Åk 6–9

English/فارسی
4.1 Multiplication
When we work with multiplicatoin of natural numbers, fractions and decimals, we use something called ”repeated addition” to start with, and this is what it looks like:

Positive number · Positive number

3 · 2 =
2 + 2 + 2 =
(+2) + (+2) + (+2) =
(+2) and (+2) and (+2)

On the number line, it would look like this:

The result is then 6.

Positive number · Negative number

You can do this the same way for multiplication of a positive number with a negative number:

3 · (-2) =
(-2) + (-2) + (-2) =
(-2) and (-2) and (-2)

This is how it looks on the number line:

The result become -6.

Another way to understand multiplication with negative numbers is to look at how the product changes when you change on of the factors. Study this example:

3 · 5 = 15
2 · 5 = 10
1 · 5 = 5
0 · 5 = 0
(-1) · 5 = -5
(-2) · 5 = -10
(-3) · 5 = -15
and so on

Negative number · Negative number

Even this can be show in several different way in order to better understand.  We begin by continuing the number series from the last example, but now decrease by a factor instead.

We assume now you know how to multiply a positive number with a negative number:

(-3) · 5 = -15
(-3) · 4 = -12
(-3) · 3 = -9
(-3) · 2 = -6
(-3) · 1 = -3
(-3) · 0 = 0
(-3) · (-1) =
(-3) · (-2) =
(-3) · (-3) =
and so on

Another way to explain multiplication of two negative numbers is to use opposite numbers.  We will look at an example:

If we take two opposite numbers, 7 and -7, and multiply them by the same number, even the products will be opposite numbers.  Look here:

 2 · 7 = 14 and 2 · (-7) = -14

The new opposite numbers are 14 and -14. We try another example with the numbers 4 and -4 but now we are multiplying with a negative number.

 (-2) · 4 = -8 and (-2) · (-4) = ?

We know that even the products become opposite number if we multiply with the same number.

(-2)
· 4 = -8
and (-2) · (-4) = 8

Multiplication with negative numbers cannot be shown on the number line, but after these examples, you should be able to make some conclusions.  Can you answer these questions?