# Combining Positive and Negative Numbers

How do we calculate an expression involving addition and subtraction?

## Calculating an algebraic sum

### From left to right

In general, we calculate from left to right.

#### Example:

To work out $(-2) + (-1.5) – (-0.3)$, first calculate $(-2) + (-1.5)$, then subtract $(-0.3)$ from the result. We can present the calculations in the following manner, showing one stage of the calculation on each line:

$A = \color{red}(-2) + (-1.5) \color{black}- (-0.3)$

$A = -3.5 – (-0.3)$

• Notice the removal of the parentheses around $-3.5$. As discussed before, there is no need to rewrite them.
• The two negatives next to each other change to a $\textbf{+}$ in the following step.

$A = -3.5 + 0.3$

$A = -3.2$

### By changing the order of the terms

It is possible to change the order of the terms in order to make calculations easier.

#### Example with two terms:

$B = (-24.8) \color{red}- (-32.5) \color{black}+ (+24.8)$

$B = (-24.8) + (+24.8) \color{red}- (-32.5)$

$B = 0 + (+32.5)$

$B = 32.5$

## Giving a simplified notation of an algebraic sum

### Rules

1. For positive numbers, writing the $+$ sign and parentheses is optional.
2. In a sum, if the first term is negative, the brackets are optional.
3. Adding a number is the same as subtracting its opposite (and subtracting a number is the same as adding its opposite)

### Examples with two terms

$(+7) – (+2)$ can be written $7 – 2$, according to rule 1.

We write: $(+7) – (+2) = 7 – 2 = 5$.

$(-3) + (+1)$ can be written $(-3) + 1$, according to rule 1, then $-3 + 1$, according to rule 2.

We write: $(-3) + (+1) = -3 + 1 = -2$.

$(+1) + (-4)$ can be written $(+1) – (+4)$, according to rule 3, then $1 – 4$, according to rule 1.

We write: $(+1) + (- 4) = 1 – 4 = -3$.