0.7 doesn’t sound like a magical number — at best, it’s a relatively obscure decimal. It’s in a fairly comfortable ‘higher than average’ zone, I suppose, if you’re looking at probabilities, but… well, it’s one of the Mathematical Ninja’s favourite numbers.

It comes up in two major places: $\ln(2) \simeq 0.693$ and $\sqrt{\frac{1}{2}} \simeq 0.707$. Rather neatly, $\ln(2)$ is about 1% lower than 0.7 and $\sqrt{\frac{1}{2}}$ is 1% higher.

These are numbers that come up all the time — $\ln(2)$ very frequently when you’re dealing with half-lives and doubling times, and $\sqrt{\frac{1}{2}}$ in trig questions (it’s $\sin(\frac{\pi}{4})$, for example).

Knowing that 0.7 is very close to $\sqrt{\frac{1}{2}}$ is a godsend: if you need to multiply one of the key values by a multiple of 7, you can simply say “oo! Each of those is about 5 (give or take 2%)” and go from there.

The really sharp ninjas are the ones who do the adjustment carefully. $0.7 \times 7$ is 2% short of 5, which means $7 \ln(2)$ is 3% short, and $7\sqrt{\frac{1}{2}}$ is just 1% short.

Putting it together, if you somehow end up with $21 \ln(2)$, you can quickly say “that’s about 15 — actually just a smidge less, call it 14.55.” You would then thump the table in disgust as it works out to 14.556 and rounds up — a moral victory at the very worst.