomatic Euclidean geometry, and I was totally blown away that the parallel postulate was just that, a postulate, and not an undisputable true fact. If upmto that point I just considered (school) math easy, from that moment I realized is incredibly beautiful. I did not look back since.

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(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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For me it was when I realized that with sine and cosine I could draw a circle!

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In my school when I learned about Cartesian coordinate system was shocking! Because that was the time I learned that was possible to make drawings with numbers.

Unlike most people here, I didn't have so much fun playing with numbers, but everything changed when I realized that I could convert numbers (more precisely, ordered pairs) in drawings over the coordinate plane.

And YES, that was a lot of fun.

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First of all I must say that I really appreciate the idea of such a book. I wish I was exposed to such a book when I was younger as it was relatively late in my life (high school)I started appreciating mathematics. Anyway here is something I consider to be beautiful and simple, that you might find of interest:

The Pigeon hole principle and it's applications. The pigeon hole principle goes something like this:

Assume that you have some pigeons and some holes, and you want to put your pigeons into the holes, then if you have more pigeons than holes at least one of the holes must contain more than one pigeon. For example if I have 3 pigeons, but only two holes then one of the holes must contain at least two pigeons. The more mathematical way to state this is that if you have a set $X$ consisting of $n$ elements and another set $Y$ consisting of $m$ elements and $n > m$ then there cannot exist an injective function from $X$ to $Y$.

Now this statement is fairly obvious and I am sure most people can understand this. But this statement shows up a lot in various disguises.

Here is an example I think is pretty cool: Suppose a group of people are at a party. Each person may introduce himself/herself and shake hands with someone else at the party. I claim that there will always be at least two persons who have shaken the same amount of hands. Here is a proof of that statement:

Suppose there are $n$ people at the party. Then a given person can either shake $0, \;1 \ldots n-1$ different peoples hands. That is $n$ different possibilities, however if there is a person who shakes $0$ hands (that is he doesn't shake hands with anyone) then there can't be a person who shakes hands with $n-1$ persons (that is he shakes hands with everyone except himself), and conversely if a person shakes hands with everyone, then it is not possible that someone else doesn't shake hands with anyone. So there are only $n-1$ possibilities but there are $n$ people, so thinking of the people as pigeons and the possibilities as holes we see that we have $n$ pigeons and $n-1$ holes so at least two pigeons must go into the same hole, that is at least two people must shake the same amount of hands at the party.

You can read more about the pigeonhole principle here: http://en.wikipedia.org/wiki/Pigeonhole_principle

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When I was maybe 8 or 9, the following trick was showed to me as a sanity check for calculation by hand.

  1. Take two numbers, let's say $358$ and $77$.
  2. Sum up the digits until you get a single digit number. $$ \begin{array}{r} 358 \to 16 \to 7\\ 77 \to 14 \to 5\\ \end{array} $$
  3. Do the same with the two sums $$ \begin{gather} 7 + 5 = 12 \to 3\quad\text{and}\\ 358+77 = 435 \to 12 \to 3 \end{gather} $$
  4. You get the same result? Try with other numbers. Be Amazed!
  5. Best of all? It also works with products: $$ \begin{gather} 7 \times 5 = 35 \to 8\quad\text{and}\\ 358 \times 77 = 27566 \to 26 \to 8 \end{gather} $$

I could not believe this always worked, it looked at the same time so beautiful and magic! A few years later, I was finally able to prove it by myself. I was so happy!

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I like cars and automotive racing and such. What got me real interested in it were two things:

The first, to a great extent, in Calculus:

  • $\displaystyle \frac{d}{dt}\ \text{Displacement} = \text{Speed}$
  • $\displaystyle \frac{d}{dt}\ \text{Speed} = \text{Acceleration}$
  • $\displaystyle \frac{d}{dt}\ \text{Acceleration} = \text{Jerk}$

It all made sense to me after that!

Then there was a problem in my Cal. book about calculating the force of a piston in an engine. I can't quite remember it, but it was basically:

$\text{Force} = \text{RPM}^3$

or something similarly extreme. It relates to the automotive aphorism: Power doesn't kill motors, RPM does.

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I was completely baffled when I learned the approach of C.F. Gauß for summing 1+2+3+...+100. Of course I would have gone for the hard way as well and I was deeply impressed when I learned that this equates to 1+100 + 2+99 + 3+98 + ... = 50*101 = 5050.

The next big thing for me was when I discovered that you can reduce multiplication to looking up squares by the identity

a*b = ((a+b)(a+b)-(a-b)(a-b))/4

However by that time I was already hooked.

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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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Many years ago, before I knew multiplication, I wrote numbers 1 to 10 in a row:

1  2  3  4  5  6  7  8  9 10

Then I wrote a second row, just for the fun of it, starting with 2, increasing each number by 2:

2  4  6  8 10 12 14 16 18 20

And then a next row, starting with 3, with an increment of 3, and so on, until I got:

 1  2  3  4  5  6  7  8  9  10
 2  4  6  8 10 12 14 16 18  20
 3  6  9 12 15 18 21 24 27  30
 ...
 9 18 27 36 45 54 63 72 81  90
10 20 30 40 50 60 70 80 90 100

I showed this to my parents, and they told me it was this thing called the "multiplication table" and explained how it worked. I was amazed.

Still today I'm very proud that I reinvented the multiplication table :)

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The following riddle blew my mind when I was a kid.

Three men went into a hotel. At the front desk they were told that the room would be \$30, so they each gave \$10.

After the men went to their room the manager realized they booked a room that was only \$25, so he gave the bell boy \$5 in ones to take back to the men.

On his way, he thought, "5 can not be evenly divided by 3 men", so he pocketed two and gave the other three to the men, one to each.

So, effectively each man paid \$9 for the room, a total of \$27. Remember, the bell had \$2 in his pocket. \$27 the men paid + \$2 the bell kept = \$29. Where did the extra dollar they paid go?

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I guess this is not as special as the other ones, but this is how mathematics amazed me for the very first time:

I just turned 4 years old (I still vividly remember this), and my mother bought 4 cartons of eggs, a dozen per carton. My mother, trying to challenge me, asked how many eggs we bought in total, and after a short while I said 48 (I've always had a knack for arithmetic and I guess intuitively I knew it was $12 \times 4$). My mother was amazed, and she asked me how I did it. At this point I wasn't formally introduced to any mathematics (no multiplication and division, just basic addition and subtraction using our hands). So when I tried to show using my fingers how I got to 48 by taking 12 four times, it took me a lot longer, and my mother decided to teach me multiplication right then and there. This was the beginning of my interest.

The more I think about this story, the more beautiful it gets. I implement the lesson I learn everytime someone says mathematics is useless!! Ask them to do $12+12+12+12$ with their fingers.

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Though a lot have been said (I too worked out Pascals triangle as a kid) no one has (yet) mentioned Gauss' method for adding sequential numbers.

It may be apocryphal but the story I heard was that a teacher wanted busy work, so she told the class to add the numbers 1-100, thinking that would take forever. Gauss was smart, he knew that the pair 100+1 was the same as the pair 99+2, the same as the pair 98+3... and now that he paired these numbers off, he now had 100/2 or 50 pairs of them. 50 pairs of 101 was 5050. He told the teacher the answer way before it was expected, and shocked them.

The coolness of the story is that it's probably at the level of your audience, something they can do and experiment with. and the guy's a legend too.

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My two favorite mathy things (not mentioned in other answers) were the powers of 11 

11 ^ 0  = 1  (1)
11 ^ 1  = 11  (1, 1)
11 ^ 2  = 121  (1, (1+1), 1) 
11 ^ 3  = 1331 (1, (1+2), (2+1), 1)
11 ^ 4  = 14641 (1, 1+3, 3+3, 3+1, 1) 
11 ^ 5  = 161051  (1, 1+4, (bump 1 to left) 4+6, (bump 1 to left) 6+4, 4+1, 1)
11 ^ 6  = 1771561 (1,1+6, 6+1, 1+0, 0+5, 5+1, 1)

and the estimated relationship btwn powers of 2 and powers of 10 and how they diverge (hard drive manufacturers think this is beautiful :P ) 

2^10 ~= 10^3 (1024, 1000) "kilo"
2^20 ~= 10^6 (1048576, 1000000) "mega"
2^30 ~= 10^9 (1073741824, 100000000) "giga"
2^40 ~= 10^12 (1099511627776 , 1000000000000) "tera" 
2^n ~= 10^floor(n/3) (where n is a multiple of 10)
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Solving for an unknown. 2x = 4 so x = 2. Beautiful.

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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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When I saw my first list of mathematical axioms (algebraic in this case). This was when I was 11.

  1. If the same quantity or equal quantities be added to equal quantities, their sums will be equal.
  2. If the same quantity or equal quantities be subtracted from equal quantities, the remainders will be equal.
  3. If equal quantities be multiplied into the same, or equal quantities, the products will be equal.
  4. If equal quantities be divided by the same or equal quantities, the quotients will be equal.
  5. If the same quantity be both added to and subtracted from another, the value of the latter will not be altered.
  6. If a quantity be both multiplied and divided by another, the value of the former will not be altered.
  7. If to unequal quantities, equals be added, the greater will give the greater sum.
  8. If from unequal quantities, equals be subtracted, the greater will give the greater remainder.
  9. If unequal quantities be multiplied by equals, the greater will give the greater product.
  10. If unequal quantities be divided by equals, the greater will give the greater quotient.
  11. Quantities which are respectively equal to any other quantity are equal to each other.
  12. The whole of a quantity is greater than a part.

It was an almost religious experience, as in "if you take these on faith, the rest can be proven". I compared these to axioms of religious faith. There was, is, nor will there ever be any comparison.

In short, seeing this list sold me on rationality forever.

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The simple and commonly used sum, and divide of apples. I was really bad at math, and using objects instead of numbers really teached me how to love (math, LOL). It's amazing how math can be used on anything.

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One of the things I really like in math is the probability. One of the best examples that I like is on the film 21.

You are in a program show and you have three doors:

One: With the prize, and the other two with monsters;

The presenter tells you to pick up a door. When you finally choose a door, he asks "Are sure about it?

Then for some reason he decides to open one of the wrong doors and asks you: "Are you going to stay with your door or change it?" and he says "But remember I know where the prize is".

So what should have you do?

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Not sure if it was the first, but one very early one for me was realizing if you knew the square of an integer, you could easily step to the next one by adding the known square, the original integer and the next integer together.
Know $2^{2}$
Want $3^2$

$3^2 = 2^2 +2 +3 $
$3^2 = 4+2 +3$
$3^2 = 9$

or $(n+1)^2 = n^2 + n + (n +1)$

Another although much later point for me was when Calculus just clicked (4th time taking Calc 1). It was like cracking The Matrix.

I could see derivatives and integrals in everything around me and the relationship between the trig functions suddenly made sense.

Also the magic of Fourier and Laplace transformations.

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The fact that Gabriel's horn has infinite surface area, but finite volume, hence you can "fill it with paint, but you can never cover the whole surface".

Gabriel's horn (also called Torricelli's trumpet) is the graph of $y =1/x$ for $x\geq 1$ rotated around the $x$ axis.

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I found a book by Isaac Asimov called "Asimov On Numbers" which is a compilation of his essays related to math and numbers.

It was all very fascinating - things like why Roman numerals are inefficient, why zero was such a groundbreaking number to invent, and things like that. You might want to see if you can find that book to get some inspiration.

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I would have to say that it was the square root. There was (ans still is) something very fascinating about being able to recover the number that was multiplied by itself. If I know that $x^2 = 9$ then I knew that $x$ could be $3$ (just thinking about positive numbers here). And I thought that it was crazy how one could also take square roots of numbers that aren't actually squares themselves.

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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.

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I discover that math is beautiful again and again, I suspect there is no end to this. Some of things that blew my mind:

  • The very first thing that hooked me to science in general was the ability to model reality. When I was a child my father showed me how differential equations (by then I had no idea what it was) could simulate reality (it was masses connected by springs and dampers with some simple graphical representation).
  • Prime numbers. The natural numbers are intrinsic to the world and so a prime numbers. If there is a sentient alien race, they are aware of prime numbers (or they will be someday).
  • The numbers $e$ and $\pi$, their interconnection, and importance in both continuous and discrete worlds.
  • Fourier transform (both continuous and discrete). There is no denying: Fourier transform is just awesome, the sheer number of applications, implications, similar transforms, etc. speaks for itself.
  • The 3D proof of Desargues' theorem. It's one of those illuminative cases where considering a harder case simplifies the problem.
  • The probabilistic method. The first time I saw it, it was very inspiring, also the person of Paul Erdős is very inspiring too (and funny, e.g. the title of "Permanent Visiting Professor").
  • Randomness, randomized algorithms, and the P vs. NP problem. There is something incredible in the fact that for some problems the only fast solutions we know are randomized, any known deterministic approach is way slower (the canonical example being checking if the symbolic determinant of a matrix containing variables is zero).
  • ... this ever-growing list continues...

Cheers!

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In the elementary school, when I was learning about maps on the geography lessons, I was amazed by the concept of the scale of a map.

It might sound silly now but I was very happy when I understood the relation between ratios of distances and ratios of areas (and ratios of volumes:).

It somehow provoked me to thinking about what length, area and volumes really are, how to define them. And how to define what a map is.

Of course I got familiar with precise definitions much, much later :)

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A Fibonacci spiral and the way that at large enough scales it converges on the golden ratio.

Also the golden ratio.

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Not an example of my own youth I've followed a small seminaryseminar on how to teach math a few years ago, and one of the things the teacher mentioned was that counter-intuitive results were more likely to mark the kids in a way they would start to try to understand why the results wasn't what they expected.

The example he gave us was fairly simple:

Imagine you ran a rope around Earth's diameter, lying on the ground. Then, add 1 meter to the length of the rope, keeping its shape as a circle (let's forget mountains and pretend Earth is just a ball for a while) - at what distance of the ground will the rope be?

For most people, adding one meter to such a long rope is negligible so that there's simply no way it would be far from the ground. Convincing them that it's actually nearly 16cm above the ground is fun to do.

As far as I remember, that example was extracted from a book, full of such examples and historical references which are also useful to show math isn't just a boring school obligation; but I can't find the name of the book right now.

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From an interview with Vladimir Arnold (NOTICES OF THE AMS, Vol. 44, No. 4):

Please tell us a little bit about your early education. Were you already interested in mathematics as a child?

...

The first real mathematical experience I had was when our schoolteacher I. V. Morozkin gave us the following problem: Two old women started at sunrise and each walked at a constant velocity. One went from A to B and the other from B to A. They met at noon and, continuing with no stop, arrived respectively at B at 4 p.m. and at A at 9 p.m. At what time was the sunrise on this day?

I spent a whole day thinking on this oldie, and the solution (based on what is now called scaling arguments, dimensional analysis, or toricvariety theory, depending on your taste) came as a revelation.

The feeling of discovery that I had then (1949) was exactly the same as in all the subsequent much more serious problems—be it the discovery of the relation between algebraic geometry of real plane curves and four-dimensional topology (1970) or between singularities of caustics and of wave fronts and simple Lie algebra and Coxeter groups (1972). It is the greed to experience such a wonderful feeling more and more times that was, and still is, my main mo- tivation in mathematics.

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For me, it was topology, and beautiful Klein bottle and Möbius strip.

enter image description here

Related to this was the realisation that a coffee cup is topologically identical to a doughnut:

Coffe cup to doughnut

This still fascinates me to this day despite not being involved in advanced maths at all.

Coincidentally, I learnt about this from a maths book for children written ~30 years ago :)

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I felt like an Einstein and was really interested in mathematics when I myself discovered the truth behind a^0 =1. That is, a^0 = (a)^(1-1) = a^1/a^1 = 1

Yeah, I know this is simple.. But generally it is taught as a formula. Instead this one can be used to change the way of thinking...

Also, multiplication is repeated addition... This used to fascinate me a lot...

2 * 3 = 6 that is, 2 + 2 +2

4 * 3 = 4 + 4 + 4

5 * 8 = 5 + 5 + 5 + 5 + 5 + 5 + 5 + 5

And then in the end you can say that, for very big numbers, you cant sit adding all of them and hence, multiplication is the shortcut to add all of them :)

I am not a writer... But probably you can take some god examples to explain what I am trying to say here... I think this will be really interesting approach to teach multiplication! All the best for your book. Do let us know the name of the book. We will also cherish it... :)

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As a child, I liked drawing.

When I realized that there was an easy way of telling whether it is possible to draw a given figure in a single stroke, I was intrigued.

I read this in a popular mathematics book and it can be easily explained to a child.

(if there is 0 or 2 intersection with odd degree, the figure can be drawn in a single stroke)

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Compared to most answers this is certainly not going to blow anyone away, but at the time it did amaze me. Our maths teacher asked us how long it would take us to get home if, we only walked half the way home, and then half the way of what was left, and then half the way of what was left, etc, etc. The realisation that if you kept dividing something by two (no matter how many times), you would never get to zero.

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Mine was the discovery of sets in higher order math classes, and how all the lower math classes including physics theories were strictly derived from higher order calculus, and all of the formulas I had ever learned became such simple child's toys.

I don't think those belong in a children's text, however.

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When I was in second grade we memorized the times tables through $9$. At the very end of the year, our teacher taught us simple single-digit division. I was floored: "We can reverse multiply?!?!"

I think that got me to pay more attention in math. The first thing that really cemented my love of math was learning set theory in seventh grade (widely reviled as "the new math" by parents and politicians in the U.S.). I wasn't hooked for life until 11th grade when we were given the definition of a relation as a subset of the cartesian cross-product between two sets. I still remember getting chills when I understood that.

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I remember reading a magazine when I was a kid that asked this question. If you have 6 pieces of spaghetti that extend as long as you please, and you cross them so that they create as many overlaps as 6 sticks allow. How many cross points do you have. Then it asked how many would it be for 17 spaghetti sticks, could you figure it out for any number of spaghetti?

And I remember concluding that $\frac {n(n-1)}2$ is the formula for finding the answer. I was excited at the time. Looking back now I see how elementary that was.

Here is a visual way to see it: enter image description here

So, I started by drawing out a strand of spaghetti assuming that they could be as long as you please and as thin as you please, then I start with one and work my way up to 5 counting how many times they cross.

enter image description here

So then for 5 sticks of spaghetti I labeled all of the crossings. I did this for 6 as well just to see what was happening. I noticed that the number of crossings on each strand of spaghetti was the number of total spaghetti - 1 because it didn't cross itself. So from now on I will refer to the number of spaghetti as n. So to count the number of crossing I knew it was $n-1$ crossings for every stick and $n$ sticks so the total number of crossings was $(n-1)n$ and I noticed that each crossing occurs on two separate sticks, because one crossing is the crossing of two sticks to the total number of crossings is half of the number in the diagram so it was $\frac 12 (n-1)n = \frac {n^2-n}2$

P.S. sorry for using the words crossing and sticks instead of points and lines. It was something that stuck because of the spaghetti analogy in my head. I didn't realize I was doing it until it was too late.

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For me, the result that really captured my imagination was the divergence of the harmonic series:

$$ 1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+\frac{1}{6}+\ldots=\infty $$

It combines some wonderful ideas about the infinite and the infinitesimal, and it seemed (at the time) completely absurd to me that adding infinitely small numbers could result in an infinitely large one.

As an illustration of this idea, say we have a big pile of 1-foot square boards. We stack the first board on the second, hanging half-way (6 inches) over the edge. Then we stack the third on top the second, hanging 1/3 of the way (4 inches) over the edge. The forth is stacked on, hanging 1/4 of the way (3 inches) over the edge. The fifth...you get the idea. At first glance, one might think that our pile can only extend horizontally a finite distance - we might take bets that it gets at most 2 feet, or maybe 5 or 10 feet horizontally. But it turns out that if we have enough boards (negligibly thin, say), we could build a bridge across any river, any ocean, in fact we could build a bridge across the entire universe this way.

Here is a Wolfram demonstration of this, although their stack is upside-down from how I have described it: http://demonstrations.wolfram.com/OverhangingCards/

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Like most people, my most amazing discovery was tables. How 2+2+2 was six and how 2 times 3 was also six. And then I could count the number of chocolates lying on a table when they were paired. And then, even if chocolates were not grouped, I could mentally take a base of 2 and count 2, 4 6, 8.. chocolates and always be the first one to count the number of chocolates/things on a table. Most recently I was extremely fascinated by a model at display in the science/maths museum in Cambridge. The model was describing accuracy in probability. It was two sheets of glass standing between which there were random rods connected to the sheets in a certain way. On the glass was drawn a graph (like a parabola or a sine wave) which was a prediction of how the end graph would look like and to shape the graph there were little balls dropped over a period of 10 minutes or so between the sheets. What it proved was the 100% accuracy in the probability of a certain shape of a graph being formed with random balls thrown for a certain period of time. It just blew my mind away and I was standing there with little children for 30 minutes watching this over and over and was awed everytime the same graph was formed. I searched a lot on the MIT museums website but am not able to find this exhibit mentioned. It may more have been a physics thing.

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I remember my own observation about Pythagorean triples. I already knew that $3^2+4^2=5^2$ and $5^2+12^2=13^2$, and realized that the same trick can be done starting with any odd number $n$, and the other two will be serial numbers that add up to $n^2$.

For example, starting with $n=7$, we get $24+25=7^2$, and finally $7^2+24^2=25^2$.

1
On

$$\sqrt{\sqrt{\dotsb\sqrt{x}}} = 1$$

(or its more precise version: $lim_{n \rightarrow \infty} \sqrt[n]{x}$, for x positive)

As a kid I would always type in a number in my calculator and then keep hitting the square root key until the display went to 1. I would also do this with other keys on the calculator to see what would happen (some would blow up past the capacity of the floating point storage and some would go to 0, some to 1).

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If I wrote a book, a few pages would be dedicated to visualizing square root through blocks like this:

square root blocks

A kid can put cards on a table and count the edge rectangles to figure out the approximate square root of any number. With the help of some legos you can even teach cube root!

3
On

I remember that when I was five, I made this reasoning "if I can write the digits $0, 1, 2, 3, 4, 5, 6, 7, 8, 9 \space$ then I will be able to write all the numbers".

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I recall being told about binary numbers when I was about 7 or 8 years old, and the idea that numbers could be represented otherwise than in base 10 must have fascinated me. Later in school I was mildly disappointed to learn that $\pi$ cannot be expressed in any simple way, as a ratio or using any of the mathematics I knew at that time.

Modular arithmetic is something that I more or less found out about on my own, surely prompted by its usefulness in handling operations on the twelve pitch classes.

It is a very entertaining practical experiment to fold a Möbius strip with paper and tape, then cut it once, and why not twice. It's not very intuitive what is going to happen!

At some point I remember trying to figure out how to generalize the factorial to real numbers. Of course I failed, and it took a few years before I saw the Gamma function in some book.

Huge numbers may provoke curiosity. After addition and multiplication there is exponentiation, and then towers. Just showing that you can construct numbers such as $x^{a^{b^{\ldots}}}$ can be interesting, and even more that some towers with infinite numbers of terms converge (but that is certainly fairly advanced).

For more reading I recommend Lakoff and Núñez, Where Mathematics Comes From.

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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...

1
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In elementary school, my math teacher taught us this trick for the 9 multiplication values:

enter image description here

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When I was still pretty young (I don't remember my exact age) I was very proud that I could already compute with decimal fractions which nobody I knew in my age could at the time. Around that time my aunt had a student for a visit in her home, and he talked to me about math, and asked me to compute $1/3+2/3$. I asked to how many digits and he said as many as you like. So, I sat down and computed it to 10 digits or something: \begin{align} 0.3333333333\\ \underline{+0.6666666666}\\ 0.9999999999 \end{align} Proudly, I presented my result. He said well done, but it's way easier \begin{align} \frac13+ \frac23= \frac{1+2}3= \frac{3}3=1. \end{align} The beauty in this impressed me a lot and kind of got me started in math.

2
On

The possibilities of abstraction. This liberated me. Until I was about 13, I always had trouble with solving problems involving proportionality and inverse proportionality. Until I learned about variables. When I realized that you could just put a symbol instead of the number you don't know and just perform computations with it until everything simplifies in a way you can find back the number, I had a feeling of unstoppability.

The power of abstraction is so great that I'm very saddened by our current educational system in which it has nearly disappeared. All the students I get are struggling with symbols there were I have always seen them as my friends.

1
On

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.

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I don't find it beautiful, but I still find the idea expressed by the following something of a psychological curiosity:

How can it be that when some algebraists say "AND" and "OR" they mean exactly the same thing?

OR means this that "false or false" is false, "false or true", "true or false" as well as "true or true" are true, or more compactly:

    F  T
 F  F  T
 T  T  T

AND means this:

    F  T
 F  F  F
 T  F  T

But, since NOT(x OR y)=(NOT x AND NOT y) and NOT(T)=F and NOT(F)=T, OR and AND, to an algebraist, mean exactly the same thing!

0
On

As others have mentioned, kids love $\pi$. Prime numbers are also good, if they have a good handle on division. I think the fundamental theorem of arithmetic is intuitively true once you understand it (at east it was to me).

It would be great to mention some unsolved problems, like the twin prime conjecture or the Collatz conjecture.

For me, one thing that I remember being fascinated about at an early age was the fact that multiplication is commutative. That $3+3+3+3+3=5+5+5$ (or if you want, five baskets with three apples each is the same as three baskets with five apples each) was not immediately obvious to me, and the fact that it worked for any two numbers amazed me. Once you understand the geometric "square of dots" proof it makes sense, but I think that before that it doesn't.

Knuth up arrow notation is worth mentioning. Kids love that multiplication is repeated addition and that powers are repeated multiplication, and would be interested to see that idea taken further.

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Thanks to @FacebookAnswers for suggesting Conway's Game of Life, a cellular automaton devised by John Conway in 1970.

A Gosper Glider Gun

Gosper Glider Gun

With its patterns, oscillators, spaceships, glider guns (the minimalist Gosper Gun is shown above), breeders, Turing Machines, and the many derivatives, this "game" has spawnd much thinking and imagining.

A generation $\approx 10^{28}$ Turing Machine in Golly

A 10^28 generation Turing Machine in Golly

Of course it's a challenge to replicate the wonders in a static book, but there's great potential for the CD, ebook, or website.

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On

The commutative law doesn't hold for some series. I think this is an amazing fact to teach.

http://www.math.tamu.edu/~tvogel/gallery/node10.html

The example in the link amazed me.

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