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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

153 Answers 153

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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When I was 11, my math teacher asked us whether we thought that any three points in the plane that do not lie on a single line would lie on a single circle, and I remember being amazed to see that this was true.

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

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I 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 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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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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This recalls the introduction to Wigner's The Unreasonable Effectiveness of Mathematics in the Natural Sciences. –  Matt Aug 23 '14 at 12:44

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.

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Congratulations for finding that out as a child. –  azimut Jun 21 '14 at 13:26
Dry humor? (Hope so.) –  Did Aug 21 '14 at 10:10

My story is not that impressive - as a kid I've observed on several simple examples that $(a-b)(a+b)$ is $a^2-b^2$, and that decided my fate.

The moral is I think that there are many ways one might share joy of math with kids, and showing them its beauty is probably not the universally efficient one. At least in my case, nothing compares to the feeling that accompanied an independently initiated discovery of something out of the material world yet undoubtedly as real as anything material, maybe even more so.

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There is a nice and simple theorem that still was not mentioned here (maybe because it is in the beautiful Paul Lockhart "A mathematician's lament" you already read?). Summing the first odd numbers we see a curious regularity: $$ 1+3=2^2 $$ $$ 1+3+5=3^2 $$ $$ 1+3+5+7=4^2 $$ $$ 1+3+5+7+9=5^2 $$ ...and so on. This is charming. Furthermore it is instructive for a kid to see that any numerical calculus like this above can't prove that this rule will work "forever", but the power of a creative proof reach this goal (THE proof is a graphical one, it is sufficient consider the angular "L" of square of pebble to "see the light").

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

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

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

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

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I was always good at maths as a child, and took to reading extension maths books for fun (other kids thought I was weird). When I was about 10 I was completely hooked when I saw Euclid's proof for an infinity of primes. I had been given it as a question in one of the books I was reading. I spent about an hour desperately trying to prove it . . . then I looked at the solution - I was stunned by its elegant simplicity. Another thing I really enjoyed was finding cool facts about numbers in kids maths cartoon books and proving them. I loved to show WHY things always worked, that is perhaps my favorite thing about maths.

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For me it was Monty Hall problem:

Suppose you're on a game show, and you're given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say No. 1, and the host, who knows what's behind the doors, opens another door, say No. 3, which has a goat. He then says to you, "Do you want to pick door No. 2?" Is it to your advantage to switch your choice?

I saw this problem when I was 15 year old. I answered correctly (I probably used some kind of math intuition), but I thought that probability in the second case is $1/2$. Actually it is $2/3$. The proof is beautiful, as well as the answer. This fact amazed me. Even now, at 18, I suppose it is quite a beautiful problem.

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My first think of infinity was going from one corner of square to opposite corner. Start with route (0,0) -- (0,1) -- (1,1), then take (0,0) -- (0,½) -- (½,½) -- (1,½) -- (1,1) and so on. Path will come visually closer to diagonal, but lenght will stay at 2.

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Draw any triangle. On each side of the triangle, draw an equilateral triangle such that the new equilateral triangle shares a side with the original triangle. Connect the midpoints of your three new triangles - the result is another equilateral triangle!

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A story which I heard when I was in Primary School motivated me to understand the Power of Exponentials. The story goes like this--..........A Brahmin Priest presented the King with the Chess Board and explained to him how to pay War Games on this board . The King was pleased and asked the priest , what he wanted as a reward ..........The priest asked the King , that as he was very poor ,he needed some grains to feed his family .He asked the King to put one grain of rice in the first square of the chess board. , then put two grains in the second square ,four grains in the third square----and continue this way doubling the number of grains in the next square --till he reaches the end--the 64th square ."I will take whatever grains are there on the Chess Board..that will be sufficient for my needs"..said the Brahmin...........The king tried to satisfy the needs of the Brahmin ,but soon found out that all the grains in the Kingdom will still fall short of his needs ....... . The King was pleased with the priest's intelligence and appointed him as the Royal Astronomer & Astrologer .

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This is a new interesting 4x4 Magic Sqaure, which I believe will be interesting to School Children. Here each element of the square is a square number. This was provided by Dr. Geoffrey Campbell

509020 is the sum of rows and columns

  29^2 |  191^2 |  673^2 |  137^2 || 509020 
  71^2 |  647^2 |  139^2 |  257^2 || 509020 
 277^2 |  211^2 |  163^2 |  601^2 || 509020 
 653^2 |   97^2 |  101^2 |  251^2 || 509020    
509020 | 509020 | 509020 | 509020 || 509020 
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The $\sqrt{-1}$, complex analysis, and how real world problems could be solved "by these objects which don't exist."

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What's it matter that $i$ doesn't exist? All numbers don't exist. (WARNING: The preceding statement is opinion.) –  Akiva Weinberger Nov 13 '14 at 3:26

The symbol:

$$\int_{a}^{b} f(x) dx$$

It was cool. But then came along contour integration!

$$\int_{-\infty}^{\infty} \frac{\sin(x)}{x} dx$$

Using a contour integral

$$\oint_{C} f(z) dz = \int_{-\infty}^{\infty} f(x) dx + \int_{\Gamma} f(z) dz$$

And Residue Theorem,

Complex Analysis and Real analysis proofs.

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I have spent many decades studying why so many highly intelligent people are so mystified by mathematics. Lockhart's view is very serious and cannot be negated by the personal experiences of mathematically inclined people. My study has clearly shown that the best advice is to be simple and sensible. For example, our place number system is an ingenious solution to the problem of too many different names and shorthand symbols for quantities. The solution is not sensible if the problem is not clear. Addition is immensely useful regardless of how it is done, including by a calculator. So is subtraction. multiplication is a wonderfully ingenious way to count when the items counted come in fixed size packages. Division is also very useful, again completely aside from how to do it. Our conventional emphasis on HOW is terribly off-putting. In this electronic age, "how" is far less important anyway. Mathematics is not a skill and should not be identified as one. Numbers and numerical operations and functions and condition equations and so on, and the properties of all of these, are completely real and sensible and have nothing to do with so-called "reasoning" or "rigor" or "skill" etc. Everything sensible involves reasoning. And rigor is the concern of mathematicians, not lay appreciators and users of mathematics. And "skill" is vastly overrated. It is easy to develop skill if you understand what the subject is about. It is the latter that is missing in our education.

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Telescoping series. Double counting to prove combinatorial identities. All the paterns in Pascal's triangle. The medians of a triangle always intersect at one point. Using roots of unity filter to solve combinatorics problems.

Guage invariance over Floer homologies for conformal Khovanov manifolds in $n$-dimmensional geometries.

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

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protected by Zev Chonoles Mar 7 '13 at 22:43

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