Floating Point Representation of Numbers
In the decimal number system, very large and very small numbers are expressed in scientific notation as follows: 4.69x 1023 and 1.601x10-19. Binary numbers can also be expressed by the floating point representation. The floating point representation of a number consists of two parts: the first part represents a signed, fixed point number called the mantissa (m); the second part designates the position of the decimal (or binary) point and is called the exponent (e). The fixed point mantissa may be a fraction or an integer. The number of bits required to express the exponent and mantissa is determined by the accuracy desired from the computing system as well as its capability to handle such numbers. For example, the decimal number +6131.789 is represented in floating point as follows:
Sign sign
0 0.6132789 0 04
mantissa exponent
The mantissa has a 0 in the leftmost position to denote a plus. Here, the mantissa is considered to be a fixed point fraction. This representation is equivalent to the number expressed as a fraction 10 times by an exponent that is 0.6132789 x 10+04. Because of this analogy, the mantissa is sometimes called the fraction part.
Consider, for example, a computer that assumes integer representation for the mantissa and radix 8 for the numbers. The octal number +36.754 =36754x 8-3 in its floating point representation will look like this:
Sign sign
0 36754 1 03
mantissa exponent
When this number is represented in a register in its binary-coded form, the actual value of the register becomes 0 011 110 111 101 100 and 1 000 011.
Most computers and all electronic calculators have a built-in capacity to perform floating-point arithmetic operations.