ما معنى Floating-point number, machine epsilon, unit in the last place؟

A binary floating-point number is $\pm m \times 2^e$ with an integer significand $m$ of $p$ bits and an exponent $e$ in a fixed range; IEEE 754 double precision (binary64) has $p = 53$ and exponents from $-1022$ to $1023$. Machine epsilon is the gap between 1 and the next larger number, $2^{1-p} = 2^{-52} \approx 2.2 \times 10^{-16}$; the unit roundoff $u$ is half of it (some texts, and some libraries, call $u$ itself machine epsilon). A unit in the last place (ulp) of $x$ is the gap between the two floating-point numbers around $x$.