Number Representation Systems

Binary

Most switching devices, such as a light switch, only have two possible states - on and off. The same is true with the millions of transistorised switches inside a computer. These two possible states are represented mathematically as 1 or 0. The number system based upon these two states is known as BINARY (or base 2).

With any number system we need a “place” value. Decimal numbers increase from right to left from “units” to “tens” to “hundreds”, etc. Each is an increasing power of 10, ie 10¹ then 10², then 10³ . The same applies with binary numbers but in this case we are dealing with powers of 2.

This system will therefore have a range in binary from 00000000 to 11111111 that would be 0 to 255 in decimal. Each of the Binary digITs (a 1 or a 0) is given the name BIT. Eight bits are normally referred to as a BYTE (but it could be the number of bits required to represent a character. If a computer system accesses data in larger chunks it is termed a WORD. 1024 bytes is a kilobyte.

If we wish to represent more than 255 (in decimal) more bits and bytes will be required. For instance, 16 bits could represent the numbers up to 65535. If a computer stores data in 16 bit “chunks” then it is said to have a “word-length” of 16 bits. Modern PC’s now work with a word-length of 32 bits and even 64 bits.

To convert from decimal to binary there are a number of methods eg repeated division by 2, with the remainder being the binary digit.

example

Convert 123 to binary

123/2 = 61 remainder 1

61/2 = 30 remainder 1

30/2 = 15 remainder 0

15/2 = 7 remainder 1

7/2 = 3 remainder 1

3/2 = 1 remainder 1

1/2 = 0 remainder 1

So, reading from bottom to top gives:

1 1 1 1 0 1 1

Normally this would be converted to 8 bits by adding a leading zero. ie

0 1 1 1 1 0 1 1

An alternative method would be to repeatedly subtract, if possible, 128, 64, 32, etc from the original number and then the numbers subtracted from the remainders. If subtraction can take place then the bit it set to 1, if not it is set to 0.

So far we have only dealt with "pure" binary numbers - we have treated them as a number without any sign.  The alternative is to use something called Sign and Magnitude.  In this system we have the most significant bit (or MSB) which is the left-hand one that tells us if the number is positive or negative.  If there is a zero in there it is positive, a one represents negative.  So

0 0 0 0 0 0 1 1   would mean +1

1 0 0 0 0 0 1 1  would mean -1