Number Conventions

The F-format introduced in the previous section has a redundancy problem: the same value can be written many different ways. For example:

$0.12345 \times 10^3 \;=\; 0.012345 \times 10^4 \;=\; 0.0012345 \times 10^5$

All three are identical in value but occupy different slots in the discrete floating-point set, wasting representable values. A convention eliminates this redundancy by requiring a specific canonical form.

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Convention 1: Standard Form

Rule: The first mantissa digit $d_1$ is always 1.

$X = \pm\,(0.\underbrace{1}_{d_1}\,d_2\,d_3\,\ldots\,d_m)_\beta \;\times\; \beta^{\,e}$

Because $d_1 = 1$ is fixed, it carries no information and so it need not be stored. This is the idea behind the implicit leading bit used in IEEE 754.

Example: Binary, $\beta=2, m=3, e_{\min}=−1, e_{\max}=2$

With $d_1 = 1$ fixed, only $d_2$ and $d_3$ are free (2 bits → 4 choices):

Mantissa	Binary value	Decimal value
$0.100$	$2^{-1}$	$0.5$
$0.101$	$2^{-1}+2^{-3}$	$0.625$
$0.110$	$2^{-1}+2^{-2}$	$0.75$
$0.111$	$2^{-1}+2^{-2}+2^{-3}$	$0.875$

Largest number (largest mantissa × largest exponent):

With and without negative support, the largest number is:

$$X_{\max} = (0.111)_2 \times 2^2 = \frac{7}{8} \times 4 = \mathbf{3.5}$$

Smallest positive number:

Without negative support (smallest mantissa × smallest exponent):

$$X_{\min} = (0.100)_2 \times 2^{-1} = \frac{1}{2} \times \frac{1}{2} = \mathbf{0.25}$$

With negative support -(largest mantissa × smallest exponent):

$$X_{\min} = - (0.111)_2 \times 2^{2} = - \frac{7}{8} \times 4 = \mathbf{-3.5}$$

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Convention 2: Normalized Form

Rule: The first digit of the mantissa is always 1; and $d_1 = 1$ is shifted to the right by 1 position.

$X = \pm\,(0.1d_1\,d_2\,\ldots\,d_m)_\beta \;\times\; \beta^{\,e}$

Now all $m$ digits $d_1, \ldots, d_m$ are free, giving twice as many distinct mantissas as Standard Form.

Example: Binary, $\beta=2, m=3, e_{\min}=−1, e_{\max}=2$

$d_1, d_2, d_3$ are all free → $2^3 = 8$ mantissas, ranging from $0.1000$ to $0.1111$:

$0.1000 = 0.5 \qquad 0.1111 = 0.5 + \tfrac{1}{4} + \tfrac{1}{8} + \tfrac{1}{16} = \tfrac{15}{16}$

Largest number (largest mantissa × largest exponent):

With and without negative support, the largest number is:

$$X_{\max} = (0.1111)_2 \times 2^2 = \frac{15}{16} \times 4 = \mathbf{3.75}$$

Smallest positive number

Without negative support (smallest mantissa × smallest exponent):

$$X_{\min} = (0.1000)_2 \times 2^{-1} = \frac{1}{2} \times \frac{1}{2} = \mathbf{0.25}$$

With negative support -(largest mantissa × smallest exponent):

$$X_{\min} = -(0.1111)_2 \times 2^2 = -\frac{15}{16} \times 4 = \mathbf{-3.75}$$

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Convention 3: Denormalized Form

Rule: Instead of starting with $0.1$, the number starts with a whole 1 in front of the decimal point, followed by the $m$ free fractional digits.

$X = \pm\,(1.d_1\,d_2\,d_3\,\ldots\,d_m)_\beta \;\times\; \beta^{\,e}$

Example: Binary, $\beta=2, m=3, e_{\min}=−1, e_{\max}=2$

Just like Convention 2, all 3 bits ($d_1, d_2, d_3$) after the decimal are free to change. This gives us $2^3 = 8$ possible mantissas, ranging from $1.000$ to $1.111$.

$1.000 = 1.0 \qquad 1.111 = 1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} = \frac{15}{8} = 1.875$

Largest positive number: (largest mantissa × largest exponent)

$$X_{\max} = (1.111)_2 \times 2^2 = \frac{15}{8} \times 4 = \mathbf{7.5}$$

Smallest positive number

Without negative support (smallest mantissa × smallest exponent):

$$X_{\min} = (1.000)_2 \times 2^{-1} = 1.0 \times 0.5 = \mathbf{0.5}$$

With negative support -(largest mantissa × largest exponent):

$$X_{\min} = -(1.111)_2 \times 2^2 = -\frac{15}{8} \times 4 = \mathbf{-7.5}$$

Side-by-Side Comparison

Feature	Conv. 1 (Standard)	Conv. 2 (Normalized)	Conv. 3 (Denormalized)
Format	$0.1\,d_2\,\ldots\,d_m$	$0.1\,d_1\,d_2\,\ldots\,d_m$	$1.d_1\,d_2\,\ldots\,d_m$
Free bits	$m - 1$	$m$	$m$
Unique combinations (for $m=3$)	$2^2 = 4$	$2^3 = 8$	$2^3 = 8$
Max value ($X_{\max}$)	$3.5$	$3.75$	$7.5$
Min positive ($X_{\min}$)	$0.25$	$0.25$	$0.5$

Note: Convention 3 allows us to reach much larger maximum numbers because the whole number portion starts at 1 instead of 0.

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The Zero Problem

Danger

In these conventions, zero cannot be represented exactly. Every mantissa starts with 0.1... or 1.0..., so the smallest positive value is $\beta^{-1} \times \beta^{e_{\min}} > 0$. In practice, zero is either approximated by a very small number (e.g. $(0.100\ldots)_2 \times 2^{-50} \approx 0$) or handled as a special reserved bit pattern, which is exactly what IEEE 754 does. We will see it shortly.

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