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Binary numbers, represented by 1s and 0s, are used in electrical and computer applications. They correspond to a power of 2 and can be easily represented with a two-position electrical switch. Binary arithmetic can be performed using simple rules, and logical operators can be interchanged with mathematical operations. A block of eight bits is called a byte, and a group of bits is called a word, with 16-, 32-, and 64-bit words being the most common.
The word “binary” refers to a system made up of two parts, such as a binary star. Binary numbers are no different from the numbers you are used to; they are simply represented differently, with only 1s and 0s. While binary numbers are used in numerous fields, they are most commonly used in electrical and computer applications.
The most common system for representing numbers is not the binary system; it is the decimal system. Also known as base 10, the decimal system using ten digits, from 0 to 9. Each position within a number corresponds to a power of 10. Therefore, the decimal number 546.23 equals:
(5 x 102) + (4 x 101) + (6 x 100) + (2 x 10-1) + (3 x 10-2)
However, there are many other number notation systems; the binary system, also known as base-2, is one. Binary numbers use only the digits 0 and 1. Each position in the number corresponds to a power of 2. Thus, the binary number 11100, for example, would be represented in the following decimal format:
(1 x 24) + (1 x 23) + (1 x 22) + (0 x 21) + (0 x 20) = 16 + 8 + 4 + 0 + 0 = 28
Clearly, the decimal system is a more compact notation system than the binary system. However, the binary system has some unique properties that make it very useful for certain operations, including those used by digital computers. Since each binary digit, or bit for short, has only two possible states, it is easily represented with a two-position electrical switch. The number “1” represents the switch on, or “yes”, while the number “0” represents the switch off, or “no”.
Binary arithmetic can be performed using a limited number of simple rules, allowing numbers to be computed using only a handful of electrical gates. For example, to multiply two digits together, you just need to remember the following:
0 x 0 = 0
0 x 1 = 0
1 x 0 = 0
1 x 1 = 1
It can also be seen that the two value system for representing binary numbers corresponds to the two truth values used in symbolic logic. Consider the following truth tables using the logical operator “AND:”
F E F = F
F E T = F
T E F = F
T E T = T
If you replace “F” with “0” and “T” with “1”, it becomes clear that the logical operator “AND” is equivalent to the multiplication sign in binary arithmetic. The other mathematical operations can also be interchanged with logical operations. Since logical operators are easy to represent in computer circuits, it becomes possible to build an electrical device capable of performing arithmetic operations. Doing math this way is known as “Boolean algebra” after its discoverer, 19th-century mathematician George Boole.
In computer memory, a block of eight bits is called a byte. One byte can represent the numbers from 00000000 to 11111111, i.e. from 0 to 255 in the decimal system. Different computing architectures can handle different numbers of bits in a single computation; such a group of bits is called a word. A word is often a multiple of eight bits, with 16-, 32-, and 64-bit words being the most common.
