In the two previous post in Mathematics, we have discussed how to add and subtract signed numbers. In this post, we are going to learn how to multiply signed numbers particularly integers. Signed means positive and negative.
Positive Integer x Positive Integer
Clearly, the product is positive. We had been multiplying positive integers since Grade school and we all know that the product is positive.
Positive Integer x Negative Integer
When you multiply, notice that you are actually adding repeated. When we say 2 x 3, we are actually saying twice three or 2 groups of 3 or 3 + 3. When we say, thrice 11, we are saying 11 + 11 + 11. With this in mind, 3 x – 5 = -5 + -5 + -5. Since we are adding integers which are negative, the sum is also negative or -15. This means that 3 x -5 = -15. If we generalize this, we can say that the product of a positive integer and a negative integer is negative.
Negative Integer x Positive Integer
If you can remember, multiplication is commutative. This means that the order of the number you multiply does not matter, their product will always be the same. For example, 4 x 3 x 5 is equal to 5 x 4 x 3 or 3 x 4 x 5 or any other arrangement using the three numbers. This means that 3 x -5 = -5 x 3. So, a negative integer multiplied by a positive integer is also negative.
Negative Integer x Negative Integer
For multiplication of two negative integers, we can use patterns to know their product and generalize.
-3 x 2 = -6
-3 x 1 = -3
– 3 x 0 = 0
Now, what is -3 x -1?
If we look at the pattern in the product, we are actually adding by 3 each step so the next number is 3. All other numbers from -1, -2, -3 and so on will be positive (Why?). Therefore, the product of two negative numbers is positive.
Summary: Rules on how to Multiply Signed Numbers
From the above discussion, we summarize the multiplication of integers.
Positive Integer x Positive Integer = Positive Integer
Positive Integer x Negative Integer = Negative Integer
Negative Integer x Positive Integer = Negative Integer
Negative Integer x Negative Integer = Positive Integer
We can also say that if we multiply two numbers with the same sign, the answer is positive. If we multiply two numbers with different signs, the answer is negative.