Floating Point System and Machine Epsilon in Numerical Algorithms
Definition
Floating Point System 1
A floating point system consists of numbers $x$ represented as follows. $$ x = m \cdot \beta^{e} , \quad m = \pm d_{0} . d_{1} d_{2} \ldots d_{t} $$ Here $m$ is called the mantissa and $e$ the exponent. For every $k = 0 , \cdots, t$, $d_{k}$ is a digit satisfying $0 \le d_{k} < \beta$ with respect to the base $\beta \in \mathbb{N}$. Usually the mantissa is normalized so that $1 \le |m| < \beta$ and the exponent so that $L \le e \le U$, and the floating point system is represented by the ordered tuple $F = \left( \beta , t, L, U \right)$.
Machine Epsilon 2
For a floating point system $\left( \beta , t, L, U \right)$, the unit roundoff $\epsilon$ is defined as follows. $$ \epsilon = {\frac{ 1 }{ 2 }} \beta^{1-t} $$
Explanation
Floating point is usually mentioned in numerical computations that use computers, and since as of 2026 ordinary computers use binary, the case $\beta = 2$ is the most common. In this case the machine epsilon becomes $\epsilon = 2^{-t}$, which can be thought of as roughly the smallest number in magnitude among the numbers representable in that floating point system. In a similar sense, $\epsilon$ is also defined as the smallest $\varepsilon > 0$ satisfying $1 +_{F} \varepsilon > 1$. Here $+_{F}$ denotes the addition operation in the floating point system $F$.