Powers of 2

This is a really quick method I’ve found for estimating what 2^n is.

As we can see on the left there is some kind of pattern as to when we move up an order of magnitude

  • If your n ends in a 0 a 4 or a 7 then you will be on the first entry for the next order of magnitude up
  • The next useful thing to see is that every time we add 10 we increase the order of magnitude by 1000
  • If the power is between the ends with a number between 0-4 then you want to start with 1 and double as necessary
  • If it ends with a number between 4-7 you want to start with 1.6
  • if it ends with a number between 7-0 you want to start with 1.3

Example #1: 2^32:

  • We can see n is in the 30’s so we know the number will be greater than 1 x10^9
  • Since it ends between 0-4 we start with 1 and our order of magnitude will be x10^9
  • Then we must double it 3 times
  • This would give us a guess of 4×10^9
  • Actual answer is 4.3×10^9

Example #2: 2^16

  • n is in the 10’s so our order of magnitude will be > x10^3
  • Since it ends between 4-7 we start with 1.6 and our order of magnitude will be x10^4
  • Then we must double it 3 times
  • This would give us a guess of 6.4×10^4
  • Actual answer is 6.55×10^4

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