Undiscovered Math

One day in fifth grade, I was playing with numbers, scribbling down products and quotients—you know, typical cool-kid stuff—when I noticed a pattern. Take any pair of numbers that are two apart (like 6 and 8, or 9 and 11). Multiply them together. Then add one.

You’ll get the square of the number in between them!

This blew my mind. The numbers were hiding secret alliances, passing coded messages amongst themselves, and I’d somehow broken inside. I was a number spy.

Tapping into what little I knew of algebraic notation, I rewrote my discovery in abstract terms:

I carried that formula with me for the next three years. I wore it like a locket and recited itlike an oath. It was pure and crystalline and true. Never mind that I hadn’t proved it. It was right, and it was mine.

When I hit Algebra 1, and learned not merely to write algebraic symbols but to manipulate them, I found that my formula wasn’t just known. It was a trivial consequence of the distributive property.

I suffered ten seconds of stunned disappointment that my treasured conjecture was nothing but old hat. Then I smiled. When you’re standing on a mountain, you don’t care that others have stood on that same spot before. You’re just enjoying the view.

From time to time, the formula and my “discovery” drifted back into my thoughts, like an old phone number or a favorite lyric. Then, just recently, I was toying with the idea of multiplication as arrays. It’s an old, simple idea, one that my sister teaches to 2nd graders. I thought about square numbers:

Then I pictured subtracting one:

Then I pictured taking the last column, turning it on its side, and sticking it below the bottom row:

Suddenly the pattern threw off its disguise, and there it was, shining up at me, a diamond in a field of dots. It was a new proof of my formula. I’d never imagined the elegance and simple insight that the geometric perspective might bring.

There is nothing novel or historic or remotely significant about my observation. It is a fact that has been discovered and re-discovered thousands of times, across all the continents. It is old news. It is—to use the mathematician’s favorite word of sneering derision—trivial.

But it is also a pattern grander and deeper than the minds that stumble upon it, so this fact belongs to no one and everyone at once. It is mine, just as it is the intellectual property of some Babylonian who derived it millennia ago and murmured, “Trivial… but interesting.”

It’s like literature. I don’t believe that the great writers invent new themes. The human condition is only so complicated, and the messages which strike deepest at our hearts (“Power corrupts”; “Love wins out”; “We are all alone at death”) have been circulating in our plays and poetry for centuries. The job of the writer is not necessarily to pour new ideas into the reservoir of human thought. It’s to draw water from that reservoir, and pour it into the vessels of character and story and phrase. The job of a writer isn’t to invent or discover truth. It’s to bring truth to our lips, so we can drink.

So it is sometimes with math. As the X-Files promised, the truth is out there. And to discover it makes us mathematicians, no matter whether others have already walked the same path.

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38 thoughts on “Undiscovered Math

  1. Heh, I have stories like these once in a while.
    Like, back in 8th grade, I found out that the difference between two consecutive squares went like this:
    (x+1)^2 – x^2 = 2x + 1
    Back then I knew not how to prove it. I already knew reasonable algebra, but I was unsure on how to simplify this.
    Then I figured out how to apply the distributive. It all became simpler.
    This isn’t the only example. There are many things that took me a bit to figure out because I was missing one piece of the puzzle.
    I always end up figuring it out though.
    Or most times anyway. But if I can’t figure something out I’ll look it up anyway.

    • I like that one – surprisingly similar in nature to the one I was playing around with.

      It can be a good motivation to push forward in math, actually: to find the explanations for patterns you’ve already noticed.

  2. If you have or can get hold of Asimov on Numbers, you might want to look at his article “The Asimov Series”, which is about his discovering various prettier infinite series for e.

    • Thanks – I’ll add it to my list. I’ve been meaning to go back and read more of his fiction anyway; I read the first two Foundation books in high school and enjoyed ‘em a lot.

  3. This has happened and still happens to me. I stumble upon a seemingly significant oberservation that upon further inspection turns out to a trivial consequence of the patently obvious. When I first make such an observation I am overcome by the seemingly profound and utterly mysterious truth I have found, but upon further research I am quickly embarrassed at how trivial it is. Nevertheless, I think those occasional Eureka! moments, even if they originate from trivialities, make it worth maintaining a curiosity and an attitude of wide-eyed experimentation toward mathematics. One day, perhaps, one of those observations may turn out not to be trivial.

    • Definitely. I heard a great story at the Joint Math Meetings, about a master trumpet player who was giving a lesson to four very-good trumpet players. He listened to them play and gave each a constructive critique. He always asked, “How long do you play the basic exercises?” (which apparently ever trumpet player knows).

      “About 5 minutes,” each of them said. “Then I practice other stuff for 2-4 hours.”

      “Spend more time on the basic exercises,” the expert said.

      At the end, a bystander (Michael Starbird, actually–the guy giving this talk) asks, “Why did you tell them to spend more time playing these simple exercises, when obviously that’s childishly easy for them?”

      The expert asks one of the students to play the exercises. It sounds far too simple and childish to be worth his time.

      Then the expert plays the exercises himself, and they sound GORGEOUS. One note melts into the next. It’s breathtakingly beautiful.

      Moral of the story: Time spent on the fundamentals is never time wasted.

  4. I once discovered the formula for the sum of consecutive integers. I remember the moment well, as I was sitting in the back of the car playing with the sums of integers and their products in my head. I remember thinking at first that it only worked for even numbers, and then testing it with odd numbers, and realizing that the division by 2 still resulted in a whole number (and realizing why; because one of n and n+1 will always be even).

    It is very likely the reason why I eventually chose to do a mathematics degree.

    • It’s funny how powerful those moments can be.

      I remember going through the same thought process when I first encountered that formula: “Won’t we need to deal separately with odd and even?” “Ah, no, because n(n+1) will always be even.”

  5. This is the joy of number theory, or of the “higher arithmetic.” This is why we always return to the deceptively simple concept of number.

    • Agreed. I loved abstract algebra in college, wish I’d gotten around to taking number theory. I’ve got Berlinski’s “One, Two, Three” sitting on my shelf – been meaning to give it a read.

  6. Wonderful read. I’d love to share this article with some of my elementary school students to see if I can elicit that same joy you felt.

    Your point about the sense of wonder also rings true with me – even something that’s been discovered countless times by countless people can bring elation to a budding explorer and push them to continue pursuing new revelations. Another wonderful idea to present to young students!

  7. I was a little disappointed when I read this article. I was hoping for a series of examples of math we know is out there, but we haven’t figured out yet. I understand why you titled the article this way, but I think “Oft-discovered Math” would have been better. Maybe you could do an entry on strange math correlations that aren’t fully understood another time?

    I’m a visual learner, and I drew out the 2×4 and the 3×5 example from the first slide to understand why it was true before I even clicked through from my email to see the rest of the article. I found it interesting that this was my first instinct and not something you’d done until recently. I think this shows how you can potentially reach greater understanding faster by working with people with a variety of learning and investigative styles.

    Thanks for a fun read.

    • Thanks – that’s a good suggestion. I was keeping a list somewhere of surprising open conjectures in elementary number theory; maybe I’ll try to polish that off into a post.

  8. My first very cool “math discovery” went something like this

    9*1=9
    9*2=18
    9*3=27
    9*4=36
    9*5=45…
    9*11=99

    Huh…
    0+9=9
    1+8=9
    2+7=9
    3+6=9
    4+5=9
    9+9=18, 1+8=9

    Every multiple of 9′s digits add up to a multiple of nine.

    You can look at any number, no matter how big, and if you add up the digits in that number and get a multiple of 9, that number is divisible by 9. 415908 is a multiple of 9. ta-da!

    I’m sure that’s been discovered lots of times before, too, but I still discovered it on my own :)

    I think I was in 5th grade?

  9. I was 11 when I was playing around with some Pythagorean triplets, and I noticed that you could make a triplet with any odd number:

    1) Take the odd number (say, 5)
    2) Square it.(25)
    3) Halve it. (12.5)
    4) Subtract 0.5 to get the second number in the triplet. (12)
    5) Add 1 to the number obtained in the previous step (13).

    Then, next year, I found that I had not discovered it for the first time. That was a real disappointment, and I still hold some resentment towards Euclid for discovering it first.

  10. I had two growing up:

    First, if you know a number squared, and you want to know the next number squared, then you take the first number squared and add that number and add the next number and you get the next number squared (i.e. (n+1)^2 = n^2 + n + (n+1)) — this I figured out while waiting for my brother’s swim class to be over, I think, and then I was super excited to prove it a few months later when we got to multiplying polynomials in algebra.

    Second, if you write a fraction that’s an n-digit number over an n-digit number, and an equivalent fraction that’s an m-digit number over an m-digit number, you can just put them together to get another equivalent fraction (e.g. 3/5 and 36/60, put together for 336/560)! This one took me much longer to figure out why it wasn’t magic for some reason, Ax/Ay; Bx/By; ((A*10^m+B)x)/((A*10^m+B)y) — this one I figured out when I had written 2/3 right next to 4/6 on my math homework scratch and accidentally read it as 24/36….WOW!!!!!

    It’s the small things that are so pretty…

    • Wow, I’m first year college, software engineering, and still this second one looks like magic to me.

      The first one I also discovered on my own, and I actually use it when calculating squares in my head.

    • I’m with Obrens – I really love that second one. It’d never occurred to me to create equivalent fractions by concatenating like that.

      Interesting, also, how many of these moments of discovery revolve around squaring. I think it speaks to how deep and fundamental the polynomials are, given that so many of our earliest “Aha!” moments involve quadratics.

      • My “discovery” was about squaring, too! Each Square is equal to the sum of the previous squares (starting with 1) plus 1. For example, 4^2 = 3^2 + 2^2 + 1^2 +1. I thought it was super cool until I realized that I was just adding 1 to “full columns” in binary (111 + 1 = 1000, 1111 + 1 = 10000, etc) and this was no more exciting than saying that in base 10, 999 + 1=1000. But I still like it.

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  12. I know that feeling of discovery, too. I discovered this cool thing about 2-digit numbers and their “reverse”: 74 – 47 = 9(7 – 4). In general, XY – YX = 9(X – Y). Later I proved it, making it all that more special. (Start with “XY” = 10X + Y and the result comes out pretty quick.) Whenever I see reversed numbers, I smile about “my formula”, my preciousssss :)

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  14. Fabulous. LOVE the matrix-based proof–super satisfying.

    Closest to time travel I ever got was doing matrix problems for statistics (which require you to think in 3-D). Sat down to work on them in a basement office in the morning, looked up at the clock in what I assumed was about 30 minutes, it was NIGHT TIME. I’d been mentally in matrix land ALL DAY, and the clock spun around while I was in math space. Apparently, it is absorbing work. =)

  15. I like the graphical proof, but I’ve always liked those. Here’s one that’s a bit difficult to understand as a static picture, but works in my head as an animation: https://plus.google.com/103146034564209742520/posts/2sqyewiPSh2

    I think my favorite mathematical insight ever came as an animated mental image of a hypercube passing corner-first through our three-dimensional space: https://plus.google.com/103146034564209742520/posts/JQt5s1NpXPb

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