Multiplying Negatives - Introduction
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.
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Some of the topics which have been explained in the Multiplying Negatives - Signs, Examples, Rules, Solved Examples and FAQs PDF are as follows:
Signs
Rules for Multiplying Negatives
Division of Negative Numbers
What Happens When We Multiply Negatives with Matrices?
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 the 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 as 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.
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.
Two negative numbers give a positive result at all times.
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.
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FAQs on Multiplying Negatives
1. What are the simple rules for multiplying negative numbers?
When multiplying integers, the rules for signs are straightforward. Here’s a quick guide:
- A negative number multiplied by a negative number gives a positive product. For example, (-5) × (-3) = 15.
- A negative number multiplied by a positive number gives a negative product. For example, (-5) × 3 = -15.
- A positive number multiplied by a negative number also gives a negative product. For example, 5 × (-3) = -15.
2. Why does multiplying two negative numbers result in a positive answer?
Think of multiplication as scaling. Multiplying by a negative number means flipping the direction on a number line. When you multiply (-4) by (-2), you first take the number 4, flip it to -4 (multiplying by -1), and then you flip it again (multiplying by the other -1). Flipping the direction twice brings you back to the positive side, resulting in +8. It's like 'the opposite of the opposite' is the original position.
3. What is the rule for multiplying three or more negative numbers together?
The rule extends from the basic one. The sign of the final product depends on how many negative numbers you are multiplying:
- If you multiply an even number of negative numbers (like two, four, or six), the final answer will be positive.
- If you multiply an odd number of negative numbers (like one, three, or five), the final answer will be negative.
For example, (-2) × (-3) × (-4) = -24, because there are three (an odd number) negative numbers.
4. What happens when a negative number is multiplied by zero?
Any number, whether it is positive, negative, or a fraction, when multiplied by zero, results in zero. This rule is absolute. For instance, (-150) × 0 = 0.
5. Where would someone use the multiplication of negative numbers in real life?
A common real-life example is calculating debt changes. If you have a debt of ₹50 (which is -50) and you triple that debt, you are multiplying (-50) by 3 to get a total debt of ₹150 (which is -150). Another example is in physics, where negative velocity multiplied by negative time can indicate a position in the positive direction.
6. How is multiplying by a negative number different from just adding a negative number?
These are two very different operations. Adding a negative number is the same as subtraction and moves a value to the left on the number line. For example, 10 + (-3) = 7. Multiplying by a negative number is about scaling and flipping the value's direction. For example, 10 × (-3) = -30, which is a much larger change and on the opposite side of zero.
7. Do the same rules for multiplying negatives apply to fractions and decimals?
Yes, absolutely. The rules for signs are universal for all real numbers. Whether you are multiplying negative integers, negative fractions, or negative decimals, the outcome of the sign follows the exact same logic: two negatives make a positive, and a negative and a positive make a negative.
