The Mathematical Ninja took one look at the list of the 10 coolest numbers and scoffed. “Those,” he said, “are not the ten coolest numbers.”

“What are, then?” said the student. “Remember, nothing for cultural significance, so you can’t have 42. And no physical constants, so $g$ is out.”

“I wouldn’t dream of it,” growled the Mathematical Ninja. “Here are the ten coolest numbers.”

10. 6

“Six is the smallest semi-prime,” said the Mathematical Ninja, “and the smallest perfect number.”

“I’ve heard of those,” said the student. “The factors - 1, 2, and 3 - add up to the number itself!”

“Correct,” said the Mathematical Ninja. “And a semi-prime is a number with only two prime factors - factorising them is one of the big unsolved problems in cryptography. Plus, it’s the size of the smallest non-Abelian group - where $ab$ doesn’t necessarily equal $ba$.”

“That Abel - smart guy, huh?”

“Abel? No good,” said the Mathematical Ninja.

9. $\frac{1}{2}\sqrt{3} \simeq 0.866025$…

“You’re obsessed with triangles!”

The Mathematical Ninja frowned. “This is not obsession. This is focus. And this particular number is the height of an equilateral triangle with unit sides. The simplest non-trivial shape, I would contend.”

The student, hearing the tone in his voice, decided not to argue.

8. $i = \sqrt{-1}$

“You and your imaginary friends!” said the student.

The Mathematical Ninja’s eyebrows lowered like Roger Moore acting. “I see,” he growled, “what you did there. I’ve gone for $i$ because it epitomises the triumph of notation over instinct: when they were first solving cubics, this $\sqrt{-1}$ showed up in some of the equations. Some said ‘bah! can’t be done;’ others said ‘let’s roll with it,’ and found the answers worked out.”

“Keep going, faith will come?”

“Precisely.”

7. 16

“That’s clearly a cultural reference,” said the student, wishing he’d brought his QI klaxen. “You haul 16 tonnes and what do you get?”

“Depends on the coefficient of friction,” said the Mathematical Ninja. “But no: I’ve picked 16 because it’s $2^4$ and $4^2$, the only integer that’s both $a^b$ and $b^a$ for $a \ne b$.”

“I prefer another day older and deeper in debt.”

6. 3003

“Oo, I know this one!” said the student. “It’s in Pascal’s Triangle, isn’t it?”

“It is,” said the Mathematical Ninja. “Eight times, no less. As far as we know, it’s the only number that appears more than six times. It’s $^{14}C_{6},~^{14}C_{8},~^{15}C_{5},~^{15}C_{10},~^{78}C_{2},~^{78}C_{76},~^{3003}C_{1}$ and $^{3003}C_{3002}$.”

“Phew! That Pascal, pretty smart guy, huh?”

“Pascal? No good,” said the Mathematical Ninja. “Singmaster, no good either. ((David? If you’re reading this… no hard feelings.))”

5. $\sqrt{2} \simeq 1.414214$

“The deadliest surd,” said the Mathematical Ninja, “and one of the few, possibly the only, that anyone’s been murdered over.”

“There was that lottery scandal where you used to teach.”

“We do not talk about that. Instead, we talk about poor Hippasus, who proved - legend has it - to Pythagoras that $\sqrt{2}$ is not a rational number. Pythagoras - who believed order in the universe came from whole numbers and fractions - took the huff and threw him into the sea.”

“You can’t write it as a fraction because they both need to be even?” The student knew, if he didn’t mention this, he’d suffer a similar fate to Hippasus. “Pythagoras - pretty smart guy, huh?”

“Pythagoras? No good,” said the Mathematical Ninja.

4. $\ln(2) \simeq 0.693147$…

The Mathematical Ninja shrugged. “This one just comes up a lot. And it’s easy to do sums with.”

“It’s 0.7, less 1%,” said the student. “Very close to $\frac{1}{2}\sqrt{2}$.”

“Straying into the cultural there, my friend,” said the Mathematical Ninja, perhaps projecting a little. “But correct.”

3. Graham’s number

“Graham’s number holds the world record for the largest finite number ever used in a proof,” said the Mathematical Ninja. “It’s too large to be written using normal mathematical notation, and it’s an upper bound on a constant in Ramsay Theory that’s believed to be about 13.”

“It must be pretty close to infinity,” said the student, unthinkingly.

“Infinity looks at Graham’s number and says ‘molest me not with this pocket calculator stuff’,” said the Mathematical Ninja.

The student nodded, chastised. “It’s turtles all the way up. So that Graham, pretty smart guy, huh?”

“Graham? No good,” said the Mathematical Ninja. “Pythagoras? No good. Pascal? No good. Abel? Singmaster? I spit me of them. No mathematicians anywhere any good except me. Paul Erdos and Colin Beveridge not bad. Not good, but not bad. ((Total rip-off of the The best joke in all of P.G. Wodehouse. But I’ve acknowledged it now.))”

2. $e^{-1} \simeq 0.367879…$

“Beveridge’s first law of MathsJam probability states ‘37%’ is normally a good guess.”

“So that Beveridge… oh, you said he wasn’t bad.”

The Mathematical Ninja nodded. “37% is usually a good guess because MathsJam probabilities often work out to $\frac{1}{e}$. That’s why it’s the second-coolest number of them all.”

1. $\tau \simeq 6.2831852…$

“It’s the angle in the middle of a circle,” said the Mathematical Ninja. “You may know it as $2\pi$, but $\tau$ is better yet.”

“I thought it was 360!” said the student.

“Get out of my shop,” said the Mathematical Ninja.