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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

149 Answers 149

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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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.

I was granted immunity from this idea at a very young age.

When I was in 3rd grade (possibly 4th, my memory is a little fuzzy) my father showed my sister and I knotplot (http://knotplot.com/), a piece of software that one of his friends was working with. He explained that his friend was a knot theorist, a kind of mathematician who studied knots.

I had no idea what that really meant, of course. All I knew was that studying knots was apparently a thing that mathematicians did. It wasn't a lot, but it was enough. Whenever the North American math curriculum tried to trick me into thinking that math was about numbers, my brain would reply "You say that... but what about knots?" It was my vaccine.

It also helped that my physicist father insisted on teaching us the interesting bits of math whenever it seemed appropriate. He told us about negative numbers the moment we learned about subtraction (I then obnoxiously quizzed my classmates on the playground. "What's $5-6$?", "Zero?", "Nope! Minus one." It's a wonder I didn't have more friends...). He showed us imaginary numbers when we learned about squares (the mysterious $i$ became another mental vaccine.) When I mentioned $\pi$, he countered by telling me about $e$.

The thing about math is, once you have a general curiosity, you'll start discovering interesting things all over the place. Once that starts happening, you're doomed. Math will never be boring again.

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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

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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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

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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Beremiz, an Arab mathematician, arrives on foot to a bedouin camp, accompanied by a friend that rides a camel, only to find three boys having a dispute about the testament of their recently deceased father. In the will, the father had left his herd of camels to be divided among his sons, in the following proportion: the older son should receive half of the camels, the middle son should receive a third of the camels, the younger son should receive one in each nine camels. The problem is that his herd was composed of 35 camels... So, the boys reckoned that, in order not to fail their father's instructions, they would have to reserve at most 17 camels to the older one, at most 11 camels to the middle one, and at most 3 camels to the younger one. This looked bad, not only because the divisions all left remainders, but because in fact there would still be a few camels attributed to no one after the three sons received their share! To help them find a fair solution to their predicament, Beremiz then asks his friend to give his own camel to the brothers, a suggestion to which his friend reluctantly assents. As a result, the division now goes smoother, and the three boys are happier, having respectively received 18, 12 and 4 camels. They are so happy, indeed, that they give the 2 remaining camels to Beremiz and his friend!

I once had to teach young children (and some not so young!) to sum fractions, and it was easy to generate a whole set of problems based on the same approach (a collection of fractions that do not sum to 1). I believe this will give rise to straightforward illustrations for your book, Liz.

Malba Tahan's book "The Man Who Counted" may be claimed to be the way in which any Brazilian younger than 80 has first gotten in touch with the beauty of math. Its romanticized recreational presentation makes it easily accessible and great fun for children already in their early school years. "Splitting 35 camels" is the first mathematical problem which the readers are confronted with, in this book.

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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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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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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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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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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.) –  columbus8myhw 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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The first thing for me is the working of an equation. it is, to me, like a stanza of a poem that tells us many things in minimum words. No one would have ever thought of describing a geometrical figure. Every one used to draw it before math's entry in the real world. It's awesome for a mathematician to say that write me a circle, ellipse etc.

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

" Practice is hollow without understanding ". - Sufyan Sheikh

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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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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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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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I don't find it beautiful, but I still find the idea expressed by the following something of a psychological curiosity:

How can it be that when some algebraists say "AND" and "OR" they mean exactly the same thing?

OR means this that "false or false" is false, "false or true", "true or false" as well as "true or true" are true, or more compactly:

    F  T
 F  F  T
 T  T  T

AND means this:

    F  T
 F  F  F
 T  F  T

But, since NOT(x OR y)=(NOT x AND NOT y) and NOT(T)=F and NOT(F)=T, OR and AND, to an algebraist, mean exactly the same thing!

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Your answer implies that $\neg ( \perp \lor \top) \iff ( \neg \perp \land \neg \top) \iff ( \top \land \perp ) \iff ( \perp \lor \top)$. Your truth table for $\land$ is wrong. –  Andrew Salmon Mar 23 '13 at 21:10
@AndrewSalmon Thanks, I don't know how I did that. –  Doug Spoonwood Mar 24 '13 at 2:45

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

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