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.

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 $b-a$.

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

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

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

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

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

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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

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

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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:

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

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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 $A-B?$

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

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