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

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

However, I myself am mathematically unsophisticated.

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

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

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For me Euclid's proof of the infinitude of primes was the first thing that made me realize the beauty of mathematics. –  Manjil P. Saikia Mar 7 '13 at 7:02
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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

153 Answers 153

There is a series of math children books in Russian by
Владимир Артурович Лёвшин. To list some:
Магистр рассеянных наук (translates roughly as Master (as in M.Sci) of the absent-minded
sciences, though google translates it as Master scattered Sciences),
Новые рассказы Рассеянного Магистра (New stories of the absent-minded Master),
Путешествие по Карликании и Аль-Джебре (google transtales it as Travel Karlikanii and Al Gebre),
Черная маска из Аль-Джебры (The Black Mask from al-ğabr(=al-gebra)).
More of them at http://www.koob.ru/levshin/ (in Russian).

I am a Bulgarian (presently working in New York), and as a child (could have been anything between 6 yr old and 9 yr old) read the Bulgarian translation of Путешествие по Карликании и Аль-Джебре (or it might have been one of the other books listed above).

I was fascinated. At hindsight mathematically the book is fairly simple or even routine (goes on to set and solve an equation, must have been a quadratic one, though it might have even been linear), so once you know how to solve such equations it might appear boring. But amazingly it does it in a way that unfailingly keeps the readers attention. It is written like a detective story (the $X$ with the black mask was enchanted and was to be freed by the Master, and its assistant the Нуличка, i.e. the Naught or the Null), with characters to relate to, number system and operations introduced and, thus, developing in parallel, the necessary math background and an intriguing story to follow and enjoy. I motivated myself to understand the math details (it might have been that we had not yet covered that material in school), so I could keep reading. I was also interested in logic-puzzle books at about the same time. I cannot single out a particular math piece that is exciting in this book (generally it is about real numbers or perhaps integers, setting and solving equations), it is the whole process of taking an ignorant (but intrigued) child and making him willing to follow the story, and to eventually learn algebra (at the level of quadratic equations) on their own, and make them feel great about it (and at the same time to not know at all what exactly feat they have done, that is, there was no feeling at all that I was being educated in "accurate reckoning for inquiring into things, and the knowledge of all things, mysteries...all secrets", to borrow from Ahmes, even if there is no direct relation).

I do not know if an English translation is available of this series (google doesn't seem to know about it, and google knows everything, unless I do not know how to find it). I think this is a great book and would recommend it to anyone who could read it (or, well, would certainly recommend it to children, since adults would be spoiled with what they already know, and might not enjoy it). In my opinion this book has the spirit of adventure, and it might make an interesting reading (it reminds me of another Russian (or Soviet) well-known "adventure" book, The Twelve Chairs, with sequel The Little Golden Calf, though both format and subject are very different, but perhaps one could feel that both represent Russian culture ... don't know what the author(s) would have thought of this alleged affinity though).

I also remember that as a child I had a problem understanding infinity (безкрайност in Bulgarian, literally endlessness), and kept thinking about it. It is not clear if I understood it (it looked like a winding road that kept going no matter what), but at some point I stopped thinking about it. Nowadays I presumably know that there are different related notions: limitless/ boundless / infinite, and different parts of math might have use of one or the other (e.g. manifold without a boundary like the circle, vs the real line which extends both ways, or transfinite numbers which go just one way but could be used for counting). I could not quite accept that infinity exists (and strictly speaking nobody could prove that it does, but anyway it is an accepted convention), my point is that I forced myself to imagine that winding road never-ending, to try to get an idea of infinity, but I don't think I ever convinced myself. I could not see the whole thing, even if any time I approached the end, it kept extending (as I would just generate one more piece of it in my mind and put it there for the sake of the argument), so for me infinity was something that I cannot exhaust by way of observation, but cannot comprehend either. I could not rule out its existence, so I live with that, but I never saw it, so I can't vouch for it. (Much later at university I had a dream, almost a nightmare when I was supposed to pick a rifle from what seemed an endless field of identical rifles, and I couldn't make a choice. Eventually I picked one, its virtue was that is was exactly the same as all the others, but fulfilled the task of picking one. Somehow I tend to relate this to the Axiom of Choice, though strictly speaking you do not need AC to pick just one element of just one set. And I have no idea why it was rifles, and not, say, apples. Also, that was a multiset, not a set, so I don't know what it had to do with AC, I guess I had to come up with something familiar when I woke up, and we had already studied AC. Or perhaps someone could indeed relate this to AC in a meaningful interpretation.)

And of course, I do appreciate things like Euclid's proof that there are infinitely many primes, or that (Pythagoras or Hippasus) $\sqrt{2}$ is irrational (these were some of the first things I enjoyed introducing to my students last semester in a History of Math class), but for me these came "later" when I was already a converted mathematician (or I thought of myself so). I can't tell when this conversion happened, but it might have been in early school. I was good at math (so my teachers were happy and my schoolmates sought my help, and for that matter everyone would keep telling me that my grandmother, whom I newer saw, was a famous (or infamous, because she would uncompromisingly fail bad pupils) math teacher in my home-town), but it is not just about being good at it (as I realized, there were better students than me, once I got into university, in particular some of those coming out from so-called matematicheska gimnazia - a high-school emphasizing math, science and languages), so it is not so much about being good at it, as about being attracted by math, willingness to keep working/discovering, and the adventure and meaning that comes with it.

Please let us know when your book is published :)

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In the age four or five i knew: $$2*5=5*2$$
I hope that you understand how this result is wonderful for me on this age, because yet i didn't use to commutativity of multiplication on $\mathbb N$!
In addition i didn't generalize this fact to another numbers!

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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 is rather recent (Less than a year ago), but, since I am 14, I suppose it should still apply. I remember that I was bored in some class, and that I took out my calculator and started playing with it, writing "hello" with numbers upside down. Then I saw this button (this was a scientific calculator) that said "log," and so I pressed it. At first I received "error" for log(0) = -infinity (well, close enough), but then I tried other numbers, 1,2 10. Then I saw that at 10 it would blurt out 1, and at 100 2. I then realized that what log did was find the exponent of a number from a base number (of course, I didn't know that terminology then) but it was still pretty amazing. (I also learned later on that all calculators are log base 10)

Edit: is there something wrong with this answer? Why was it down voted?

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The first thing for me is the working of an equation. it is, to me, like a stanza of a poem that tells us many things in minimum words. No one would have ever thought of describing a geometrical figure. Every one used to draw it before math's entry in the real world. It's awesome for a mathematician to say that write me a circle, ellipse etc.

In order to tell people that math is not only concerned to problem-solving, I have produced my own quote.

" Practice is entirely hollow without understanding ". - Sufyan Sheikh

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The most wonderful thing I've recently seen is this (sorry it's in French) form of the Sieve of Eratosthenes and of course your question too.

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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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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

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 you realize that taking derivatives is so simple, you look back and realize, "I can't believe people use this as an example of difficult mathematics!" –  Joe Z. Mar 13 '13 at 20:25
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I had a similar moment of realization for reducing polynomials, back in middle school when my Sunday School teacher used a really long rational polynomial expression as an example of a "problem that's too hard for you to solve" (it was part of a teaching package). She had to resort to using trigonometry and asking me how I would calculate $\tan 35^\circ$, which I didn't know at the time. The polynomial ended up being something contrived, but it did actually reduce quite a bit. –  Joe Z. Mar 13 '13 at 20:32
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Of course, now when I look back at it, I think, that wasn't actually hard! –  Joe Z. Mar 14 '13 at 14:20

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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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 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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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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Goats aren't monsters! D: –  Joe Z. Mar 7 '13 at 21:33
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Of course, switch. The first time I have 1/3 chance of being right. The other two doors 2/3. Now, one wrong door is opened, meaning that the remaining door has the unified 2/3 chance of being right. I am flappergasped how many people do not see this. This is where I demand them to envision this example with 1 million doors, you chose one at random. Then the moderator opens 999.998 other doors leaving you with your first choice, and the one that was not openened. Do you really think not switching would be smart? –  k0pernikus Mar 8 '13 at 13:07
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Isn't this called the Monty Hall Problem? –  funkymushroom Mar 8 '13 at 16:28
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I remember my teacher telling us this problem, but she had trouble explaining why it was that way. Most of my classmates didn't believe it was better to switch. But I wrote a Basic program on my Commodore 128 running 1000 iterations and found out that it really resulted in 50% chance of winning when switching doors. Several years later I stumbled upon a good explanation of why it is better to switch. –  Anlo Mar 14 '14 at 17:51

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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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. –  NikolajK 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
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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

Solving for an unknown. 2x = 4 so x = 2. Beautiful.

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It becomes hideous when forced to solve endless amount of them. –  Phonics The Hedgehog Jun 2 '13 at 2:44

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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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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Your first sentence is incorrect. :) I had the "grid" epiphany, too! –  columbus8myhw Nov 13 '14 at 1:23

My favourite maths book when I was little was 'Magic House of Numbers' by Irving Adler.

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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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1  
In English these are sometimes known as cryptarithmetic puzzles. –  Will Mar 7 '13 at 16:41

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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Sounds like Zeno's Dichotomy paradox. –  BobStein-VisiBone Mar 7 '13 at 15:28

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 white black

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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.

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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.

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This kind of joke was very commom im my school ^^ –  Integral Mar 7 '13 at 15:43
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@drjimbob: You're missing the "add the number you thought at the beginning" line, which makes the result be $(9x+3) + x = 10x + 3$. –  ShreevatsaR Mar 24 '13 at 11:43

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.

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  1. 17 + 20 = 8
  2. 17 − 20 = 26
  3. 17 · 20 = 21
  4. 17^(−1) = 12 (inverse of 17)

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

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

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

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

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

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

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

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

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

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

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

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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

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

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The 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

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