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

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

However, I myself am mathematically unsophisticated.

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

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

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

162 Answers 162

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

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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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Continuous? Do you mean repeated? – Asaf Karagila Mar 8 '13 at 11:17
Yaa, repeated..!! – mk.. Mar 8 '13 at 11:19
y down vote? You dint like it, thats y?? Funny.. – mk.. Mar 8 '13 at 11:21
@Sandy, I liked it +1 – user58512 Mar 8 '13 at 19:05

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 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
See which researches the earliest versions of this story. They have rather little in common with your version. – Robert Israel Mar 7 '13 at 19:02

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):

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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I don't get the question at all, are we assuming they left at different times due to the earth being curved.. while also assuming they walk in a straight line as if the earth was flat? – user58512 Mar 9 '13 at 20:23
@caveman: No, they just walk at different speeds. – ShreevatsaR Mar 24 '13 at 11:45

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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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

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.

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When I was 10, I read a math booklet, that talked about Euler characteristic. There were drawings of all Plato's polyhedrons, and I counted, and realize that their Euler characteristic was always 2. I was amazed, asked my mom, math teacher, if she knew anything about it, and she told me she didn't. Now I'm 21, and I am just starting to be math-mature enough to understand this theorem. Maths are beautiful :D !

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Oh, no, you'll be crushed when I show you a "donut-ish" shaped polyhedron! Euler characteristic is 0! (Unless you've seen this already.) :) – Akiva Weinberger Nov 13 '14 at 3:05

If you're writing a children's book on mathematics, please start by reading some excellent children's books dealing with mathematics. Here are some books I have fond memories of:

  • The Man Who Counted
  • The Phantom Tollbooth
  • Flatland
  • Alice in Wonderland / Through the Looking Glass
  • Everything by Martin Gardner
  • Godel, Escher, Bach: An Eternal Golden Thread
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I'm not sure if this is suitable, but for me, the power of Mathematics lies in the absoluteness of its proofs. This is the only discipline where you can prove something to be true and it will stand up to the test of time, where no textbooks need replacing and facts are always right. (I'm assuming we don't make fundamental changes in axioms and what not!) This cannot be found in any other human endeavour and I find this to be very reassuring!

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For me, it was the beauty of the number 1, how it can be multiplied with anything , and it won't change the number it is being multiplied with, also how it can be represented as any number divided by itself such as 4/4=1 I would also love to share this beautiful poem by Dave Feinberg that is titled "the square root of 3" and was also featured in a Harold and Kumar Movie, it renewed my love for math and is and always has been one of my favorite poems! :

I’m sure that I will always be A lonely number like root three

The three is all that’s good and right, Why must my three keep out of sight Beneath the vicious square root sign, I wish instead I were a nine

For nine could thwart this evil trick, with just some quick arithmetic

I know I’ll never see the sun, as 1.7321 Such is my reality, a sad irrationality

When hark! What is this I see, Another square root of a three

As quietly co-waltzing by, Together now we multiply To form a number we prefer, Rejoicing as an integer

We break free from our mortal bonds With the wave of magic wands

Our square root signs become unglued Your love for me has been renewed

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The realization that you can go on counting forever.

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How simple! How beautiful! Kids may love this. – TobiMcNamobi Aug 22 '14 at 10:26

One of my most memorable moments in mathematics was when I was attempting to prove the formula for the volume of a sphere on my own. I hadn't been taught calculus yet and had no idea about it, but I was convinced I could solve the problem. I used an infinite amount of small disks and added their volume ( essentially the limit of a riemann sum, an integral, but I didn' know that at the time) I made the disks a certain height, worked out the sum using sums of consecutive squares and then made the height equal zero. And voila, I got the right volume! Later I found out I had re-discovered a part of calculus. The realisation that different people can independently discover mathematical truths and techniques was beautiful to me.

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At about 10 or 11 I discovered that the area of a circle was half the circumference multiplied by the radius.

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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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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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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 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.

    T E N
    T E N


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
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In English these are sometimes known as cryptarithmetic puzzles. – Will Mar 7 '13 at 16:41
@Will Thank you very much!, sir. ;) – Qwerty Mar 7 '13 at 18:21
+1! As a kid I loved the Sideways Arithmetic From Wayside School books because they were full of these fun puzzles.… – Mattias Andersson Mar 9 '13 at 0:36

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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Haha, when I had to 'teach' my fellow 8th graders Algebra, I would commonly answer their question by saying "One zebra plus two zebras equals 3 zebras" and I would mess around with the numbers as needed. Also, when asked what $x+y$ equaled, I might reply "[that] a zebra plus an elephant does not equal a horse", implying that perhaps you simply can't add $x$ and $y$ together and must keep them separate. My friends quickly caught on. :D (accomplishments) – Simple Art Jan 24 at 1:07

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
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
Of course, now when I look back at it, I think, that wasn't actually hard! – Joe Z. Mar 14 '13 at 14:20

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

The example in the link amazed me.

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Similarly: finite sums of rationals are rational, but infinite sums need not be—for a similar reason to why the commutative law doesn't work for infinite series! – Akiva Weinberger Nov 13 '14 at 3:07

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

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

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As for the "Huge numbers" you may wish to check the "big numbers" tag or Googology. – Simple Art Jan 24 at 1:08

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...

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