# 34 Rounding

## 34.1 Floor (Next Lowest Integer)

Rounding to the next lower integer, denoted $$\lfloor x \rfloor$$, is defined as

$\lfloor x \rfloor = max\{m \in \mathbb{Z} | m \leq x\}$

## 34.2 Ceiling

Rounding to the next larger integer, denoted $$\lceil x \rceil$$, is defined as

$\lceil x \rceil = min\{m \in \mathbb{Z} | m \geq z\}$

## 34.3 Nearest Integer

Rounding to the nearest integer is a common operation without a notation with well established concensus. It may be represented as $$\lbrack x \rbrack$$, $$\lfloor x \rceil$$, $$||x||$$, $$nint(x)$$ or $$round(x)$$. Here, we will use the $$nint(x)$$ notation so that we may extend the $$round(x)$$ notation beyond just rounding to the nearest integer..

A mathematical representation of rounding would be really nice to have right here

## 34.4 Nearest Multiple

Rounding to a nearest multiple may be obtained through a rescaling of the value $$x$$ into a integer scale based on multiples of $$m$$. One available notation for this operation is $$mround(x, m)$$.

$mround(x, m) = nint \Big( \frac{x}{m} \Big) \cdot m$

## 34.5 Nearest Place (Base 10)

It is common in mathematical operations to round a value not to an integer, but to a decimal place. This is no different than rounding to a multiple of ten. We define the following conventions.

• Let $$x \in \mathbb{R}$$
• Let $$p \in \mathbb{Z}$$ where $$p$$ represents the negative power of ten of the desired precision of the result.

Thus, when $$p = 0$$, we with to round to the $$10 ^ {-0} = 10^0 = 1$$, or ones/integer position. When $$p = 1$$, we round to the $$10 ^ {-1} = \frac{1 / 10} = 0.01$$, or tenths position. And when $$p = -1$$, we round to the $$10 ^ 1 = 10$$, or tens position.

We define the operation $$round(x, p)$$ to be the operation of rounding to the nearest decimal place.

\begin{aligned} round(x, p) &= mround(x, 10 ^ {-p}) \\ &= round \Big( \frac{x}{10 ^ {-p}}, 0 \Big) \cdot 10 ^ {-p} \\ &= \frac{round(x \cdot 10 ^ p, 0)}{10 ^ p} \end{aligned}

Under this definition, $$nint(x)$$ is a special case of $$round$$ where $$nint(x) = round(x, 0)$$.

## 34.7 References

(I really need to be better than this)