Counting Numbers in a Range
Asked how many numbers run from 7 to 15, most people subtract and answer 8. The true answer is 9, because subtraction counts the gaps between numbers rather than the numbers themselves.
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Asked how many numbers run from to
, most people subtract and answer
. The true answer is
. Subtraction counts the gaps between numbers, not the numbers themselves — and that single mismatch is behind a whole family of test questions.
Theorem
The two rules
From to
, including both ends:
. Strictly between
and
, excluding both:
.
The wording decides which rule applies. "From … to …" and "inclusive" keep the endpoints; "between" throws both away. The two answers always differ by exactly .
Concept
Why you add one
Count the numbers from 
to 
. Subtracting gives 
, but listing them gives 
— five numbers.
The subtraction measured the four gaps. The first number sits before any gap begins, so it never got counted. Add to include it.
This is the fence-post idea: a fence with five posts has only four spaces between them. Posts always outnumber gaps by one.
Example
Including the endpoints
How many whole numbers are there from to
, including both?
Apply the inclusive rule: 
Subtract first: 
Add the endpoint back: 
Check by listing: . Nine of them, as the rule promised.
Example
Excluding the endpoints
How many whole numbers lie strictly between and
?
Apply the exclusive rule: 
Subtract first:
Remove one more: 
A quick cross-check: the inclusive count was , and dropping the two endpoints leaves
. The two rules agree, as they must.
Note
Read the wording first
| Wording | Endpoints | Formula |
|---|---|---|
| "from |
Included | |
| "inclusive" | Included | |
| "between |
Excluded |
Summary
- Subtraction alone counts gaps, not numbers.
- Including both endpoints, the count is
.
- Excluding both endpoints, the count is
.
- The two answers always differ by
— one endpoint at each end.
- "From … to" and "inclusive" keep the endpoints; "between" drops them.
- Fence posts always outnumber the gaps between them by one.