## Order of operations

When calculations with negative numbers have more than one operation, we have to follow rules for the order of operations:

**Step 1:**

Do all operations that lie **inside the parentheses** first:

$$−3−8×(2+3) = −3−8×5$$

**Step 2:**
From left to right, do all

**multiplication and division**.

$$−3−8×5 = −3−40$$

**Step 3:**
From left to right, do all

**addition and subtraction**.

$$−3−40 = −43$$

## Adding and subtracting negative numbers

If a number has no sign it means that it is a positive number (positive number are higher than 0). We can write it with a plus sign, e.g. 5=+5.
We add and subtract, as we normally do. The sum of two positives numbers is always positive (higher than zero).

Subtraction of two positive numbers can take us beyond zero in some cases: 3−10=(−7) To add and subtract with negative numbers, at first add positive signs to positive numbers, e.g. 20−5=20−+5

For each combination of two signs follow these rules:

**Rule** | **Example** |

** ** Two like signs (++ or −−) become a positive sign (+). | 5 + (+ 3) = 5 + 3 = 8 −5 + (+ 3) = − 5 + 3 = − 2 2 − (− 3) = 2 + 3 = 5 −4 − (− 3) = − 4+ 3 = −1 |

** ** Two unlike signs (+− or −+) become a negative sign (−). | 4 + (− 3) = 4 − 3 = 1 −8 + (− 4) = − 8−4 = −12 1 − (+ 3) = 1−3 = −2 −5 − (+ 4) = − 5 − 4 = − 9 |

## Multiplying and dividing negative numbers

To multiply and divide two numbers follow these rules:

**When the signs are different the result is negative:**

$$(−1)×3=(−3)$$

$$(−6)÷3=(−2)$$

$$8/{(−4)}=(−2)$$

**When the signs are the same the result is positive:** $$(−1)×(−3)=3$$

$$4×3=12$$

$${(−2)}/{(−3)}=2/3$$

If there is a long multiplication with some negatives, just cancel off "minus" signs in pairs:

E.g. simplify:

$$(–1)×(–2)×(–1)×(–3)×$$

$$×(–4)×(–2)×(–1)=?$$

At first count up the minus signs: seven minus signs.

So there are three pairs that can be eliminated, with one left over. As a result, the final answer will be negative: (−48)

In other words, if there is an **even number of negative factors** (2 factors, 4 factors, 6 factors, etc.) , the result is **positive**. If there is an **odd number of factors with negative sign**, the result is **negative**.

*Example:*

$$(–1)×(–2)×4×(–1)×(–3)=?$$

There are four negative numbers → result will be positive:

$$(–1)×(–2)×4×(–1)×(–3)=bo24$$

The same applies for division:

$$ {4 × (–3) × (–2)}/{6×(–2)} = ? $$

There are three negative numbers → result will be negative:

$$ {4 × (–3) × (–2)}/{6×(–2)} =bo–bo2 $$