Arithmetic

Arithmetic studies elementary operations on numbers.

A real number is an element of the complete ordered field $\mathbb{R}$.

Every real number in $[0,1]$ is a sum ${\sum }_{j\ge 1}{\epsilon }_j{2}^{-j}$ with binary digits ${\epsilon }_j\in \{0,1\}$.

A dyadic rational has the form $m/2^k$. Its terminating binary expansion has an alternative expansion ending in infinitely many $1$ digits.

Addition combines two numbers or elements of an additive algebraic structure into their sum.

The additive inverse $-x$ satisfies $x+(-x)=0$.

A sum combines a finite or infinite sequence of terms by repeated addition. The symbol $\sum$ denotes summation over an index.

For positive weights $w_i$, the weighted mean isIt lies between the smallest and largest $x_i$; the same result holds for integrals with a positive weight function.

The prefix sums $P_m={\sum }_{i=1}^m{a}_i$ can all be computed by the recurrence $P_m=P_{m-1}+a_m$ in linear time. Any contiguous sum is then $P_b-P_{a-1}$, so it takes constant time after preprocessing.

Subtraction adds the additive inverse: $a-b=a+(-b)$.

Multiplication combines two factors into their product.

A multiplicative inverse of $x$ is an element $x^{-1}$ satisfying $x{x}^{-1}=x^{-1}x=1$.