Floating Point Representation

Fixed-point representation breaks down whenever a problem involves both very large and very small quantities simultaneously. Floating-point solves this by separating a number into a significant digits part and a scale part, much like scientific notation.

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The Floating-Point System F

The complete set of floating-point numbers for a given set of parameters is:

$F = \left\{\; \pm\,(0.d_1\,d_2\,\ldots\,d_n)_\beta \;\times\; \beta^{\,e} \;\right\}$

Components

Symbol	Name	Description
$\pm$	Sign	Positive or negative
$0.d_1 d_2 \ldots d_n$	Mantissa	The fractional part carries all significant digits
$\beta$	Base (radix)	The number system base (2 for computers, 10 for decimal)
$e$	Exponent	An integer that scales the mantissa by a power of $\beta$

Constraints

All parameters are integers ($\beta,\, d_i,\, e \in \mathbb{Z}$), subject to:

$0 \;\le\; d_i \;\le\; \beta - 1 \qquad \text{and} \qquad e_{\min} \;\le\; e \;\le\; e_{\max}$

Note

The mantissa is always written with an implicit leading 0., so $d_1$ is the first significant digit after the radix point. The exponent $e$ then shifts the effective magnitude of the number.

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Converting a Number to F-Format

The goal is to rewrite a number so that its integer part becomes 0 and the significant digits appear after the radix point, while $e$ accounts for the shift.

Example 1: Base 10

Convert $123.45$ to floating-point:

$$123.45 \;=\; 12.345 \times 10^1 \;=\; 1.2345 \times 10^2 \;=\; \mathbf{(0.12345)_{10} \times 10^3}$$

So $d_1=1,\; d_2=2,\; d_3=3,\; d_4=4,\; d_5=5$ and $e=3$.

Example 2: Base 2

Convert $(1001.11)_2$ to floating-point:

Shift the radix point 4 places to the left (each left-shift multiplies $e$ by one factor of 2):

$1001.11_2 = \mathbf{(0.100111)_2 \times 2^4}$

So $d_1=1,\; d_2=0,\; d_3=0,\; d_4=1,\; d_5=1,\; d_6=1$ and $e=4$.

Note

Shifting the radix point left by $k$ positions increases $e$ by $k$. Shifting right by $k$ positions decreases $e$ by $k$.

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Precision vs. Range

Every floating-point system makes two independent choices:

Parameter	Controls	Effect of increasing
$n$ (mantissa digits)	Precision	Finer granularity: smaller gaps between adjacent numbers
$e_{\max} - e_{\min}$ (exponent span)	Range	Wider range of representable magnitudes

In hardware, both are fixed by the standard. The most important standard is IEEE 754, covered in Section 5.

Caution

Key Takeaway: A floating-point number is a scaled fraction: $\text{mantissa} \times \beta^{\text{exponent}}$. Because both the number of mantissa digits and the exponent range are bounded integers, only a finite, discrete set of real numbers can be represented exactly. Everything else must be rounded, which is the source of all floating-point error.

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