Binary Numbers

Our familiar decimal number system is based on powers of 10. The number 123 is actually 100 + 20 + 3 or 1 x 10<sup>2</sup> + 2 x 10<sup>1</sup> + 3 x 10<sup>0</sup>.

The binary number system is based on powers of 2. The number 100101<sub>2</sub> (that is, ``100101 base two'') is 1 x 2<sup>5</sup> + 0 x 2<sup>4</sup> + 0 x 2<sup>3</sup> + 1 x 2<sup>2</sup> + 0 x 2<sup>1</sup> + 1 x 2<sup>0</sup> or 32 + 4 + 1 or 37.

We usually speak of the individual numerals in a decimal number as digits, while the ``digits'' of a binary number are usually called ``bits.''

Besides decimal and binary, we also occasionally speak of octal (base 8) and hexadecimal (base 16) numbers. These work similarly: The number 45<sub>8</sub> is 4 x 8<sup>1</sup> + 5 x 8<sup>0</sup> or 32 + 5 or 37. The number 25<sub>16</sub> is 2 x 16<sup>1</sup> + 5 x 16<sup>0</sup> or 32 + 5 or 37. (So 37<sub>10</sub>, 100101<sub>2</sub>, 45<sub>8</sub>, and 25<sub>16</sub> are all the same number.)

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