Ask Uncle Colin: Two Numbers Close To 1

Dear Uncle Colin,

How do I tell which is larger, $2^{-{10}^{-20}}$ or $1-2^{-{10}^{20}}$?

- Unexpectedly Narrow Interval… Thank You!

Hi, UNITY, and thanks for your message!

As you’ve doubtless realised, both of those are “pretty much 1”. The question is, which is closer? As usual, there are several methods.

$N$th powers

The first way that jumped to mind was to let $N={10}^{20}$ to make the sums nicer, and then see what might spring out. We’re comparing $2^{-1/N}$ with $1-2^{-N}$.

If we take the $N$th power of the first of those, we get $\frac{1}{2}$.

If we take the $N$th power of the second, we get, binomially, $1-N\times 2^{-N}$ + small terms. Now, $N$ is much smaller than $2^N$, so this one is certainly bigger than a half.

That means $1-2^{-N}$ is the larger of the two values.

Logs

If we take logs of the first, we get $-\frac{1}{N}$.

If we take logs of the second, we get, Maclaurinally, $-2^{-N}$ + small terms.

Again, $\frac{1}{N}$ is much larger than $2^{-N}$, so the second one is closer to zero.

Hope that helps!

- Uncle Colin