Representing Numbers

Numbers are an abstract mathematical concept. Numbers themselves can be represented in many different forms. Some common ways are:

• Denary (the number system we use day to day)
• Binary
• Roman Numerals

Numbers can be represented physically using machinery. For example, the number of teeth on cogs in the old mechanical computers can represent different numbers and calculations can subsequently be carried out on them. An abacus is another form of physical representation.

Numbers can also be represented electronically by using any of the following:

• Electric current - current / no current
• Voltage - positive / negative
• Voltage - high / low
• Pitch (or frequency) - high / low
• Varying voltage - 1V, 2V, 3V, 4V etc...

Using a binary system with wires between circuits, each wire can be set to 1 of 2 states. The combination of multiple wires enables the representation of larger numbers that just 0 or 1 for example.

To store data in memory, the storage component needs signals for the actual data itself which is to be stored, where to store it (its address or location), and whether the data is being written or retrieved.

The storage speed of a storage component depends on its word size and how many operations can be carried out per second. The word size means the number of bit combinations that can be written at one time (i.e. the more wires connected to the memory component, the higher the word size).

For example, if 64 bits could be written at one time, and 400,000,000 processes were carried out per second the data transfer rate (in GB/sec) would be:

64 / 8 (8 bits on a byte)
8 x 400,000,000 = 3.2 GB/s

Representing Numbers in Binary

• Positive / negative numbers
• Fractions
• Currency
• Very large numbers
• Very small numbers

Twos Complement is used to represent negative numbers. Each bit of the number is inverted (i.e. each 0 changes to 1, and each 1 changes to 0) and then one is added (so that the bit to the far right is changed). This representation technique simplifies the processes of addition and subtraction.

Floating point numbers can be used to represent very large or small numbers. They consist of a mantissa, an exponent, a base number (which stays fixed).

For example, the number 456,450,980,917 can be represented by 4.5645 x 10¹¹ where 4.5645 (mantissa) and 11 (exponent) are the only two numbers stored in the floating point number. As already mentioned, the base number (in this case 10) is assumed to be the same for each number stored. If very small numbers are to be stored, the exponent will be stored as a negative number.

Try a Computer Systems Quiz in the Computing Students Computing Quizzes Section to test your knowledge.

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