Fixed Point Representation

Before exploring how modern computers store real numbers, we start with the simplest scheme: fixed-point representation. Here the position of the radix point (the generalisation of the “decimal point” to any base) is predetermined and never changes.

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Notation

A fixed-point number is written as:

$X = \pm\,(d_1\,d_2\,\cdots\,d_{k-1}\,.\,d_k\,\cdots\,d_n)_\beta$

Symbol	Meaning
$\beta$	Base (radix) of the number system
$d_1, d_2, \ldots, d_n$	Digits, each satisfying $0 \le d_i \le \beta - 1$
.	The radix point which separates the integer part (left) from the fractional part (right)
$\pm$	Sign of the number

Note

For base 2 (binary), every digit is either 0 or 1. For base 10 (decimal), digits range from 0 to 9. The radix point plays exactly the same role as the familiar decimal point, just generalised.

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How to Evaluate a Fixed-Point Number

Each digit is multiplied by the base raised to its position’s power. Positions to the left of the radix point have non-negative powers; positions to the right have negative powers.

$X = \pm\,\sum_{i} d_i \times \beta^{\,p_i}$

where $p_i$ is the power at position $i$.

Example: Binary to Decimal

Convert $(10.1)_2$ to base 10.

Position	$2^1$	$2^0$	.	$2^{-1}$
Digit	1	0	.	1
Value	2	0	—	$\frac{1}{2}$

$$1 \times 2^1 \;+\; 0 \times 2^0 \;+\; 1 \times 2^{-1} \;=\; 2 + 0 + \tfrac{1}{2} \;=\; \mathbf{2.5}$$

So $(10.1)_2 = (2.5)_{10}$.

Example: Signed Decimal

The number $-(12.3)_{10}$ evaluates as:

$$-\bigl(1 \times 10^1 + 2 \times 10^0 + 3 \times 10^{-1}\bigr) = -(10 + 2 + 0.3) = -12.3$$

In base 10 this is simply the familiar signed decimal $-12.3$.

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The Fundamental Limitation

Because the radix point is fixed, a single representation must commit to how many bits it allocates for the integer part versus the fractional part. This creates an unavoidable trade-off:

• A system with many integer digits can store large numbers, but cannot represent small fractions precisely.
• A system with many fractional digits provides fine resolution near zero, but overflows on large values.

Example: With 8 binary digits split 4 integer + 4 fractional (e.g. IIII.FFFF):

Quantity	Representation	Exact?
$15.9375$	1111.1111	✓
$0.0625$	0000.0001	✓
$100$	—	✗ overflow
$0.001$	—	✗ underflow

This inflexibility motivates a much more powerful scheme where the radix point is free to “float” to any position.

Caution

Key Takeaway: Fixed-point representation is simple and fast, but its range is severely limited by the fixed position of the radix point. Real-world numerical computation requires the more flexible floating-point format covered in the next section.

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