Subtracting as Adding the Opposite on a Number Line

The additive inverse of a number is the number you add to it to get $0$. It is also called the number's opposite. On a number line, a number that is not $0$ and its additive inverse are the same distance from $0$, on opposite sides.

• The additive inverse of $3.5$ is $-3.5$, because $3.5 + (-3.5) = 0$.
• The additive inverse of $-\frac{3}{4}$ is $\frac{3}{4}$, because $-\frac{3}{4} + \frac{3}{4} = 0$.
• The additive inverse of $0$ is $0$, because $0 + 0 = 0$.

For any rational number $q$, the symbol $-q$ means the additive inverse of $q$, so $q + (-q) = 0$ every time. Be careful: $-q$ is not always a negative number. When $q = -\frac{3}{4}$, its additive inverse is $-q = \frac{3}{4}$, which is positive.

Why do $2 - 3.5$ and $2 + (-3.5)$ name the same number? Start from what subtraction means. Subtraction undoes addition, so $2 - 3.5$ is the number that gives $2$ when you add $3.5$ to it.

Test whether $2 + (-3.5)$ is that number by adding $3.5$ to it:

$(2 + (-3.5)) + 3.5 = 2 + ((-3.5) + 3.5) = 2 + 0 = 2$

The first step changes which two numbers are added first. That is allowed because regrouping the numbers in a sum never changes its value (the associative property of addition). The second step works because $-3.5$ and $3.5$ are additive inverses, so they sum to $0$. The last step works because adding $0$ changes nothing.

The result is $2$, so $2 + (-3.5)$ passes the test. That means $2 - 3.5 = 2 + (-3.5)$.

Now picture addition as movement along a horizontal number line, where numbers grow to the right and one unit is the distance from $0$ to $1$. You need one more term: the absolute value of a number $q$, written $|q|$, is its distance from $0$. For example, $|1.5| = 1.5$ and $|-1.5| = 1.5$.

For any rational numbers $p$ and $q$, the sum $p + q$ is $|q|$ units from $p$: to the right when $q$ is positive and to the left when $q$ is negative. When $q = 0$, the sum is $p$ itself.

This rule shows again why opposites sum to $0$: from $1.5$, adding $-1.5$ moves you $1.5$ units left, exactly onto $0$.

The ticks (small marks) on the number line below are $0.5$ apart, so moving from one tick to the next is a move of $0.5$. Three values are marked. Start at $2.5$.

• $2.5 + 1.5$: the number added is positive, so move $1.5$ units right. That is $3$ ticks, and you land on $4$. This is the rewritten subtraction $2.5 - (-1.5)$ from the first worked example.
• $2.5 + (-1.5)$: the number added is negative, so move $1.5$ units left. That is $3$ ticks, and you land on $1$.

Adding a number that is not $0$ and adding its additive inverse move you the same distance in opposite directions.

Takeaway. For any rational numbers $p$ and $q$, subtracting $q$ gives the same result as adding its additive inverse: $p - q = p + (-q)$. On a number line, the sum $p + q$ is $|q|$ units from $p$, to the right when $q$ is positive and to the left when $q$ is negative, and it is $p$ itself when $q = 0$. So you can handle any subtraction of rational numbers in two stages: rewrite it as an addition, then start at the first number and move.

Next, you will add and subtract positive and negative decimals and fractions.