Hi everyone! Welcome back to another session where we’ll delve into the intriguing world of directed numbers. Today, we’ll focus on something many students find tricky – multiplying and dividing negative numbers. So let’s jump straight into it!

## Understanding Multiplication with Negative Numbers

When we multiply negative numbers, it can be a bit confusing. Let’s take a straightforward example:

Imagine we have ( -2 X 3 ).

This means we have three lots of (-2).

### Visualising Negative Multiplication

To visualise this:

- We have (-2)
- Another (-2)
- And one more (-2)

So, altogether, we get:

[-2 + -2 + -2 = -6]

Therefore, ( -2 X 3 ) equals ( -6 ). Essentially, we’re just adding (-2) three times.

To reinforce this concept, if we switch the order and write ( 3 X -2 ), the result remains the same. It becomes three lots of (-2):

[3 X -2 = -6]

### Multiplication Laws

Remember, multiplication has certain laws. The order of the numbers doesn’t matter; you’ll still get the same product. So, whether it’s ( -2 X 3 ) or ( 3 X -2 ), the answer remains (-6).

#### Another Example

Let’s consider another example with larger numbers:

[-5 X -4]

This time, we have a negative number multiplied by another negative number. Think of it as ( 5 X 4 ) which equals 20.

However, since we are multiplying two negative numbers, the rule goes that ‘a negative times a negative is a positive’. So, the answer becomes:

[-5 X -4 = 20]

This might seem counterintuitive, but it’s because they cancel each other out to result in a positive number.

“Remember, a negative times a negative results in a positive.”

## Division with Negative Numbers

Now, let’s shift our focus to division. Just as with multiplication, division involving negative numbers can be quite straightforward once you grasp the concept.

### Division Example

Suppose we have:

[-20 ÷ 4]

We know that 20 divided by 4 is 5. But since we started with a negative number:

[-20 ÷ 4 = -5]

We can visualise this division by thinking about it as distributing 20 negatively into 4 parts.

Similarly, if we had:

[20 ÷ -4]

We’re essentially looking for the negative of ( 20 ÷ 4 ):

[20 ÷ -4 = -5]

### Dividing Two Negatives

What if we have to divide two negatives? Consider:

[-20 ÷ -4]

Here, dividing a negative by another negative cancels out the negatives, resulting in a positive number:

[-20 ÷ -4 = 5]

The negative signs neutralise each other, just like in multiplication.

## Key Takeaways

To sum up, here are the essential points to remember when dealing with negative numbers:

**Multiplying Two Negatives:**- The result is positive because the negatives cancel each other out.

**Multiplying a Negative with a Positive:**- The result is negative because you’re effectively adding the negative value multiple times.

**Dividing Two Negatives:**- The result is positive, as the negative signs cancel each other out.

**Dividing a Negative by a Positive or Vice Versa:**- The result is negative due to the direction of division.

### Practice Problems

For better understanding, try solving these on your own:

- ( -3 X 5 )
- ( -7 X -2 )
- ( 15 ÷ -3 )
- ( -24 ÷ -6 )

By practising these problems, you’ll enhance your understanding of how negative numbers behave in multiplication and division. With patience and regular practice, you’ll master these concepts in no time!