# How do you find a midpoint of two numbers ?

Mar 23, 2017

Find their average: take their sum, and divide it by 2.

#### Explanation:

The midpoint $m$ of two numbers $a$ and $b$ is, by definition, a number that is the same distance from $a$ as it is from $b$.

Using this, we can get a formula for the midpoint $m$ in terms of $a$ and $b$.

Picture $a$ and $b$ as two points on a number line, with $a$ to the left of $b$. Now, $m$ must be between them, so $m$ is greater than $a$ but less than $b$. So if the distance from $a$ to $m$ is the same as the distance from $m$ to $b$, we can set these two distances to be equal, like this:

$m - a = b - m$

This equation says: "The difference between $m$ and $a$ is the same as the difference between $b$ and $m$."

Now, all we do is rearrange this to equation to solve for $m$:

m-a color(green)(" "+ m)=b"          " (add $m$ to both sides)

$\text{ "2m" "=bcolor(green)(" "+a)" }$ (add $a$ to both sides)

$\text{ "m" "=(b+a)/color(red)2" }$ (divide both sides by 2)

And there it is—the formula for the midpoint! This is often called the average of the two numbers $a$ and $b$, and written as

$m = \frac{a + b}{2}$

## Example:

What is the midpoint of 3 and 5? Well, the answer is intuitively 4, but with the formula, we get:

$m = \frac{3 + 5}{2} = \frac{8}{2} = 4$

which matches our intuition.

Mar 23, 2017

The method Geoff K gave also works with a combination of negative and positive numbers

#### Explanation:

Suppose we have $- 3 \mathmr{and} + 3$ then the mid point is $\frac{- 3 + 3}{2} = 0$

$\text{ }$or if you prefer you can write this as $\frac{+ 3 - 3}{2} = 0$

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Suppose we have $- 3 \mathmr{and} + 5$ then the mid point is

$\frac{- 3 + 5}{2} = \frac{2}{2} = 1$