Algebra Know-It-ALL

The binary system

When engineers began to design electronic calculators and computers in the twentieth

century, they wanted a way to count up to large numbers using only two digits, one to

represent the “off ” condition of an electrical switch and the other to represent the “on”

condition. These two states can also be represented as “false/true,” “no/yes,” “low/high,”

“negative/positive,” or as the numerals 0 and 1. The result is a base-two or binary numera-

tion system.

Figure 1-6 shows how numerals in the binary system are put together. Instead of going

up by multiples of ten, eight, or sixteen, you double the value of each digit as you move one

place to the left. Numerals in the binary system are longer than numerals in the other systems,

but binary numerals can be easily represented by the states of simple, high-speed electronic

switches.

Every binary numeral has a unique equivalent in the decimal system, and vice versa.

When you use a computer or calculator and punch in a series of decimal digits, the machine

converts it into a binary numeral, performs whatever calculations or operations you demand,

14 Counting Methods

0 0 0 0 110 110

No higher-valued digits

1 “one”

1 “two”

No “fours”

1 “eight”

No “sixteens”

1 “thirty-two”

Multiple of five hundred twelveMultiple of two hundred fifty-sixMultiple of one hundred twenty-eightMultiple of sixty-fourMultiple of thirty-twoMultiple of sixteenMultiple of eightMultiple of fourMultiple of twoMultiple of one

Figure 1- 6 In the binary system, large numbers are represented by

building up numerals digit-by-digit from right to left, giving

each succeeding digit twice the value of the digit to its right.

Note the absence of commas in this system.