Floating Point/Floating-Point Numbers

This page is going to introduce floating point numbers, and explain some key terminology.

Terminology

base

The base of a floating-point number is the number to which the exponent is applied. In our decimal counting system, the base is usually 10. In a binary system, however, the base is usually 2. In the following number: $1.2345\times 10^{6}$, 10 is the base.

precision

The precision of a number is the amount of places available after the decimal point. In the scientific-notation number: $1.2345\times 10^{6}$, the precision is 4 because there are 4 digits after the decimal point.

significand

The significand is the value of the non-exponent part of the number. In the number $1.2345\times 10^{6}$, 1.2345 is the significand. This is also called the mantissa.

exponent

The exponent is the number to which the base is raised. In the number $1.2345\times 10^{6}$, 6 is the exponent.

$m\times b^{e}$

Where: m is the significand b is the base e is the exponent

Biasing

In computer science, a number is called biased, if it has associated with it a certain additive offset value. For instance, if we have a 3 bit number (from 0 to 111₂, which is 7 in decimal), we can define a bias of 4, so that every number value has 4 subtracted from it to provide the final value. Here are some examples:

unbiased number	biased number
0	0 - 4 = -4
7	7 -4 = 3

This means that our number, now that it has been biased can go from -4 to 3. Biasing is only one of many methods for allowing binary values to express negative values.

Variables

There are a number of variables that we will use throughout this book, and we are going to explain a few of them here:

b

b is the base number.

e

We will use the letter e to represent the number of bits in an exponent. This means that the exponent can take all integer values from 0 to $b^{e}-1$.

p

This is the precision value of the significand.