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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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Nice question, but should probably be community wiki? –  mrf Mar 7 '13 at 7:02
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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

134 Answers 134

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

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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If you look closely, the first few powers of 11 are rows of Pascal's Triangle. This is not a coincidence. –  Joe Z. Mar 8 '13 at 20:16

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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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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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 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

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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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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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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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.)

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This might seem very elementary: but amazed me when I was a child.

The fact that $a\times b = b \times a$.

I would keep drawing boxes on the number line of different lengths and then discovering that they fit snugly into one another. Still seems amazing.

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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.

enter image description here

In a flash, I gave it two half-turns. One upside-down, the other left to right. This is what came out:

enter image description here

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:

enter image description here

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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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  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}

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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 teach me how to love(math, lol). its amazing how math can be used on anything.

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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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  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
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What if I am 4? –  Jack Aidley Mar 7 '13 at 14:04
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What if I am 100? –  Trevor Wilson Mar 7 '13 at 16:32
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What if I'm 27.9? –  Asaf Karagila Mar 9 '13 at 20:28
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^ 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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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
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@AsafKaragila Sir, I'd like to present to you your well deserved award, 'worst joke ever'! –  OmnipresentAbsence Mar 7 '13 at 18:10
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@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
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@AsafKaragila I know, I'm just kidding. I actually chuckled because of how cheesy the joke was –  OmnipresentAbsence Mar 7 '13 at 18:15
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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
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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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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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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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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.

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I think one of my early favourite mathematical things was simply "proof by contradiction" -- any of them.

I think its appeal is that you nearly have proof by example, except that you're proving a negative.

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When I was a kid, I remember that I used to make 'little discoveries', although after I got all excited I realised other people already knew this and I didn't discover anything new, I was always really proud when I made one of those discoveries, Here are a few that I made and approximately how old I was when I did make the discovery (just saying I am 13 years old right now).

  • An odd number plus an odd number is even, an even number plus an even number is even, and odd plus even is odd (approximately age 4-5)
  • The rule for the Fibonacci sequence (I remember I saw it on the board or something in my classroom and then I just found the pattern) (age 6, I remember this one accurately bc it was during the first year of my primary education)
  • If you rip a piece of paper in half, and then both in half again, and again etc, the number of pieces of paper you have when you ripped it $n$ times is $2^n$ (age 10)
  • The fact that you can split a triangle into two right triangles and apply trig ratios to them (I later learned the Sine and Cosine Laws so yeah) (age 11)
  • I was learning differential calculus, and I didn't know integral calculus yet. My dad told me that a integral is the inverse of the derivative, Then I figured out the rule that offsets the power rule and I figured out that the integral of nothing is an arbitrary constant (When i officially learnt integral calculus I found the proper formulas and I was just using the wrong symbols) (age 12)
  • I proved to myself that the cardinality of the set of Reals and the set of Complex numbers are the same (age 13)
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I started math reasonably late, so I'm not sure this is a perfect example for a children book. I was totally amazed that the number $\pi$ is encountered in completely unrelated situation. Of course, I knew it's a ratio of circumference of the circle to the diameter, but then I learnt of the Normal probability density function.

So if you stretch a bell-shaped curve (Gaussian function) from $-\infty$ to $\infty$ the area under it $converges$ to $\sqrt{\pi}$?? How is it possible that this area has something to do with the square root of the ratio of circumference of the circle to the diameter? To be honest, even now, after learning the related proofs and derivations I still find it quite baffling.

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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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