## Sunday, February 2, 2014

### 16: Exponents and powers of ten.

Following on an earlier question from Day 7 ...

If I were to tell you that \$2^{100} - 100^2 = 1,267,650,600,228,229,401,496,703,195,376\$,
can you tell me what would change if I hadn't subtracted \$100^2\$ ?

What is \$2^{100}\$?

How do you know?

Just for the record, what -illion is that?

1. Since 2^10 = 1024 > 10^3, it follows that 2^100 = (2^10)^10 is GREATER than (10^3)^10 = 10^30.

I'll now show that that 2^100 is LESS than 3 x 10^30.

Note that 2^100 = (1024)^10 = (1000 x 1024/1000)^10 = (1000)^10 x (1024/1000)^10 = 10^30 x (1024/1000)^10.

Therefore, 2^100 is (1024/1000)^10 times 10^30, and thus the problem comes down to estimating how big (1024/1000)^10 is.

To do this, I'll try to get something of the form (1 + 1/n)^n to show up (the same method I used in a comment in your Day 7B post) and then use the fact that (1 + 1/n)^n is less than e = 2.71828... [Recall from precalculus math that compounding n times is less than continuous compounding.]

1024/1000 equals (1000 + 24)/1000 equals 1 + 24/1000, which is LESS than 1 + 100/1000 = 1 + 1/10. Therefore, (1024/1000)^10 is LESS than (1 + 1/10)^10, which in turn is less than e = 2.71828... Hence, 2^100 is LESS than 10^30 times 2.71828..., which is LESS than 3 x 10^30.

Thus, we have found that 2^100 is between 1 x 10^30 (1 nonillion) and 3 x 10^30 (3 nonillion).

2. 2^100 = 1,267,650,600,228,229,401,496,703,185,376
100^2 is 10,000 so only the thousands changed from to 195 to 185.