Binary Place Value: Why 101101 Equals 45

Read from the right

Binary is base 2, so each position is worth twice the position to its right. Starting at the rightmost digit, the values are 1, 2, 4, 8, 16, 32 and so on. A 1 includes the place value; a 0 does not.

For 101101, the six positions are 32, 16, 8, 4, 2 and 1. Keep 32, 8, 4 and 1. Their sum is 45. The binary string therefore represents decimal 45, not decimal one hundred one thousand one hundred one.

Convert 45 back to binary

Repeatedly divide by 2 and record the remainders: 45 gives quotient 22 remainder 1; 22 gives 11 remainder 0; 11 gives 5 remainder 1; 5 gives 2 remainder 1; 2 gives 1 remainder 0; 1 gives 0 remainder 1.

Read those remainders in reverse order to obtain 101101. Reading them in the order they were produced is a common mistake. The first remainder is the rightmost, lowest-value bit.

Leading zeros do not change the value

00101101 and 101101 represent the same positive integer. The leading zeros can be useful when a system stores a fixed number of bits, but they add no quantity. This is different from a code where the entire string may be an identifier.

Number JD displays positive integers without unnecessary leading zeros. Each page includes the active binary place values, giving an independent check on the conversion.

Check the mathematical definitions

For fractions, percentages and worked arithmetic, see OpenStax Prealgebra. For number definitions, see OpenStax on prime and composite numbers and Wolfram MathWorld on digital roots. Worked examples on this page are calculated for this guide.