02 Multiplication and Division of Rational Numbers
Multiplication
Now that we have dealt with adding and subtracting positive and negative rational numbers, let’s try to see how it works for multiplication and division.
Multiplication of negative numbers is the same as that of whole numbers. The only difference is the presence of the negative sign (obviously!). While multiplying negative numbers, you just multiply the absolute values and then add the sign accordingly. These negative signs decide whether the answer is positive or negative.
Let’s use a different example than temperature to understand it this time. Example of taking a loan and depositing money at the bank.
You decide to make a deposit of $5, three times – which gives you 5 + 5 + 5. This is the repeated addition model that gives you multiplication of 5 three times. Since making a deposit is considered positive, you add 5 three times.
3 x 5 = 15
Here, the deposit (+) is added (+) three times or the deposit is multiplied 3 times positively. So you get a positive result (+10).
You remove a deposit of $5, three times, leaving you with money that you owe the bank. This gives you a negative number since you now owe the bank money.
-3 x 5 = -15
The deposit (+) is reduced (-) three times or the deposit is multiplied negative 3 times. So you get a negative result (-15). Note that -15 $ simply means you need to give the money, not that you have negative money (which does not make sense).
You decide to take the same amount of loan (-5), three times. Taking a loan is considered negative, so we can write it as -5. This will give you thrice the amount of loan you need to pay the bank. Since you are taking out a loan in total, the result has to be negative.
3 x -5 = -15
Your loan (-) is added thrice (+) or loan is added three times positively. So you have a negative result (-10).
The bank decided to cancel your loan. They paid three times the loan amount of $5. This means that -5 has been removed three times, giving an overall positive result.
-3 x -5 = 15
Your loan (-) is reduced (-) twice, or your loan is removed 3 times negatively, giving you a positive result.
The basic idea that can be gained from here is as follows: Multiplication of 2 positive numbers gives a positive. (+)x(+) = (+) Multiplication of 2 negative numbers gives a positive. (-)x(-)=(+) Multiplication of a positive and negative number gives a negative. (-)x(+)=(-)
Division
[[Need to edit this a bit]]
(We might need to remove context here since it makes it harder to understand)
To understand division of using rational numbers (negatives), we can use the very helpful idea of division being the inverse of any division problem is actually a multiplication problem:
Because we know how to multiply signed numbers, that means we know how to divide them.
The sign of a positive number divided by a negative number is always negative. The sign of a negative number divided by a positive number is always negative. The sign of a negative number divided by a negative number is always positive. A number that can be used in place of the variable that makes the equation true is called a solution to the equation. For example, for the equation , the solution is -10, because it is true that .
100/20 = 5 If you had to make a deposit of 100 with each deposit being $20, how many times would you need to deposit the amount? You would need to deposit it 5 times to get 100. So your answer is 5. You are dividing a positive by a positive number, which gives you a positive answer.
-100/20 = -5 How many times would you need to deposit 20 to get a loan of -100? That would not be possible! Rather, you would need to take loans of 20 five times to get a total loan of -100. Since taking a loan is considered negative, you get -5.
100/-20 = -5 How many loans, each of magnitude 20, would you need to take to make a total deposit of 100? This one doesn’t make sense either. You would rather need to make deposits 5 times. Depositing is a positive task and since we are talking in terms of loan, we need to write the answer is negative 5.
-100/-20 = 5 How many loans, each of magnitude 20, would you need to get a total loan of -100? You would need to take 5 loans, each time 20. Here, taking a loan is a negative, but we are already talking in terms of loan, so the answer is a positive 5. (If you look at it as a multiplication problem, taking a loan of 20 (-20) five times (5) will give you a loan of 100)
This is confusing right? Rather, an easier way to find answers for such questions is by simply ignoring the sign, carrying out the necessary division and then adding the sign back.
So, in the case of -100/20, think of it as 100/20, which gives us 5. Now, we add the negative sign remaining, and we get -5. The same applies for 100/-20. What about if we have negative in both divisor and dividend? In the case of -100/-20, we get 5 after removing the sign, and adding them gives us - (-5). This is the same as multiplying -5 by -1, which we know gives us a positive answer. So two negatives will give us a positive.
The basic idea to be gained from here is as follows: Division of 2 positive numbers gives a positive. (+)/(+) = (+) Division of 2 negative numbers gives a positive. (-)/(-)=(+) Division of a positive and negative number gives a negative. (-)/(+)=(-)
The idea of multiplication and division of negative numbers applies to decimals and fractions as well.
What do you think is -¾ x ⅚? We can multiplying the magnitudes and then add the negative sign, giving us -⅝.
What about if we have two negative signs? -½ x - 3/7?
We get the magnitude as 3/14 and adding the signs gives us -(-3/14), which is like multiplying -3/14 with -1 which gives us positive 3/14!
Note that -1/2 is the same as 1/-2. In both cases, we can isolate the negative sign to give us -(½).
Division works in the same way!
-3/4 divided by 6/4 is written as -¾ x 4/6 = -½ (once we have written division in the form of multiplication, we follow the same steps).