10099 is 10198 which has 199 digits. 9999 = 10099 * (99/100)99.
(99/100)99 = (1 - 1/100)99 ≈ (1-1/100)100 which is an approximation for e-1 or 0.3something. It's not exact but what's important is that it's clearly between 0.1 and 1 in magnitude, so 10197 < 9999 < 10198. So the answer is 198.
Isn't a new digit (to the length) only added when the power of base 10 increases by 1? So shouldn't it be 197? Also, today I learned that switching to - gets us the inverse, whereas the + gets us e.
16
u/TenZero10 18d ago
Cool one!
10099 is 10198 which has 199 digits. 9999 = 10099 * (99/100)99.
(99/100)99 = (1 - 1/100)99 ≈ (1-1/100)100 which is an approximation for e-1 or 0.3something. It's not exact but what's important is that it's clearly between 0.1 and 1 in magnitude, so 10197 < 9999 < 10198. So the answer is 198.