Dear Uncle Colin,

How would you work out $\br{\frac{4}{5}}^{10}$ in your head?

- Probability Estimation Needed, Relic Of Slide-rule Era

Hi, PENROSE, and thanks for your message!

That’s really two questions: how *I* would do it, and how *the Mathematical Ninja* would do it.

## Me

Me, I’d probably use the fact that $\br{1 - \frac{1}{n}}^n$ goes to $\frac{1}{e}$ as $n$ gets large. What we have is $\br{1 - \frac{1}{5}}^{10}$, which should be approximately $e^{-2}$, which is about 0.135.

## The Mathematical Ninja

“Ha! I don’t believe I have to tell you this, but *five is not large*.”

“Depends on what you compare it with, sensei.”

“You can do much better by saying $\ln\br{\frac{4}{5}} \approx -\frac{2}{9}$.”

“Why is that?”

A sigh. “$\ln(1+x) \approx x + \frac{1}{2}x^2$. $\ln(1-x) \approx -x+\frac{1}{2}x^2$. So $\ln\br{\frac{1-x}{1+x}} \approx -2x$, give or take a cubic term.”

“… and the ninth?”

“That’s not a ninth. *That’s* a ninth. Oh, er… yes, if you solve $\frac{1-x}{1+x} = \frac{4}{5}$, you find $x = \frac{1}{9}$ (shortcut: it’s the difference, 5-4 over the sum, 5+4).”

“OK, so I buy that $\ln\br{\frac{4}{5}} \approx -\frac{2}{9}$.”

“Then the logarithm of the thing we want is roughly $-\frac{20}{9}$, so we want $e^{-\frac{20}{9}}$.”

“I suppose you happen to know what that is?”

“Well, $e^{2.20} \approx 9$, so $e^{-2.222…}$ will be about 2% less than a ninth. A smidge less than 0.11 is about as close as I’d go.”

“0.1073, says the calculator I’m totally not looking at right now.”

Hope that helps!

- Uncle Colin (and The Mathematical Ninja)

## Colin

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.

## CyberChalky

I would have gone with some simple factor shifts;

4/5 = 8/10.

Having a 10 in the denominator makes for simple place value manipulations

8/10 = 2^3/10

changing the 8 to a power of 2 makes it easy with binary calculations

thus, 2^30/10^10 = (2^10)^3 / 10^10,

which leads quickly to a rough estimation of 1/10. More precision available easily via doublings.

## Colin

I put this to the Ninja, who furrowed an eyebrow and said “I was wondering who’d be the first to spot that.”