Operation On Integers

Question 1

______ of a number is its distance from zero

Answer 1

absolute value

Question 2

this tells you how far a number from zero is. We use the symbol | | to indicate the

Answer 2

absolute value

Question 3

What is the absolute value of 3?

Answer 3

3
herefore, the absolute value of 3 is equal to 3.

In symbols, | 3 | = 3.

Question 4

What is the absolute value of – 4?

Answer 4

Using a number line, you can verify that – 4 is 4 units away from zero.

Hence, the absolute value of -4 is equal to 4.

In symbols, | – 4 | = 4

Question 5

It is important to note that the absolute value of a number is always

Answer 5

nonnegative

Question 6

You can easily determine the absolute value of a number without drawing a number line. You just need to follow these rules:

Answer 6

Rule 1: If the number is positive, the absolute value of the number is itself.
Rule 2: If the number is negative, just drop the negative sign.

Question 7

the absolute value of -16 is

Answer 7

16

Question 8

can you determine the absolute value of 0, – 321, 1500, and -9000?

Answer 8

| – 321 | = 321

| 1500 | = 1500

| – 9000 | = 9000

0 | = 0

Question 9

The first thing you need to consider before adding integers is to determine

Answer 9

determine whether the given integers have the same or different signs.

Question 10

To add integers with the same signs (either both are positive or both are negative):

Answer 10

Step 1: Add the absolute values of the given integers

Step 2: Put the common sign to the number you have obtained from Step 1.

Question 11

Example 1: 15 + 32 = ?

Answer 11

Solution:

Step 1: Add the absolute values of the given integers.

The absolute value of 15 is 15 while the absolute value of 32 is 32. We add their absolute values: 15 + 32 = 47

Step 2: Put the common sign to the number you have obtained from Step 1.

Since both 15 and 32 are positive integers, then their common sign is positive. The number we have obtained from Step 1 was 47. Therefore, the sign of 47 must be positive.

Indeed, 15 + 32 = 47

Question 12

Example 2: What is the sum of – 210 and – 172?

Answer 12

olution:

Let’s use the steps on adding integers with the same signs since – 210 and – 172 are both negative integers (same signs).

Step 1: Add the absolute values of the given integers.

The absolute value of – 210 is 210 while the absolute value of – 172 is 172. We add their absolute values:

210 + 172 = 382

Step 2: Put the common sign to the number you have obtained from Step 1.

Since – 210 and – 172 are both negative integers, then their common sign is negative. Therefore, we put a negative sign to the number we have obtained from step 1 which is 382.

Hence, the sum of – 210 and – 172 is – 382.

Question 13

Now, what if the given integers have different signs? What if one integer is positive while the other is negative and vice-versa.

Just follow these steps to add integers with different signs easily:

Answer 13

Step 1: Subtract the absolute values of the given numbers.

Step 2: Put the sign of the integer with a larger absolute value to the number you have obtained from Step 1.

Question 14

Example 1: Add -19 and 25.

Answer 14

-19 is a negative number and 25 is a positive integer. They have different signs. Hence, we will use the steps above on adding integers with different signs.

Step 1: Subtract the absolute values of the given numbers.

The absolute value of – 19 is 19. Meanwhile, the absolute value of 25 is 25.

Subtracting the absolute values (larger – smaller): 25 – 19 = 6

Step 2: Put the sign of the integer with a larger absolute value to the result you have obtained from Step 1.

Note that the absolute value of 25 is larger than the absolute value of – 19. Also, 25 is a positive number. Therefore, the result we have obtained from Step 1 (which is 6) must be a positive integer.

Hence, – 19 + 25 = 6

– 19 | = 19 and | 25 | = 25.

Question 15

Example 2: Add – 32 and 15.

Answer 15

he given integers have different signs. Let’s use the steps on adding integers with different signs.

Solution:

Step 1: Subtract the absolute values of the given numbers.

The absolute of – 32 is 32 while the absolute value of 15 is 15.

Subtracting the absolute values (larger – smaller): 32 – 15 = 17

Step 2: Put the sign of the integer with a larger absolute value to the result you have obtained from Step 1.

Note that the absolute value of – 32 is larger than the absolute value of 17. Also, – 32 is negative. Therefore, the result we have obtained from Step 1 (which is 17) must be a negative integer.

Hence, – 32 + 15 = – 17

– 32 | = 32 and | 15 | = 15.

Question 16

Example 3: Add – 90 and 32.

Answer 16

Step 1: Subtract the absolute values of the given numbers.

The absolute of – 90 is 90 while the absolute value of 32 is 32.

Subtracting the absolute values (larger – smaller): 90 – 32 = 58

Step 2: Put the sign of the integer with a larger absolute value to the result you have obtained from Step 1.

Note that the absolute value of – 90 is larger than the absolute value of 32. Also, – 90 is negative. Therefore, the result we obtained from Step 1 (which is 58) must be a negative integer.

Hence, – 90 + 32 = – 58

– 90 | = 90 and | 32 | = 32.

Question 17

There are two steps you need to follow when subtracting integers:

Answer 17

Step 1: Change the operation into addition and reverse the sign of the second integer (or the subtrahend).

Step 2: Apply the rules on adding integers.

Question 18

Example 1: What is – 19 – 5?

Answer 18

tep 1: Change the operation into addition and reverse the sign of the second integer (or the subtrahend).

The first thing you have to do is to change the subtraction sign (-) into an addition sign (+).

Afterward, reverse the sign of the second integer (or the subtrahend). The subtrahend is 5, so we reverse the sign of 5 into – 5.

– 19 + (- 5) =

Step 2: Apply the rules on adding integers.

To finish the subtraction process, we need to apply the rules for adding integers.

We have obtained – 19 + (- 5) from Step 1. This means that we need to add integers with the same signs. I hope that you still remember the rules for adding integers.

Using the rules on adding integers with the same signs:

– 19 + (- 5) = – 24

And then we’re done. The answer is – 24.

Therefore, – 19 – 5 = – 24

Question 19

Example 2: Compute for: – 32 – (-12)

Answer 19

Step 1: Change the operation into addition and reverse the sign of the second integer (or the subtrahend).

The first thing you have to do is to change the subtraction sign (-) into an addition sign (+).

Afterward, you need to reverse the sign of the second integer (or the subtrahend). The subtrahend is – 12, so we reverse the sign of – 12 into 12.

– 32 + 12 =

Step 2: Apply the rules on adding integers.

To finish the subtraction process, we need to apply the rules for adding integers.

We have obtained – 32 + 12 from Step 1. This means that we need to add integers with different signs.

Using the rules on adding integers with different signs:

– 32 + 12 = – 20

Therefore, – 32 + 12 = – 20

Question 20

Example 3: What is -18 – (- 45)?

Answer 20

Step 1: Change the operation into addition and reverse the sign of the second integer (or the subtrahend).

The first thing you have to do is to change the subtraction sign (-) into the addition sign (+).

Afterward, you need to reverse the sign of the second integer (or the subtrahend). The subtrahend is – 45, so we reverse the sign of – 45 into 45.

– 18 + 45 =

Step 2: Apply the rules on adding integers.

To finish the subtraction process, we need to apply the rules for adding integers.

We have obtained – 18 + 45 from Step 1. This means that we need to add integers with different signs

Using the rules on adding integers with different signs:

– 18 + 45 = 27

Therefore, – 18 + 45 = 27

Question 21

Multiplying integers is a lot easier than adding or subtracting integers. The rules are pretty simple:

Answer 21

If the integers have the same signs, multiply the integers and put a positive sign in the resulting integer.
If the integers have different signs, multiply the integers and put a negative sign in the resulting integer.

Question 22

Example 1: Multiply: – 3 × – 5

Answer 22

Solution:

– 3 and – 5 are both negative integers. They have the same signs so their product must be positive.

Therefore, – 3 × – 5 = 15

Example 2: Multiply: 8 × – 3

Solution:

8 and – 3 have different signs so their product must be negative.

Therefore, 8 × – 3 = – 24

Question 23

The rules in dividing integers are actually similar to multiplying integers:

Answer 23

If the integers have the same signs, divide the integers and put a positive sign to the resulting integer.
If the integers have unlike or different signs, divide the integers and put a negative sign to the resulting integer.

Question 24

Example 1: Divide -18 by -2

Answer 24

-18 and -2 have the same signs. So, we just divide the integers and the answer must be positive.

-18 ÷ (-2) = 9

Question 25

Example 2: Divide 18 by – 2

Answer 25

18 and – 2 have different signs. So, we just divide the integers and the answer must be negative.

18 ÷ (-2) = – 9

You may have wondered why Multiplication of Integers and Division of Integers almost have the same rules. The answer is simple: Dividing integers is just multiplying an integer by the multiplicative inverse or the reciprocal (we will learn the reciprocal of a number in later topics) of the other. That’s why their rules are almost similar.

Question 26

Suppose we want to multiply an integer such as – 12 by 0. What do you think will be the result?

Answer 26

Simple: The answer is 0.

If you multiply any number (real, rational, irrational, integers, fraction, or decimal) by zero, the result will always be 0. This property is called the Zero Property of Multiplication

Question 27

1) What is the sum of - 932 and - 110?
a) - 1042
b) - 822
c) 1042
d) 822

Answer 27

a

Question 28

2) Suppose that m and n are integers such that m < 0 while n > 0. Which of the following is true
about m x n ?
a) m x n is positive
b) m x n is negative
c) m x n is zero
d) None of the above

Answer 28

b

Question 29

3) Compute for 89 - (-12)
a) 101
b) - 101
c) 67
d) - 67

Answer 29

a

Question 30

4) Which of the following mathematical equations is/are true?
a) 56 - (- 32) = - 88
b) - 142 + (- 101) = 343
c) - 327 ÷3 = - 109
d) Both A and C

Answer 30

c

Question 31

5) Greg multiplied an integer by another integer and he obtained a negative integer. Which of
the following might be true about Greg’s integers?
a) Both of Greg’s integers are negative
b) Both of Greg’s integers are positive
c) One of Greg’s integers is positive while the other is zero
d) None of the above

Answer 31

d