The Mathematical Ninja and an Irrational Power

“The square root of two… I don’t even know how to say this. The square root of two to the square root of threeth power?”

“$\sqrt{2}^{\sqrt{3}}$?” said the Mathematical Ninja. “I wouldn’t bother saying it, I’d just write it down.”

“But what does it mean? I mean, I can just about get on board with fractional powers - I can see that half-powers have to lie between whole powers and that square roots make sense for them and so on - but I don’t see how that works for irrational powers.”

There’s a moment in The Thick Of It where Ollie mimes the Eastenders theme and for a moment it looks as though Malcolm is going to give him a proper chewing out, but instead he grins and says “See? Ollie gets it.” This scene went down similarly.

“It’s a limiting process. Since an irrational number is defined by sets of rational numbers either side of it, it can be shown that $\sqrt{2}^{\sqrt{3}}$ is between - for example - $\sqrt{2}^{\frac{433}{250}}$ and $\sqrt{2}^{\frac{433}{249}}$.”

“Those are, of course, the first numbers I would have picked, sensei.”

A single, infinitesimal movement of a single Ninja eyebrow communicated that the student would do well not to push his luck.

“And I suppose there’s a nice way to work it out without touching a…” whisper: “c-word?”

“It’s obviously the same as $2^{\frac{1}{2}\sqrt{3}}$,” said the Ninja, thoughtfully, “which is $e^{\frac{1}{2}\sqrt{3}\ln(2)}$.”

“Hard to argue with that.” Another eyebrow.

“Everyone knows that ${\sqrt{3}}$ is about $\frac{26}{30}$ and that $\ln(2)$ is one percent less than $\frac{7}{10}$. That makes the power $\frac{26 \times 7}{300}$ or $\frac{91}{150}$.”

“Or rather, one percent less.”

“I WAS GETTING TO THAT. (Ahem.) One percent less takes us somewhere in the vicinity of $\frac{90}{150}$, which leaves us with $e^{\frac{3}{5}}$.”

“That looks simpler. I can even have a guess at that - it’s 10% less than 2, or 1.8.”

“Not bad,” said the Ninja. “Not good, but not bad. In fact, we know that $\ln(1.8)$ is $2\ln(3) - \ln(5)$, or $2.20 - 1.61$, which is 0.59 or so.”

“So we need to adjust it.”

“We do - we’re about 1% too low. I would go to about 1.82, personally; exponentials are notoriously volatile.”

The student thought about making a comment, but thought better of it.


Colin is a Weymouth maths tutor, author of several Maths For Dummies books and A-level maths guides. He started Flying Colours Maths in 2008. He lives with an espresso pot and nothing to prove.


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