# Counting Integers in a Range

Finding how many numbers there are between two numbers might seem like a simple task, but there can be some complications: do you include the endpoints? What if you only have to include one endpoint? Some care is required to find the number of integers in an arbitrary range.

#### Contents

## Counting Integers in a Range

How many integers are there from 1 to 10? That's easy, it's 10.

How do you get the answer 10?

First of all, "1 to 10" represents the closed interval $[1, 10]$; to get 10 we simply subtract the left endpoint from the right endpoint and then add 1, i.e. $10-1+1=10$.

Generally,

In a closed interval $[a, b]$, the number of integers is $b-a+1$.

How about the number of integers from 1 to 10, excluding 1 and 10?

This is an open interval $(1, 10)$. Since we already know that there are 10 integers in the range $[1, 10]$, we simply exclude the two endpoints 1 and 10 by subtracting 2. Hence there are $10-1+1-2=8$ integers in $(1, 10)$.

Generally,

In an open interval $(a, b)$, the number of integers is $b-a-1$.

As for the last cookie, how many integers are there in a half-open interval $[a, b)$ or $(a, b]$?

Since only 1 of the endpoints is excluded, the number of integers is $b-a+1-1=b-a$.

In a half-open interval $[a, b)$ or $(a, b]$, the number of integers is $b-a$.

## Examples

How many integers are there in the open interval $(-10, 5)?$

The number of integers is $5-(-10)-1=14.$ $_\square$

How many integers are there in the closed interval $[-1, 99]?$

The number of integers is $99-(-1)+1=101.$ $_\square$

What is the total number integers in the intervals $(-15, -4]$ and $[0, 22)?$

The total number of integers is $\big(-4-(-15)\big)+(22-0)=11+22=33.$ $_\square$

What is the total number integers in the intervals

$(-4, -3), \ (-2, -1), \ (0, 1)?$

The total number of integers is

$\begin{aligned} \big(-3-(-4)-1\big)+\big(-1-(-2)-1\big)+\big(1-(0)-1\big) &=0+0+0\\ &=0.\ _\square \end{aligned}$

Which of the following intervals has the most number of integers:

$\begin{array}{c}& (-23, -7), & (-12, 3], & [0, 15), & [99, 114]? \end{array}$

The number of integers in the interval $(-23, -7)$ is $-7-(-23)-1=15.$

The number of integers in the interval $(-12, 3]$ is $3-(-12)=15.$

The number of integers in the interval $[0, 15)$ is $15-0=15.$

The number of integers in the interval $[99, 114]$ is $114-99+1=16.$Hence, the interval $[99, 114]$ has the most number of integers. $_\square$

Both $A$ and $B$ are integers such that $A<B$ and the number of integers in the interval $(A, B)$ is $99.$ What is $A-B?$

The number of integers in the interval $(A, B)$ is $B-A-1.$ Since this is equal to $99,$ we have

$B-A-1=99 \implies A-B=-1-99=-100.\ _\square$

**Cite as:**Counting Integers in a Range.

*Brilliant.org*. Retrieved from https://brilliant.org/wiki/counting-integers-in-a-range/