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

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

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

However, I myself am mathematically unsophisticated.

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

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

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

I think the first think that amazed me in this way is $\pi$. 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 Greeks discovered it, they knew it has something to do in the circumference or the area of a circle, i.e. they could manipulate it, and I find this unbelievable.

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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'.

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

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

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

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 kinda hard. This is only 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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In English these are sometimes known as cryptarithmetic puzzles. –  Will Mar 7 '13 at 16:41

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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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 prof 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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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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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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There had been a mentioning of Gauss, more than one person actually. Look closer. –  Asaf Karagila Mar 7 '13 at 18:59

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

...

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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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

yaa, 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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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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Well, you'd take the time it takes you to go halfway home, plus half of that, plus half of this next amount, and so on. –  Joe Z. Mar 13 '13 at 20:40

The fact that you can't divide by zero always amazed me. I once read the following analogy:

Imagine you go to a shop with 100 dollars in your pocket, and imagine that everything in the shop costs 1 dollar. How many things can you buy? 100. What if instead of 1 dollar, each thing costed $0.5? How many things can you buy? 200. Now imagine that everything is free. How many things can you buy? Obviously, this question doesn't make sense anymore, because things are free, so you can take 0, or 1, or 2, or... - The first time I heard that 3 times 5 is the same as 5 times 3, I was really intrigued, and I've been hooked ever since. It is pretty weird when you think that five groups of three people is as many as three groups of five. - I found the formula$(a+b)^2=a^2 + 2ab + b^2$that my father told me at a young age fascinating. (And also that$(a+b)(a-b)=a^2-b^2$.) Overall, it seems that a parents duty is to teach his children two of the following: (a) to ride bicycles, (b) To play chess, (c) The formula for$(a+b)^2$, and my father took (b) and (c). My mother let me read her high-school calculus book (incidentally one of the authors had the same last name as mine) and there what I found really fascinating (but I could not understand) is that you can add a variable to a triangle. (This was a misunderstanding of what$f(x+\Delta)$means.) - I must have been very small, around three of four, when I suddenly dashed out of my room, full of excitement, wanting to show my dad something that had made a great impression on me. I held a book, it's front cover facing me. In a flash, I gave it two half-turns. One upside-down, the other left to right. This is what came out: I held my breath, as the trick wasn't over yet. Sure enough, two same quick moves and -lo and behold- the front cover was facing me properly again, just as in the beginning. "Look! Dad!" :) That surely must have been my first conscious encounter with symmetry. I held the memory dearly close for a number of years but then forgot about it completely. It came back to me, only very recently, after going through the first pages of Nathan Carter's Visual Group Theory and seeing this image: - I first discovered that math was beautiful upon learning the divisibility rules. At that point I was just like "IT WORKS IT WORKS! HOW DID PEOPLE KNOW THAT?!" I remember I once stayed up to test the divisibility rule of dividing by$8$(if the last three numbers in the dividend are divisible by$8$then the whole number is divisible by$8$). - Maybe the fact that the homotopy category of a model category is equivalent to the full subcategory of fibrant-cofibrant objects with homotopy classes of morphisms. - Congratulations for finding that out as a child. – azimut Jun 21 at 13:26 Dry humor? (Hope so.) – Did Aug 21 at 10:10 The realization that you can go on counting forever. - Tupper's self-referential formula is a self-referential formula defined by Jeff Tupper that, when graphed in two dimensions, can visually reproduce the formula itself. It is used in various math and computer science courses as an exercise in graphing formulae. The formula was first published in his 2001 SIGGRAPH paper that discusses methods related to the GrafEq formula-graphing program he developed. The formula is an inequality defined by: $${1\over 2} < \left\lfloor \mathrm{mod}\left(\left\lfloor {y \over 17} \right\rfloor 2^{-17 \lfloor x \rfloor - \mathrm{mod}(\lfloor y\rfloor, 17)},2\right)\right\rfloor$$ where$\lfloor \cdot \rfloor$denotes the floor function and$\mathrm{mod}$is the modulo operation. Let k equal the following 543-digit integer: 960 939 379 918 958 884 971 672 962 127 852 754 715 004 339 660 129 306 651 505 519 271 702 802 395 266 424 689 642 842 174 350 718 121 267 153 782 770 623 355 993 237 280 874 144 307 891 325 963 941 337 723 487 857 735 749 823 926 629 715 517 173 716 995 165 232 890 538 221 612 403 238 855 866 184 013 235 585 136 048 828 693 337 902 491 454 229 288 667 081 096 184 496 091 705 183 454 067 827 731 551 705 405 381 627 380 967 602 565 625 016 981 482 083 418 783 163 849 115 590 225 610 003 652 351 370 343 874 461 848 378 737 238 198 224 849 863 465 033 159 410 054 974 700 593 138 339 226 497 249 461 751 545 728 366 702 369 745 461 014 655 997 933 798 537 483 143 786 841 806 593 422 227 898 388 722 980 000 748 404 719 If one graphs the set of points$(x, y)$in$0 \le x < 106$and$k \le y < k + 17$satisfying the inequality given above, the resulting graph looks like this (note that the axes in this plot have been reversed, otherwise the picture comes out upside-down): - 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. - 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. Professor explained, to do network communication you will need to understand this. I found maths is awesome after dealing with these. What we are normally learning is can not help for always (its real numbers mathematics). But the best things deal with Fields. Therefore below is the explanation of 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} - The simple and commonly used sum, and divide of apples, i was really bad at math, and using objects instead of numbers really teach me how to love(math, lol). its amazing how math can be used on anything. - 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. - 1. Take your age, and reverse it. 2. Subtract smaller number from bigger. 3. Add the digits of subtraction. 4. You get 9. - What if I'm 22? – Kobi Mar 7 '13 at 12:06 What if I am 4? – Jack Aidley Mar 7 '13 at 14:04 What if I am 100? – Trevor Wilson Mar 7 '13 at 16:32 What if I'm 27.9? – Asaf Karagila Mar 9 '13 at 20:28 ^ Then your answer will be expressed in terms of the p-adic numbers. – Joe Z. Mar 12 '13 at 13:14 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. - 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. - So whenever I commented that someone is a jerk, I was deriving this from their acceleration? :-) – Asaf Karagila Mar 7 '13 at 17:31 @AsafKaragila Sir, I'd like to present to you your well deserved award, 'worst joke ever'! – OmnipresentAbsence Mar 7 '13 at 18:10 @OmnipresentAbsence: For this? Nah, I have had infinitely worse jokes before, and in probability$1\$ I will have infinitely many more. –  Asaf Karagila Mar 7 '13 at 18:11
@AsafKaragila I know, I'm just kidding. I actually chuckled because of how cheesy the joke was –  OmnipresentAbsence Mar 7 '13 at 18:15
d/dt jerk is called "jounce", I believe. –  Joe Z. Mar 12 '13 at 13:11

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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Seminary? There's a religion of mathematical pedagogy? –  Joe Z. Mar 13 '13 at 20:41
Not that I know, but now that you mention it, maybe it isn't such a bad idea. –  Joubarc Mar 14 '13 at 13:36

I was hooked on math by a small side note in a kid's book of mathematics about perfect numbers, numbers that are twice the sum of their factors. For example, 6 is the smallest perfect number because 1 + 2 + 3 + 6 = 2 × 6 and 28 is the next one because 1 + 2 + 4 + 7 + 14 + 28 = 2 × 28. The next perfect numbers are 496, 8,128, 33,550,336, and 8,589,869,056.

I was so fascinated by the idea that I proved that those numbers were perfect by listing out all their factors and adding them together. And to this day I wonder: somewhere up there in the vast expanses of integers, could there be an odd perfect number?

I think that James Sylvester stated it eloquently: "...a prolonged meditation on the subject has satisfied me that the existence of any one such [odd perfect number] — its escape, so to say, from the complex web of conditions which hem it in on all sides — would be little short of a miracle." Marcel Danesi, in his book The Liar Paradox and the Towers of Hanoi, stated it significantly less eloquently: "No odd perfect numbers have ever been found. They probably do not exist."

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