Multiplication of Negative Numbers

Have you ever though of the meaning of multiplication?
If we are working with positive numbers, the meaning is very apparent.

1 x 2 = 2
2 x 2 = 2 + 2
3 x 2 = 2 + 2 + 2
4 x 2 = 2 + 2 + 2 + 2

It works the same way when we multiply a positive and negative number.

1 x (-2) = (-2)
2 x (-2) = (-2) + (-2)
3 x (-2) = (-2) + (-2) + (-2)
4 x (-2) = (-2) + (-2) + (-2) + (-2)

So what about (-4) x (-2)?
How do we visualize it?
How can we understand this?

I found this question from a maths forum. And have proposed a way to understand this.

Now let us remember this simple rule.
"If we multiply -1 to any term, we are effectively adding it by twice the opposite sign of its number"

Rule:
-1 x a = a + (-2a) = -a

Example :
-1 x 1 = 1 + (-2) = -1
Where "-2" is twice the opposite of "1"


Example :
-1 x -1 = -1 + 2 = 1
Where "2" is twice the opposite of "-1"


What about -4 x -2?

-4 x -2 is actually a convenient way of writing -1 x 4 x -1 x 2

-4 x -2
= -1 x 4 x -1 x 2
= -1 x -1 x 4 x 2 ~ Rearranging the sequence of the numbers
= -1 x -1 x 8 ~ where 4 x 2 = 2 + 2 + 2 + 2 = 8
= -1 x [8 + (-16)] ~ where -16 is twice the opposite of 8
= -1 x -8
= -8 + 16 ~ where 16 is twice the opposite of -8
= 8

Multiplication and Division

Before we proceed, a review of how to work with addition and subtraction will be useful.

How can we work with multiplication and division?
Listed here are the few simple steps.

1. Identify all terms, multiplication sign and division signs (if any)

Example 1:
7x-2

There are two terms, [+7] and [-2]. There is a multiplication sign [x] between the two terms.

For term [+7] => number is 7, sign is '+'
For term [-2] => number is 2, sign is '-'

2. Compare the two terms besides the multiplication or division sign.

Do the two terms have the same sign?
If SAME :
=> Final sign after multiplication or division is ALWAYS '+'

If DIFFERENT :
=>Final sign after multiplication or division is ALWAYS '-'

3. The final number is the multiplication of the numbers in the two terms.

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Example 1:
7x-2

1. What are the terms, multiplication and division signs identified:
Ans : [+7] [x] [-2]

2. Are the signs of the two terms the same?
Ans : No.

=> Final sign after multiplication is '-'
=> Final number is obtained from multiplying 7 by 2 = 14

Result of 7x-2 = -14
====================

Example 2:
-5x-8

1. What are the terms, multiplication and division signs identified?
Ans : [-5] [x] [-8]

2. Are the signs of the two terms the same?
Ans : Yes.

=> Final sign after multiplication is '+'
=> Final number is obtained from multiplying 5 by 8 = 40

Result of -5x-8 = +40
====================

Example 3:
11 x 12

1. What are the terms, multiplication and division signs identified?
Ans : [+11] [x] [+12]

2. Are the signs of the two terms the same?
Ans : Yes.

=> Final sign after multiplication is '+'
=> Final number is obtained from multiplying 11 by 12 = 132

Result of 11 x 12 = +132
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Addition and Subtraction

Here are the two steps to systematically do addition and subtraction.
By following these two simple steps, we will be able to evaluate all addition and subtraction mathematics expression.

1. Find out all the terms and signs in the expression:

Example :
-7+8

The terms identified here are -7 and +8

For term [-7] => sign is '-', number is '7'
For term [+8] => sign is '+', number is '8'

2. Proceed with two terms and compare the signs of the terms.

Check if the two terms have the same sign?
If SAME :
=> Final number is the sum of the two numbers
=> Final sign is the sign of the two terms

If DIFFERENT:
=> Final number is the difference between the two numbers
=> Final sign is the sign of the bigger number


Example 1:
-7+8

Is the sign for the two terms the same? [-7] and [+8]
Ans : No.

What is the difference between the two numbers 7 and 8?
Ans : 1

What is the sign of the bigger number (-7 and +8)?
Ans : +

Result of -7 +8 = +1
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Lets work through more examples.

Example 2:
-2-9

1. The terms are [-2] and [-9]

For term [-2] => sign is '-', number is '2'
For term [-9] => sign is '-', number is '9'

2. Compare [-2] and [-9].

Are the two terms having the same sign?
Ans : Yes

=> Final number is sum of 2 and 9, which is 11
=> Final sign is '-'

Result of -2 -9 = -11
====================

Example 3:
-5+10

1. The terms identified are [-5] and [+10]

For term [-5] => sign is '-', number is '5'
For term [+10] => sign is '+', number is '10'

2. Compare the two terms [-5] and [+10]

Are the two terms having the same sign?
Ans : No.

=> Final number is difference of 5 and 10 , which is 5
=> Final sign is the sign of the bigger number. In this case, the bigger number is 10, sign is '+'

Result of -5 +10 = +5
====================

Example 4:
6-18

1. The terms identified are [+6] and [-18]

For term [+6] => sign is '+', number is '6'
For term [-18] => sign is '-', number is '18'

2. Compare [+6] and [-18].

Are the two terms having the same sign?
Ans : No.

=> Final number is the difference of the two number 6 and 18 , which is 12
=> Final sign is the sign of the larger number. In this case, the sign of the larger number is '-'

Result of 6-18 = -12
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To help visualize the flow: