Addition & Subtraction Of Integers

What is an integer?

An integer is a number that can be expressed without the use of fractions or decimals. 

INTEGER	NOT AN INTEGER
0	-0.1
9	$\frac{2}{3}$
-25	$5\frac{4}{7}$
1203	-23.789
-82195	903.21

The set of integers is denoted by Z, which includes:

1. Zero – it is defined as an integer that is neither negative nor positive.
2. Positive Integers – positive integers are those numbers greater than zero such as 1, 2, 3, 4, …
3. Negative Integers – negative integers are those numbers less than zero such as -1, -2, -3, -4, …

Take note:

1. There is no such thing as negative zero. 
2. No need to put + sign on a positive integer. 
3. A number with no negative or positive sign is considered as a positive integer.

How to add integers?

Generally, the term “addition” refers to the process of increasing the value of something. However, in the case of integers, the addition operation may result in an increase or decrease in the given number’s value.

Here’s a list of rules for adding integers:

• The sum of an integer and its opposite or additive inverse is 0. Say for, example, 5 + (-5) = 0.
• If we add 0 to any integer, the result will be the same number. For instance, 3 + 0 = 3. 
• If we add two positive integers,  the sum will always be a positive number that is greater than both given integers.  
• If we add two negative integers, the result will always be a negative number that is smaller than the given integers.
• If we add a positive integer and a negative integer, the process will become subtraction. Then, the greater number will determine the sign of the result.

The table below shows a summary of the rules in adding integers depending on its sign.

Greater Absolute Value	Smaller Absolute Value	Operation	Result	Example
+	+	Addition	+	13 + 4 = 17
+	–	Subtraction	+	5 + (-3) = 2
–	+	Subtraction	–	(-21) + 19 = -2
–	–	Addition	–	(-4) + (-3) = -7

Adding positive integers

When adding positive integers, we simply add the numbers – just how we add whole numbers. The result of two positive integers will always be a positive number and should be greater than the given numbers. 

The table below shows example of adding two positive integers.

Example	Answer
9 + 3 =	12
32 + 45 =	77
146 + 873 =	1019

Adding negative integers

When adding negative integers, the first thing to do is get its absolute value. Absolute value is the distance of a number from zero. For instance, the absolute value of -9 is 9. The result of adding negative integers will always result to a negative value.

To add negative integers:

1. Get the absolute value of the given numbers.
2. Add the numbers.
3. Place the negative sign on the final answer.

Example #1 

What is the sum of -10 and -25?

Solution

Addition Process	Step-by-Step Explanation
-10 + (-25) =	Set up the addition process.
| -10 | = 10 | -25 | = 25	Get the absolute value of -10 and -25. Hence, we will have 10 and 25.
10 + 25 = 35	Adding 10 and 25 will result to 35.
-10 + (-25) = -35	Use the original addition process and place the negative sign on the final answer.
	Therefore, the sum of -10 and -25 is -35.

Example #2

Determine the sum of -98 and -45.

Solution

Addition Process	Step-by-Step Explanation
-98 + (-45) =	Set up the addition process.
| -98 | = 98 | -45 | = 45	Get the absolute value of -98 and -45. Hence, we will have 98 and 45.
98 + 45 = 143	Get the sum of 98 and 45. Thus, we will have 143.
-98 + (-45) = -143	Go back to the original addition process and place the negative sign on the final answer.
	Therefore, the result of adding -98 and -45 is -143.

Example #3

What is the result of adding -103 and -567?

Solution

Addition Process	Step-by-Step Explanation
-103 + (-567) =	Set up the addition process.
| -103 | = 103 | -567 | = 567	Get the absolute value of -103 and -567. Hence, we will have 103 and 567.
103 + 567 = 670	Add 103 and 567. Thus, we will have 670.
-103 + (-567) = -670	Go back to the original addition process and place the negative sign on the final answer.
	Therefore, the result of adding -103 and -567 is -670.

Adding positive and negative integers

When adding unlike signs, subtract the absolute values of two integers and then add the sign of the number to the greater absolute value.

Case 1: When the absolute value of the greater number is positive, the result is always positive.

To add unlike signs if the absolute value of the greater number is positive:

1. Get the absolute value of the given numbers.
2. Subtract the numbers.

Example #1

What is the sum of 10 and -3?

Solution

Addition Process	Step-by-Step Explanation
10 + (-3) =	Set up the addition process.
| 10 | = 10 | -3 | = 3	Get the absolute value of 10 and -3. Hence, we will have 10 and 3.
10 – 3 =	Subtract the absolute value of numbers.
10 – 3 = 7	Get the difference of the numbers. Thus,10 – 3 = 7
10 + (-3) = 7	Write the original equation. Since the absolute value of 10 is greater than the absolute value of -3, then we will copy the sign of 10 – which is positive.
	Therefore, the sum of 10 and -3 is 7.

Example #2

Determine the result of adding -7 and 23.

Solution

Addition Process	Step-by-Step Explanation
-7 + 23 =	Set up the addition process.
| -7 | = 7 | -23 | = 23	Get the absolute value of -7 and 23. Hence, we will have 7 and 23.
23 – 7 =	Write the number with greater absolute value on the minuend and put the number with smaller absolute value on the subtrahend.
23 – 7 = 16	Get the difference of the numbers. Thus,23 – 7 = 16
-7 + 23 = 16	Write the original equation. Since the absolute value of 23 is greater than the absolute value of -7, then we will copy the sign of 23 – which is positive.
	Therefore, the result of adding -7 and 23 is 16.

Example #3

What is the sum of 98 and -42?

Solution

Addition Process	Step-by-Step Explanation
98 + (-42) =	Set up the addition process.
| 98 | = 98 | -42 | = 42	Get the absolute value of 98 and -42. Thus, we will have 98 and -42.
98 – 42 =	Write the number with greater absolute value on the minuend and put the number with smaller absolute value on the subtrahend.
98 – 42 = 56	Get the difference of the numbers. Thus,98 – 42 = 56
98 + (-42) = 56	Write the original equation. Since the absolute value of 98 is greater than the absolute value of -42, then we will copy the sign of 98 – which is positive.
	Therefore, the sum of 98 and -42 is 56.

Case 2: When the absolute value of the greater number is negative, the result is always negative.

To add unlike signs if the absolute value of the greater number is negative:

1. Get the absolute value of the given numbers.
2. Subtract the numbers.
3. Write the negative sign on the final answer.

Example #1

Determine the sum of -25 and 9.

Solution

Addition Process	Step-by-Step Explanation
-25 + 9 =	Set up the addition process.
| -25 | = 25 | 9 | = 9	Get the absolute value of -25 and 9. Hence, we will have 25 and 9.
25 – 9 =	Write the number with greater absolute value on the minuend and put the number with smaller absolute value on the subtrahend.
25 – 9 = 16	Get the difference of the numbers. Thus,25 – 9 = 16
-25 + 9 = -16	Write the original equation. Since the absolute value of -25 is greater than the absolute value of 9, then we will copy the sign of -25 – which is negative.
	Therefore, the result of adding -25 and 9 is -16.

Example #2

What is the sum of 19 and -37?

Solution

Addition Process	Step-by-Step Explanation
19 + (-37) =	Set up the addition process.
| 19 | = 19 | -37 | = 37	Get the absolute value of 19 and -37. Hence, we will have 19 and 37.
37 – 19 =	Write the number with greater absolute value on the minuend and put the number with smaller absolute value on the subtrahend.
37 – 19 = 18	Get the difference of the numbers. Thus,25 – 9 = 16
-25 + 9 = -16	Write the original equation. Since the absolute value of -25 is greater than the absolute value of 9, then we will copy the sign of -25 – which is negative.
	Therefore, the result of adding -25 and 9 is -16.

Example #3

Find the sum of -108 and 49.

Solution

Addition Process	Step-by-Step Explanation
-108 + 49 =	Set up the addition process.
| -108 | = 108 | 49 | = 49	Get the absolute value of -108 and 49. Hence, we will have 108 and 49.
108 – 49 =	Write the number with greater absolute value on the minuend and put the number with smaller absolute value on the subtrahend.
108 – 49 = 59	Get the difference of the numbers. Thus,108 – 49 = 59
-108 + 49 = -59	Write the original equation. Since the absolute value of -108 is greater than the absolute value of 49, then we will copy the sign of -108 – which is negative.
	Therefore, the sum of -108 and 49 is -59.

How to subtract integers?

Subtraction generally refers to a reduction in value. However, since we have a positive and negative integers, the difference may increase or decrease in value.

Here’s a list of rules for subtracting integers:

• Subtracting a positive integer from a positive integer will result to a decrease in number. 
• Subtracting a negative integer from a positive integer will result to an increase in number.
• Subtracting a positive integer from a negative integer will result to a decrease in number. 

Subtracting positive integers

When subtracting two positive integers, there are two cases that we need to consider. 

• Case 1: If the minuend is greater than the subtrahend, then the final result is a positive number. 
• Case 2: If the subtrahend is greater than the minuend, then the final answer is a negative number.

Case 1: When the minuend is greater than the subtrahend.

To subtract two positive integers if the minuend is greater than the subtrahend, simply subtract the numbers like subtracting a whole number. 

The table below shows example of subtracting two positive integers if the minuend is greater than the subtrahend. 

Example	Answer
11 – 9 =	2
27 – 13 =	14
821 – 95 =	726

Case 2: When the subtrahend is greater than the minuend.

To subtract integers if the subtrahend is greater than the minuend:

1. Switch the position of number of the subtrahend and minuend.
2. Subtract accordingly.
3. Write negative sign on the final answer.

Example #1

What is the difference of subtracting 6 from 4?

Solution

Subtraction Process	Step-by-Step Explanation
4 – 6 =	Set up the subtraction process.
6 – 4 =	Switch the position of the minuend and subtrahend.
6 – 4 = 2	Get the difference of the numbers. Thus,6 – 4 = 2
4 – 6 = -2	Write the original equation. Since the value of the number in the subtrahend is greater than the number in the minuend, the result should be a negative integer.
	Therefore, the result of 4 – 6 is -2.

Example #2

Find the result of 13 – 25. 

Solution

Subtraction Process	Step-by-Step Explanation
13 – 25 =	Set up the subtraction process.
25 – 13 =	Switch the position of the minuend and subtrahend.
25 – 13 = 12	Get the difference of the numbers. Thus, 25 – 13 = 12
13 – 25 = -12	Write the original equation. Since the value of the number in the subtrahend is greater than the number in the minuend, the result should be a negative integer.
	Therefore, the result of 13 – 25 is -12.

Example #3

What is the difference if we subtract 162 from 57?

Solution

Subtraction Process	Step-by-Step Explanation
57 – 162 =	Set up the subtraction process.
162 – 57 =	Switch the position of the minuend and subtrahend.
162 – 57 = 105	Get the difference of the numbers. Thus,162 – 57 = 105
57 – 162 = -105	Write the original equation. Since the value of number in the subtrahend is greater than the value of number in the minuend, then the result should be a negative integer.
	Therefore, the difference of subtracting 162 from 57 is -105.

Subtracting negative integers

When subtracting two negative integers, there are two cases that we need to consider. 

• Case 1: If the absolute value of the minuend is greater than the absolute value of the subtrahend, the final result is a negative number. 
• Case 2: If the subtrahend is greater than the minuend, then the result will always be a positive number.

Case 1: When the absolute value of the minuend is greater than the absolute value of subtrahend.

To subtract two negative integers if the absolute value of the minuend is greater than the absolute value of the subtrahend:

1. Multiply minus sign by the sign of the subtrahend.
2. Get the absolute value of the integers.
3. Subtract the numbers. 
4. Write a negative sign on your final answer.

Example #1

What is the difference between -13 and -7?

Solution

Subtraction Process	Step-by-Step Explanation
-13 – (-7) =	Set up the subtraction process.
-13 + 7	Multiplying negative by negative, the sign will be positive.
| -13 | = 13 | -7 | = 7	Get the absolute value of the integers.
13 – 7 =	Subtract the numbers. Put the larger absolute value in the minuend and write the smaller absolute value on the subtrahend.
13 – 7 = 6	Subtract the numbers.
13 – (-7) = -6	Since the absolute value of the minuend is greater than the subtrahend, it follows that the sign of the final answer should be negative.
	Therefore, the result of subtracting -7 from -13 is -6.

Example #2

What is the result of subtracting -19 from -45?

Solution

Subtraction Process	Step-by-Step Explanation
-45 – (-19) =	Set up the subtraction process.
-45 + 19	Multiplying negative by negative, the sign will be positive.
| -45 | = 45 | -19 | = 19	Get the absolute value of the integers.
45 – 19 =	Subtract the numbers. Put the larger absolute value in the minuend and write the smaller absolute value on the subtrahend.
45 – 19 = 26	Subtract the numbers.
-45 – (-19) = -26	Since the absolute value of the minuend is greater than the subtrahend, it follows that the sign of the final answer should be negative.
	Therefore, the result of subtracting -19 from -45 is -26.

Case 2: When the absolute value of the subtrahend is greater than the absolute value of minuend.

To subtract two negative integers if the absolute value of the subtrahend is greater than the absolute value of the minuend:

1. Multiply minus sign by the sign of the subtrahend.
2. Get the absolute value of the integers.
3. Switch the position of the subtrahend and minuend.
4. Subtract the numbers. 

Example #1

What is the difference between -6 and -15?

Solution

Subtraction Process	Step-by-Step Explanation
-6 – (-15) =	Set up the subtraction process.
-6 + 15	Multiplying negative by negative, the sign will be positive.
| -6 | = 6 | -15 | = 15	Get the absolute value of the integers.
15 – 6 =	Switch the position of the minuend and subtrahend.
15 – 6 = 9	Subtract the numbers.
-6 – (-15) =  9	Since the absolute value of the subtrahend is greater than the minuend, it follows that the sign of the final answer should be positive.
	Therefore, the result of subtracting -15 from -6 is 9.

Example #2

Determine the result of subtracting -51 from -29.

Solution

Subtraction Process	Step-by-Step Explanation
-29 – (-51) =	Set up the subtraction process.
-29 + 51	Multiplying negative by negative, the sign will be positive.
| -29 | = 29 | -51 | = 51	Get the absolute value of the integers.
51 – 29 =	Switch the position of the minuend and subtrahend.
51 – 29 = 22	Subtract the numbers.
-29 – (-51) = 22	Since the absolute value of the subtrahend is greater than the minuend, it follows that the sign of the final answer should be positive.
	Therefore, the result of subtracting -51 from -29 is 22.

Subtracting positive and negative integers

When subtracting integers with unlike sign, there are two cases that we need to consider:

• Case 1: If the minuend is a positive integer and the subtrahend is a negative integer, the result will be positive.
• Case 2: If the minuend is a negative integer and the subtrahend is a positive integer, the result will always be negative.

Case 1: When the minuend is a positive integer and the subtrahend is a negative integer.

To subtract two integers with unlike signs:

1. Multiply minus sign by the sign of the subtrahend.
2. Add the numbers.

Example #1

Find the difference of subtracting -19 from 31.

Solution

Subtraction Process	Step-by-Step Explanation
31 – (-19) =	Set up the subtraction process.
31 + 19 =	Multiplying negative by negative, the sign will be positive.
31 + 19 = 40	Add the numbers.
	Therefore, the difference of subtracting -19 from 31 is 40.

Example #2

What is the difference of subtracting -53 from 23?

Solution

Subtraction Process	Step-by-Step Explanation
23 – (-53) =	Set up the subtraction process.
23 + 53 =	Multiplying negative by negative, the sign will be positive.
23 + 53 = 76	Add the numbers.
	Therefore, the difference of subtracting -53 from 23 is 76.

Case 2: When the minuend is a negative integer and the subtrahend is a positive integer.

To subtract two integers with unlike signs:

1. Multiply minus sign by the sign of the subtrahend.
2. Get the absolute value of numbers.
3. Add the numbers.
4. Write negative sign on the final answer. 

Example #1

What is the result of subtracting 15 from -8?

Solution

Subtraction Process	Step-by-Step Explanation
-8 – 15 =	Set up the subtraction process.
| -8 | = 8 | 15 | = 15	Get the absolute value of the given numbers.
8 + 15 = 23	Add the numbers.
-8 – 15 = -23	Write the original equation. By rule, the final answer should be a negative integer. Hence, we will simply put the negative sign on the final answer.
	Therefore, the result of -8 – 15 is -23.

Example #2

Determine the difference of subtracting 71 from -43

Solution

Subtraction Process	Step-by-Step Explanation
-43 – 71 =	Set up the subtraction process.
| -43 | = 43 | 71 | = 71	Get the absolute value of the given numbers.
43 + 71 = 114	Add the numbers.
-43 – 71 =  -114	Write the original equation. By rule, the final answer should be a negative integer. Hence, we will simply put the negative sign on the final answer.
	Therefore, the result of subtracting 71 from -43 is -114.

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