Normalised Form of Real Numbers

Topics: Numerical Analysis - Rounding Error

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To represent real numbers on a computer, the first step is to first obtain its normalised form:

$$\pm 0.d_1{d}_2{d}_3\dots d_k\times B^e$$

…where:

• $d_1$: a digit with a value from $1$ to $(B-1)$
• $d_i$: digits with values from $0$ to $(B-1)$
• $B$: numerical base ($2,8,16,10$)
• $k$: number of significative digits (precision)
• $e$: exponent (an integer)

These normalised numbers can then be used to obtain their floating point form.

The representation of $0$ is special.

Examples

Number	Normalised form
$15.812$	$+0.15812\times 10^2$
$-0.0527$	$-0.527\times 10^{-1}$
$0.00325$	$+0.325\times 10^{-2}$
$15635.00236$	$+0.1563500236\times 10^{-5}$