NUMBER LOGIC / JD · 2026.10.19
Binary Place Value Without a Calculator
How powers of two form the base-two representation of an integer.
Only two digits are allowed
Binary numerals use 0 and 1. Each place is a power of 2: 1, 2, 4, 8, 16, 32 and so on. The decimal number 13 is 1101₂ because 8 + 4 + 1 = 13.
Converting decimal to binary
For 26, find the largest fitting power of two, 16. Subtract it, leaving 10; take 8, leaving 2; then take 2. Place 1s in the 16, 8 and 2 positions to get 11010₂.
Why leading zeros are optional
001101₂ and 1101₂ have the same integer value. Leading zeroes can be useful for storing data in a fixed width, such as 8 bits, but do not change the numerical value.
Test the method
Translate 7, 12 and 31 by hand and compare them with the number dictionary. Then reverse the process by adding powers of two for 100101₂.
The calculations are mathematical facts. Interpretations attached to popular number symbolism are a separate, non-scientific subject.