Number Representations

Let $\tilde{a}$ be an approximation to $a$, then the absolute error is given by $\left| {\tilde{a}-a} \right|$.

If ${\left| a \ne 0\right|}$, the relative error is given by $\left| \dfrac{\tilde{a}-a}{a} \right|$ and the error bound is the magnitude of the admissible error.

If $y(x)$ is a smooth function, i.e. is differentiable, then $\left\| y^{\prime} \right\|$ can be interpreted as the linear sensitivity of $y(x)$ to uncertainties in $x$.

For functions of several variables, i.e. $f : \mathbb{R}^n \rightarrow \mathbb{R}$, then

$$ \left\| \Delta y \right\| \le \sum\limits_{i=1}^{n}\left\| \dfrac{\partial y}{\partial x_i} \right\|\left\| \Delta x_i \right\| $$

where $\left\| \Delta x_i \right\| = \tilde{x}_i - x_i$ for an approximation $\tilde{x}_i$, thus ${\left\| \Delta y_i \right\| = \tilde{y}_i - y_i = f \left( \tilde{x}_i \right) - \left( x_i \right)}$.

Let $b \in \mathbb{N} \backslash \lbrace 1 \rbrace$. Every integer $x \in \mathbb{N}_0$ can be written as a unique expansion with respect to base $b$ as

$$ \begin{equation*} \left(x\right)_b = a_0 b^0 + a_1 b^1 + \ldots + a_n b^n = \sum \limits_{i=0}^{n} a_i b^i \end{equation*} $$

A number can be written in a nested form:

$$ \begin{align*} \left(x\right)\_b & = a_0 b^0 + a_1 b^1 + \ldots + a_n b^n \\\ & = a_0 + b\left( a_1 + b\left( a_2 + b\left( a_3 + \ldots + b a_n\right) \right. \ldots \right) \end{align*} $$

with $a_i \in \mathbb{N}_0$ and $a_i < b$, i.e. ${a_i \in \lbrace 0, \ldots, b-1 \rbrace }$.

For a real number, ${x \in \mathbb{R}}$, write

$$ \begin{align*} x & = \sum \limits_{i=0}^{n} a_i b^i + \sum \limits_{i=1}^{\infty} \alpha_i b^{-i} \\\ & = a_n \ldots a_0 \cdot \alpha_1 \alpha_2 \ldots \end{align*} $$

A number may be represented as

$$ x = \sigma \times f \times b^{t-p} $$

where $\sigma$ is the sign of the number, i.e. $\pm 1$, $f$ is the mantissa, $b$ is the base, $t$ is the shifted exponent (where $p$ is the shift required to determine the actual exponent.)

Normalized floating point representations with respect to some base $b$, may store a number $x$ as

$$x= \pm 0 \cdot a_1 \ldots a_k \times b^n$$

where the $a_i \in \lbrace 0, 1, \ldots b-1 \rbrace$ are called the digits, $k$ is the precision and $n$ is the exponent. The set $a_1, \ldots, a_k$ is called the mantissa. Impose that $a_1 \ne 0$, it makes the representation unique.

Alternative conventions may express $x= \pm a_1 \cdot a_2 \ldots a_{k-1} \times b^n$, with $a_1 \ne 0$. Normalization refers to specifying the digits before the decimal place.