The Mathematics Stack Exchange is like Yahoo Answers, if the people on Yahoo Answers had graduate-level STEM degrees...
On November 11, 2013, a Stack Exchange user asked a question:
“I need help with this integral: the integral from negative one to one of one over x times the square root of one plus x over one minus x times the natural log of 2x squared plus 2x plus one, all divided by 2x squared minus 2x plus one, dx.”
The idea of an integral is actually pretty simple. Picture a line on a graph. Now imagine taking a colored pencil and shading in the area beneath that line, down to the bottom axis of the graph. What we’re trying to find is the area of this colored region. For a straight line, this is super easy—it’s basic geometry. But the more complicated and curvy and weird your line gets, the more difficult it is to figure out the area underneath it.
The original poster tried using a few computer programs, but none of them could give what’s called the “closed form” of the answer—that’s a precise and concise solution.
Then, four and a half hours after the original post, there’s an answer:
“I equals 4 pi times the arccotangent of the square root of the golden ratio.”
Symbolically, I=4πarccot√ϕ.
The answer came from a user named Cleo. It was a new account with only one previous answer. Cleo provided no notes, no proof, and no explanation. In mathematics, how you got what you got, is as important as what you got. Some people considered the answer a non-answer without the proof.
Cleo’s drive-by answer had unleashed madness on Math Stack Exchange. Between 2013 and 2015, she’d go on to do this 37 more times, often popping in unreasonably quickly to solve incredibly complex integration problems with fully formed answers. She did not show even an iota of her work. Then she’d disappear into the ether again.
Experts were divided about Cleo. It was clearly someone who had a real mastery of integration techniques. She mentioned strange functions that sent tenured professors looking up Cleo's notable use of esoterical mathematics.
For so many on Math Stack Exchange, all of the fun of their shared hobby is in showing your work. It’s not a dry explanation—it’s an adventure. Take a mathematician's answer to that infamous 2013 integral.
It had like an eighth-degree polynomial in the denominator, which, under normal circumstances, would mean “No, you’re not going to be able to do this.” But it turned out that the polynomial had a lot of symmetry and I could then exploit that symmetry to deduce all the roots. I was able to reduce what I had to find from an eighth-degree polynomial to a quadratic, and from the quadratic, the golden ratio fell out.—Ron Gordon, a patent agent and former physicist in Massachusetts, also does hardcore calculus.
Then Cleo disappeared in 2015 and has never resurfaced. Some people have speculated that maybe Cleo was Stephen Hawking.
Now what is Cleo into? Here's a brief climb from grade 11 geometry to advanced calculus.
1. Foundation Basics:
- Algebra: This is like your math toolkit. You need to know how to solve simple equations, like ( 2x + 3 = 0 ). Here, you figure out what ( x ) is by moving things around until you find it. This helps you understand how numbers work together.
- Functions: Think of a function like a machine that takes an input (like a number) and gives you an output. For example, if you put in 2 into ( f(x) = 2x + 1 ), you get 5. It’s important to know how these machines operate!
2. Calculus Essentials:
- Limits: Limits help you understand what happens to a function as you get really close to a certain point. For example, when you say ( \lim_{x \to 2} (3x + 1) = 7 ), you’re finding out what value the function is approaching as ( x ) gets close to 2.
- Derivatives: This is all about change. A derivative tells you how fast something is changing. If you have a function like ( f(x) = x^2 ), the derivative ( f'(x) = 2x ) shows how the output changes when you change the input a little bit.
3. Integration Skills:
- Indefinite Integrals: This is like finding the original function from its derivative. For example, if you see ( \int x , dx = \frac{x^2}{2} + C ), you’re figuring out what function, when you take the derivative, would give you ( x ).
- Definite Integrals: This helps you calculate the area under a curve between two points. For example, ( \int_1^3 (2x) , dx ) gives you the area under the line from ( x = 1 ) to ( x = 3 ). Imagine shading that area on a graph!
Cleo awoke dusty math books and revived old dreams of grasping hard-core mathematics, in the minds of people who come upon her amazing story.
For my articles in this series, visit or bookmark the following;
Brent Antonson
Brent Antonson: Where Extraordinary Recall Sparks Insight.
