# What is a difference?

## Difference Definition

In math, a difference is the result of subtracting numbers from each other. It comes  from the Latin word "differentia" which means "carrying away" or "differing".

The first number in a difference is called the minuend and the second number is called the subtrahend.

A difference can be an unsimplified expression like the left side of this equation. Or it can be a simplified version like the right side of this equation.

A lot of elementary school teachers will say that a difference is "the answer to a subtraction problem" because the "answer" is a simplified version of the expression with the minuend and subtrahend.

## Examples with Numbers

### 12 - 5 = 7

In this example, 12 is the minuend and 5 is the subtrahend.

The simplified difference is 7

### 102 - 78 = 24

Here, 102 is the minuend and 78 is the subtrahend.

The simplified difference is 24

## Examples with Variables

### 7x - 5y = 12

In this example, 7x is the minuend and 5y is the subtrahend.

7x - 5y is the expression that represents the difference.

The difference can't be simplified because we don't know what x and y are, but 12 is equal to the difference so you could say "12 is the difference of 7x and 5y."

### 7a + 4b - 5c

In this expression, 7a 4b is the minuend and 5c is the subtrahend.

So, 7a + 4b - 5c is the difference of 7a + 4b (which is a sum) and 5c

## Differences vs. Sums

When we are working with variables, we often interpret subtraction as adding a negative.

We do this because we don't know whether the variables are positive or negative. If they are negative, then subtrahends could change into addends or vice versa.

So, the easiest way to describe the example above would be...

7a + 4b - 5c is the sum of the terms 7a, 4b, and -5c

It would be correct if you said...

7a + 4b - 5c is the difference of the minuend 7a + 4b and the subtrahend -5c, where the minuend is a sum with an augend of 7a and an addend of 4b.

But it is usually easier to interpret the subtraction as adding a negative and then call the whole thing a sum of terms instead of differentiating between augends, addends, minuends, and subtrahends.

## When will I use differences?

Differences are useful when you want to compare two numbers or values to find out how much they differ from each other.

Example: How much taller is Bilbo Baggins than Frodo Baggins?

$$Height_{Bilbo} - Height_{Frodo} = Difference$$

To find the difference in their heights, you would subtract Frodo's height from Bilbo's.

In geometry, differences are used to find the distance between points on a coordinate plane.

In algebra, they are used to factor polynomials when you have a difference of perfect squares or a difference of perfect cubes.

### Using Differences in Algebra and Geometry

Factoring Difference of Squares

Factoring Difference of Cubes

Finding the Distance Between Points