### Numbers: Negative numbers

### Multiplying negative numbers

We know that: \[\orange{3} \times \red{5}= \underbrace{\red{5}+\red{5}+\red{5}}_{\orange{3} \text{ times}}=15\]

In the same way: \[\green 3 \times \blue{-5} =\underbrace{\blue{-5}+\blue{-5}+\blue{-5}}_{\green{3} \text{ times}}=\blue{-15}\]

Since the order of multiplication can be switched, we also have #\blue{-5} \times \green{3} =\blue{-15}#.

Generally said, the following holds:

*A #\green{\textit{positive}}# number multiplied with a #\blue{\textit{negative}}# number equals a #\blue{\textit{negative}}# number.*

Consider the pattern on the right. We see that the number by which we multiply decreases by one in every row. The outcome, therefore, always increases by #4#. Hence, we see that #-4 \times -1=4#.

Generally said, the following holds:

*A #\blue{\textit{negative}}# number multiplied with a #\blue{\textit{negative}}# number equals a #\green{\textit{positive}}# number.*

\[\begin{array}{rcrrr}-4 &\times& 2&=& -8 \\ -4 &\times& 1&=&-4 \\ -4 &\times& 0&=&0 \\ -4 &\times& -1&=&4 \\ -4 &\times& -2 &=&8 \end{array}\]

In general, we can state the following calculation rules for multiplication.

The calculation rules for multiplying positive and negative numbers are: \[\begin{array}{rclll} |
\[\begin{array}{rcrrr} \\[1pt] |

We are multiplying two positive numbers, so the result is positive.

#2 \times 2=4#

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