Multiplying Negatives

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Numbers less than zero are referred to as negative numbers. Numbers above zero are positive numbers. There are rules for adding, subtracting, multiplying, or dividing positive and negative numbers. 


Signs

We know that "+" is a positive sign, "−" is a negative sign. When a sign is not denoted before a number, it usually means it's positive.

Example: 8 is actually +8

Note: To avoid confusion of signs, we can put () around the numbers. For example, 5 × −8 can be written as 5 × (−8)


Rules for Multiplying Negatives

We may have positive and negative integer values when working with integers in multiplication. There are rules for multiplying integers and dividing integers which are very similar to the rule for addition and subtraction.

  • If the signs are different, the answer is negative.

  • If the signs are the same, the answer is positive.

Refer to the description below for a better understanding.

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Plus Times Plus is Plus

Example: 2 × 5 = 10 

(We already discussed that when a number doesn't have a sign, it usually means it's positive.)

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Minus Multiply Minus is Plus 

Example: (-10) × (-5) = 50

Negative multiplied by Negative is a positive number, which means that product of two negative integers is always positive. 

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Plus Times Minus is Minus

Example: 5 x (-5) = - 25

Multiplication of Negative numbers with a positive number will always result in a Negative number. 

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Minus Times Plus is Minus

Note: These rules work in the same way for rules for dividing integers; you just have to replace "times" with "divided by". 


Division of Negative Numbers

The division of negative numbers works in the same way as that of positive numbers except that the results are sometimes negative. It depends on the two numbers involved in this division whether the answer is negative. The answer would also be negative if only one of the numbers is negative. The outcome would be positive, if both numbers are negative.


Expression 

Quotient

Example

Positive ÷ Positive

Positive

25 ÷ 5 = + 5

Negative ÷ Negative

Positive

- 25 ÷ - 5 = +5

Negative ÷ Positive

Negative

-25 ÷ 5 = -5

Positive ÷ Negative

Negative 

25 ÷ -5 = - 5


What Happens When We Multiply Negatives with Matrices?

An integer matrix is a matrix, all of which are integer entries. The negative of a matrix is obtained by multiplying it by -1. 

So, if A is a given matrix

Then, − A = − 1 [A]


Solved Examples

1. What is −6 × 3?

Ans: 6 x 3 is 18. But here we have one negative and one positive number. Hence, the sign of the answer will be minus. 

Therefore, the answer is −18.


2. What is −80 ÷ 8?

Ans: 80 ÷ 8 is 10. Again, we have a positive and negative number. Hence, the sign will be negative in the final answer. 

Therefore, the answer is −10.


3. What is −50 x −5?

Ans: 50 x 5 is 250. This time, we have 2 negative numbers. Hence, the sign will be positive in the answer. Therefore, the answer is 250.


Conclusion

Only remember 2 things when you multiply negative numbers. 

1. Two negative numbers give a positive result at all times. 

2. Your answer would also be negative if you have only one negative value. Only remember these two rules and the rest is easy to calculate.

FAQs (Frequently Asked Questions)

1. How Do You Multiply Negative Numbers?

Ans: Negative multiplied by Negative is a positive number and multiplication of Negative numbers with a positive number will always result in a Negative number. 

Examples: 

  1. -3 x -6 = +18

  2. -6 x 5 = - 30

2. Why Does Negative Multiplied By Negative Give a Positive Value?

Ans: When you multiply a negative by a negative you get a positive, because the two negative signs are cancelled out. So, negative multiplied by negative is always a positive.

3. What is the Sign of the Product of a Negative and a Positive Number?

Ans: The answer is a negative number when you multiply a negative number and a positive number. Regardless of the order in which the positive and negative numbers are multiplied, the answer is always a negative one.

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