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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
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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
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I think it's a shame that this question was voted closed... – Will Mar 10 '13 at 19:50

162 Answers 162

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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Personally, I thought math was beautiful on a number of occasions:

$$1x+2x=3x$$

$$1zebra+2zebras=3zebras$$

Applying words can really help young children understand mathematics better.

Another time I found mathematics beautiful was when I learned that almost all functions have a writable inverse, written using Lagrange's Inversion Theorem.

Another cool thing for me was big numbers. It started with infinity, then I discovered very large finite numbers, which are studied in Googology.

The discovery of infinity has led me to infinitely summations, which I found interesting that they were calculable and sometimes exerted weird solutions.

The discover of $i=\sqrt{-1}$ was cool, but even cooler was the discovery that $\sqrt{i}=\frac{1+i}{\sqrt{2}}$, making me realize that I could not make new types of imaginary numbers by square rooting further. This lead me to complex analysis and the solution to $x^i$.

By sitting down and writing out the formula for the perimeter of an $n$ sided polygon, I discovered $C=2\pi r$ by taking what I didn't know was a limit to infinity. It required a bit of help though.

My own realization that some of the solutions to $f(f(x))=x$ could be found using $f(x)=x$ and that this could be extended to any amount of iterations of $f$.

The disappointing discovery that one cannot find the inverse of the general quintic polynomial in terms of a finite amount of elementary operations. Of course, you can still approximate with root finding algorithms or Lagrange's Inversion, but they are neither exact nor finite in method of reaching the solution and sometimes they fail.

The discovery that one can find the square root of a number using algorithms was pretty impressive for me.

The discovery of the Lambert W function allowed me to solve soooo many exponential problems, but then a hit an edge, a barrier full of currently invertible exponential problems like $x^{x^x}=y$, given $y$ and trying to find $x$.

The discovery of the factorial is often a fun little thing for young students, it makes them think of the interesting ways that math can work. I personally tried to extend them to all positive reals, but, like some other answers, it appeared to be impossible for my talents.

Then I discovered the Gamma function and learned Calculus.

The definition of the Euler-Mascheroni Constant was truly amazing as it gave me a method for easily approximating the natural logarithm for positive whole numbers, which extends to all positive numbers through logarithmic properties.

And lastly, I would like to point at mathematically rigorous idea in physics where velocity affects air drag, which in turn affects velocity, which will again affect air drag, etc. The sheer confusion in all of this was mind-blowing.

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One of the biggest awes I experienced was when I could fully understand how you could prove that addition and multiplication of real numbers was commutative: trying to understand this it made me go to the basic construction of the Naturals, Integers, Rationals, and finally the reals (via the dedekind cuts approach).

I just thought that journey was lovely.

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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 ".

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This makes me think of Physics, where practice has lead to revelations that have called for mathematical expressions and intuitive understandings, much like String Theory's origin or Quantum Theory, or, now that I think about it, pretty much all of Physic's history. – Simple Art Jan 24 at 1:55
    
@SimpleArt I used a phrase "Problem-solving" which clears my intention. For example, in Chess, once you understand the purpose of the move "En passant" , you are never going to forget 1) The way to make it 2) The terms and conditions of the move, otherwise, you may forget how to execute that move. The move is, "If your Pawn is in 5th rank and opponent Pawn make its first move with 2 squares, you can capture it and occupy the square, it skipped" Condition: "You can only make this move in the first immediate opportunity". Purpose is to reduce the freedom of Pawn (In short). – Sufyan Naeem Jan 24 at 10:56
    
Well, that's nice. Reading your comment makes me realize how different our interpretations are, yet they can be described in essentially the same way. First time I fell for En passant, I was really confused. Now, I get to use it on other people, and I have never forgotten it. – Simple Art Jan 25 at 23:28
    
@SimpleArt this is another thing. Of course, when you repeatedly practice something, it takes shelter in your brain. Even an animal can learn something by repeatedly practicing it. BTW, my quote states "Practice is hollow..." and this is different from saying "Practice is useless..." or from saying "Practice is nothing...". – Sufyan Naeem Jan 26 at 6:55

I remember being fascinated by amicable numbers, the subject of my junior high science fair project in the early 1970's. I was using a huge book of factorization tables that I couldn't check out from the public library. I spent hours trying to plug prime numbers in the formulas given by Euler and Erdos.

DEFINITION: A pair of numbers x and y is called amicable if the sum of the proper divisors (or aliquot parts) of either one is equal to the other.

For a list see https://oeis.org/A259180

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The first equation:

Knowing that the selling price is $220$, and the margin is $10\%$, what is the purchase price ?

At the time I was able to derive the benefit $200\times 10\%=20$ or the selling price from the purchase price $200+200\times10\%=220$ but has no idea how to do the reverse (purchase prince from sale price) as the unknow "had to be known" to compute itself with $?=230-?\times 5\%$.

The rewrite with a symbolic quantity $P+P\times5\%=P\times(1+10\%)=220$ was a revelation !

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The first time I was fascinated by mathematics was when I read Christian Goldbach's conjecture. From that day onwards, I am trying to decode the mystery of primes, which seem to be simple at first sight but are actually very difficult to understand. That's the beauty of mathematics.

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You asked:

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

For me, it was when I was 3 years old (possibly 4), contemplating my hands and fingers. I had the sudden epiphany that 5+5 absolutely had to equal 10 every time that you added them together -- not merely that they had done so repeatedly, mind you, but that they must do so in every event. I was admittedly a little off base there, not yet knowing of quirks such as modulo, but it was so astounding that I ran to the bathroom to tell my mother.

There have been a lot of other wonderful moments in math, for me, but that initial one was like seeing into the mind of god, reading the very fabric of creation, and fully knowing that reality is comprehensible. :-)

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I tried to find the number of ways in which a number can be expressed in term of sum of two numbers and I ended up learning Partitions which showed me how everything can be expressed mathematically....

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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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3  
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
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@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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