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.
Students can download the Multiplying Negatives - Signs, Examples, Rules, Solved Examples, and FAQs PDF from the Vedantu website. Anyone can download the Multiplying Negatives - Signs, Examples, Rules, Solved Examples, and FAQs PDF for free from the website. Multiplying Negatives is a very important topic of maths from which many questions get asked in exams. A proper understanding of the topic is necessary for students to score good marks in their exams. This topic is important for competitive exams like IIT and NEET. Thus, students should make proper efforts while studying the topic. The expert faculty of Vedantu who have had a lot of experience teaching students have prepared the Multiplying Negatives - Signs, Examples, Rules, Solved Examples and FAQs PDF.
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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.
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.
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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| Example | |||
| two positives make a positive: | 3 × 2 = 6 | ||
| two negatives make a positive: | (−3) × (−2) = 6 | ||
| × | a negative and a positive make a negative: | (−3) × 2 = −6 | |
| × | a positive and a negative make a negative: | 3 × (−2) = −6 |
Yes indeed, two negatives make a positive, and we will explain why, with examples!
Signs
Let's talk about signs.
"+" is the positive sign, "−" is the negative sign.
When a number has no sign it usually means that it is positive.
And we can put () around the numbers to avoid confusion.
Example: 3 × −2 can be written as (+3) × (−2)
Two Signs: The Rules
"Two like signs make a positive sign,
two unlike signs make a negative sign"
The signs are − and + (a negative sign and a positive sign), so they are unlike signs (they are different to each other)
So the result must be negative:
(−2) × (+5) = −10
The signs are − and − (they are both negative signs), so they are like signs (like each other)
So the result must be positive:
(−4) × (−3) = +12
Well, first there is the "common sense" explanation:
When I say "Eat!" I am encouraging you to eat (positive)
But when I say "Do not eat!" I am saying the opposite (negative).
Now if I say "Do NOT not eat!", I am saying I don't want you to starve, so I am back to saying "Eat!" (positive).
So, two negatives make a positive, and if that satisfies you, then you don't need to read any more.
Direction
It is all about direction. Remember the Number Line?
Well here we have Baby Steven taking his first steps. He takes 2 paces at a time, and does this three times, so he moves 2 steps x 3 = 6 steps forward:
Now, Baby Steven can also step backwards (he is a clever little guy). His Dad puts him back at the start and then Steven steps backwards 2 steps, and does this three times:
Once again Steven's Dad puts him back at the start, but facing the other way. Steven takes 2 steps forward (for him!) but he is heading in the negative direction. He does this 3 times:
Back at the start again (thanks Dad!), still facing in the negative direction, he tries his backwards walking, once again taking two steps at a time, and he does this three times:
So, by walking backwards, while facing in the negative direction, he moves in the positive direction.
Try it yourself! Try walking forwards and backwards, then again but facing the other direction.
Play With It
But maybe you would like to see it in action? Use the sliders below:
numbers/images/number-line-mult.js
| Sam gives you three $10 notes: | +3 × +10 = you gain $30 | |
| Sam gives you three $10 debts: | +3 × −10 = you lose $30 | |
| Sam takes three $10 notes from you: | −3 × +10 = you lose $30 | |
| Sam takes three $10 debts from you: | −3 × −10 = you gain $30 |
People Run Forward, Video Normal:
All normal, people running forwards: +1 × +1 = +1
People Run Forward, but Video in Reverse:
Looks like people running backwards: +1 × −1 = −1
People Run Backward, Video Normal:
You see people running backwards: −1 × +1 = −1
People Run Backward, but Video in Reverse:
Looks like people running forwards: −1 × −1 = +1
The tank has 30,000 liters, and 1,000 liters are taken out every day. What was the amount of water in the tank 3 days ago?
We know the amount of water in the tank changes by −1,000 every day, and we need to subtract that 3 times (to go back 3 days), so the change is:
−3 × −1,000 = +3,000
The full calculation is:
30,000 + (−3 × −1,000) = 30,000 + 3,000 = 33,000
So 3 days ago there were 33,000 liters of water in the tank.
Here is another way of looking at it.
First have a play with this (explanations below):
numbers/images/mult-grid.js?min=-5&max=5
Start with the multiplication table (just up to 4×4 will do):
| 1 | 2 | 3 | 4 |
| 2 | 4 | 6 | 8 |
| 3 | 6 | 9 | 12 |
| 4 | 8 | 12 | 16 |
Now see what happens when we head into negatives!
Let's go backwards through zero:
| -4 | -8 | -12 | -16 |
| -3 | -6 | -9 | -12 |
| -2 | -4 | -6 | -8 |
| -1 | -2 | -3 | -4 |
| 0 | 0 | 0 | 0 |
| 1 | 2 | 3 | 4 |
| 2 | 4 | 6 | 8 |
| 3 | 6 | 9 | 12 |
| 4 | 8 | 12 | 16 |
Look at the "4" column: it goes -16, -12, -8, -4, 0, 4, 8, 12, 16. Getting 4 larger each time.
Look over that table again, make sure you are comfortable with how it works, because ...
... now we go further to the left, through zero:
| 16 | 12 | 8 | 4 | 0 | -4 | -8 | -12 | -16 |
| 12 | 9 | 6 | 3 | 0 | -3 | -6 | -9 | -12 |
| 8 | 6 | 4 | 2 | 0 | -2 | -4 | -6 | -8 |
| 4 | 3 | 2 | 1 | 0 | -1 | -2 | -3 | -4 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| -4 | -3 | -2 | -1 | 0 | 1 | 2 | 3 | 4 |
| -8 | -6 | -4 | -2 | 0 | 2 | 4 | 6 | 8 |
| -12 | -9 | -6 | -3 | 0 | 3 | 6 | 9 | 12 |
| -16 | -12 | -8 | -4 | 0 | 4 | 8 | 12 | 16 |
We can follow along a row (or column) and the values change consistently:
- Follow the "4" row: it goes -16, -12, -8, -4, 0, 4, 8, 12, 16. Getting 4 larger each time.
- Follow the "-4" row: it goes 16, 12, 8, 4, 0, -4, -8, -12, -16. Getting 4 smaller each time.
- etc...
So it all follows a consistent pattern!
Also see this at Real Multiplication Table.
What About Multiplying 3 or More Numbers Together?
Multiply two at a time and follow the rules.
First multiply (−2) × (−3). Two like signs make a positive sign, so:
(−2) × (−3) = +6
Next multiply +6 × (−4). Two unlike signs make a negative sign, so:
+6 × (−4) = −24
Result: (−2) × (−3) × (−4) = −24
330, 1615, 1616, 1617, 3447, 3448,331, 1618, 3170, 3171
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