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Floating Point Representation Concepts

This section introduces the terminology for describing floating point representations.

You are probably already familiar with most of these concepts in terms of scientific or exponential notation for floating point numbers. For example, the number 123456.0 could be expressed in exponential notation as 1.23456e+05, a shorthand notation indicating that the mantissa 1.23456 is multiplied by the base 10 raised to power 5.

More formally, the internal representation of a floating point number can be characterized in terms of the following parameters:

The mantissa of a floating point number actually represents an implicit fraction whose denominator is the base raised to the power of the precision. Since the largest representable mantissa is one less than this denominator, the value of the fraction is always strictly less than 1. The mathematical value of a floating point number is then the product of this fraction, the sign, and the base raised to the exponent.

We say that the floating point number is normalized if the fraction is at least 1/b, where b is the base. In other words, the mantissa would be too large to fit if it were multiplied by the base. Non-normalized numbers are sometimes called denormal; they contain less precision than the representation normally can hold.

If the number is not normalized, then you can subtract 1 from the exponent while multiplying the mantissa by the base, and get another floating point number with the same value. Normalization consists of doing this repeatedly until the number is normalized. Two distinct normalized floating point numbers cannot be equal in value.

(There is an exception to this rule: if the mantissa is zero, it is considered normalized. Another exception happens on certain machines where the exponent is as small as the representation can hold. Then it is impossible to subtract 1 from the exponent, so a number may be normalized even if its fraction is less than 1/b.)

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