Binary Numbers
Jeffrey J. McConnell, Ph.D.
Introduction
Numbers represent quantities and can be expressed in any base. We are familiar with decimal numbers, which
are expressed in base 10. Binary numbers are the cornerstone of computing and are expressed in base 2. This
document will take a quick look at decimal numbers and then look at the binary number system.
Decimal Numbers
The digits of a decimal number are between 0 and 9, which is one smaller than the decimal base of 10. The
digits of a decimal number are called, from right to left, the ones digit, tens digit, hundreds digit, thousands
digit, and so on. That is because the quantity the number represents is that digit multiplied by 1, 10, 100, 1000,
and so on. For example, the number 1547 is 7 * 1 + 4 * 10 + 5 * 100 + 1 * 1000. Each of these positions
represents higher and higher powers of ten starting with a power of zero. The ones digit is multiplied by 100,
which is 1. The tens digit is multiplied by 101, which is 10. The hundreds digit is multiplied by 102, which is
100. The thousands digit is multiplied by 103, which is 1000. In the case of decimal numbers, the quantity
represented by the number is the number itself.
Binary Numbers
The digits of a binary number are 0 and 1, which is one smaller than the binary base of 2. We don’t have names
for the digits of a binary number but the process for a binary number is the same as a decimal number. The
quantity that a binary number represents is the digits of the binary number multiplied from right to left by
increasing powers of 2. For example, the binary number 110100101 is 1 * 1 + 0 * 2 + 1 * 4 + 0 * 8 + 0 * 16 +
1 * 32 + 0 * 64 + 1 * 128 + 1 * 256.
Converting a Binary Number to a Decimal Number
One way to convert a binary number to a decimal number is to determine the quantity that the number
represents, which is the decimal equivalent. For the previous example, the binary number 110100101
represents the quantity 1 * 1 + 0 * 2 + 1 * 4 + 0 * 8 + 0 * 16 + 1 * 32 + 0 * 64 + 1 * 128 + 1 * 256, which is
421.
A second way works from left to right in the binary number. This process will double the previous total and
then add the next digit of the number. At the start, the previous total is always zero. The chart below shows
how this would work for the previous example.
Previous total * 2 + digit new total
0 * 2 + 1 1
1 * 2 + 1 3
3 * 2 + 0 6
6 * 2 + 1 13
13 * 2 + 0 26
26 * 2 + 0 52
52 * 2 + 1 105
105 * 2 + 0 210
210 * 2 + 1 421
These two processes will always give the same answer so one can be used to check the accuracy of the
other.
