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# How to add and subtract integers

Integers are whole numbers and their opposites and zero. Here you can learn how to add and subtract integers.

## Step by step guide to add and subtract integers

• Integers include: zero, counting numbers, and the negative of the counting numbers. ( <… , – 3, – 2, – 1, 0, 1, 2, 3 , …>)
• Add a positive integer by moving to the right on the number line.
• Add a negative integer by moving to the left on the number line.
• Subtract an integer by adding its opposite.

### Adding and Subtracting Integers – Example 1:

Solve. ( (- 2) – (- 6)=)

Solution:

Keep the first number, and convert the sign of the second number to it’s opposite. (change subtraction into addition). Then: ((- 2) color <+> 6=4)

### Adding and Subtracting Integers – Example 2:

Solve. ( 8 + (12 – 20)=)

Solution:

First subtract the numbers in brackets, (12 – 20= – 8)
Then: (8 + (- 8)= → ) change addition into subtraction: (8 color <-> 8=0)

### Adding and Subtracting Integers – Example 3:

Solution:

Keep the first number and convert the sign of the second number to its opposite. (change subtraction into addition). Then: ((-8) color <+>5= -3)

### Adding and Subtracting Integers – Example 4:

Solution:

First subtract the numbers in brackets, (4-8= -4)
Then: (10+(-4)= →) change addition into subtraction: (10 color <->4=6) Positive integers you already know as natural numbers, and we covered the addition and subtraction of natural numbers already, so we’ll concentrate on the negative integers instead. There are a few simple rules when it comes to the addition and subtraction of integers, and to change things up a little bit, we’ll present them as in the form of a listicle. So, here go the rules for the addition and subtraction of negative numbers.

1. A minus in front of a number changes the sign of the number.

To get a grasp of this rule, we’ll call a couple of old friends to our aid – the number line and the multiplication of natural numbers. Remember how multiplying a number by the number 1 gives you that same number as a result? Well, putting a minus in front of a number is shorthand for multiplying that number by -1. The distance from the origin point on the number line stays the same, but the minus shifts it to the opposite side of the number line. So, if we put a minus in front of a positive integer, we’ll get a negative version of that same integer. And if we put the minus in front of a negative integer, we’ll get its positive version as a result.

Using just mathematical language, that means that:

2. If a negative integer is behind an operator, it has to be surrounded by parentheses.

This one is here to avoid confusion, because the minus sign is also the operator for subtraction. If we put two operators next to each other, it is unclear if:

1. one of them is a sign, and not an operator
2. one of them is a typo, or
3. a number or a variable is missing between them.

To make things easier, a rule has been created to put negative integers into brackets (parentheses). That way, everybody knows that the minus is there on purpose and that it is a sign.

For example: \$ -3 + (-5) = -8 Rightarrow – 3 – 5 = -8\$

Although mistakes can be avoided during addition and subtraction by using rule number one, this rule will be indispensable during multiplication.

3. Adding two negative integers together will always give you a negative integer as a result.

A negative integer represents the distance from a single point positioned left of the point of origin on the number line to the point of origin itself. When we add two negative integers together, we basically get the sum of their distances. But, since both of them are positioned left of the point of origin on the number line, we keep that direction. Like this: 4. Subtracting a negative integer from another negative integer will only give you a negative integer as a result in some cases.

How come, you ask? Well, remember the first rule – a minus in front of a number changes the sign of the number. That also goes for negative integers. If we put a minus in front of a negative integer, it will turn into a positive integer. And when we add a positive integer to any number, we move to the right on the number line. So, what happens if the subtrahend (second number) is larger than the minuend (first number)? When it turns into a positive integer, we will move past the point of origin and get a positive integer as a result.

5. Subtracting a positive integer from a negative integer is basically the same as adding together two negative integers, and it will always give you a negative integer as a result as well.

Again, rule number one – the minus in front of a positive integer changes its sign. When it does, we’re basically adding two negative integers together, and we covered that in rule number two.

6. Adding a negative integer to a positive integer is basically the same process as the subtraction of two natural numbers.

This is an easy one. An expression like 5 + (-3) can be easily written as 5 – 3, and the result is the same:

The only thing we have to watch out for is if the negative number is larger than the positive number. In that case, the result will be a negative number.

7. The commutative property of addition and the associative property of addition that are valid for natural numbers are valid for integers as well.

The commutative property of addition and the associative property of addition are the same for both natural numbers and integers. Just be careful moving the signs around, and you’ll do fine.

Understanding these rules helps us to solve practical problems. Now we know how to solve the problem from the previous lesson. Let’s repeat the problem:

The air temperature today at noon was 39.2°F and by the evening the air temperature declined by 42.8°F. What was the air temperature in the evening?

Now we know that the temperature in the evening was -3.6°F.

If you want to practice a bit more, we prepared some worksheets for you. You can download them using the links below.

You already know how to add 3 + 4 and so on. But there are many ways to add integers. One way to add integers is by using a number line.

Example We always start at zero. Our first number is negative four (-4) so we move 4 units to the left. We then have plus negative three (-3) which is the same as subtracting 3 so we move 3 more units to the left. This gives us the value of negative seven, (-7).

We do the same thing if we have a positive integer, but instead we move to the right. You can also add integers and variables.

Example

When subtracting something from something we wish to find out the difference between the two numbers. When you subtract a negative number from any number the difference is even bigger. The distance from the seabed at a depth of 150ft and an airplane flying at 3000ft altitude at sea level is

\$\$3000-(-150) = 3000 + 150 = 3150ft.\$\$

Thus when we subtract negative numbers, we get:

Subtracting −3 is the same as adding 3.

If we have a plus sign before the parentheses then we will not change the signs within the parentheses

If we have a minus sign before the parentheses then we the signs within the parentheses will change.

Two negatives make one positive!

## Video lesson

Calculate -3 + (-6) on a number line

Integers are whole numbers used in counting, inclusive of negative, positive and zero numbers. The concept of integers was first started in the ancient Babylon and Egypt.

Integers can be represented on a numbers line, with the positive integers occupying the right side of zero and negative integers occupying the left side of zero. In Mathematics, integers are usually represented by ‘Zahlen’ symbol i.e. Z = <…, -4, -3, -2, -1,0,1,2,3, 4…>.

Arithmetic operations such as addition, subtraction, multiplication and division are applicable to integers. Addition and subtraction of integers helps to determine the sum or total and difference of the integers. Similarly, multiplication and division are used to compare and divide integers into equal parts. In this article, our focus is how to perform addition and subtraction with integers.

Integers are a special group of numbers that are positive, negative and zero, which are not fractions. Rules for addition and subtraction are the same for all, whether it is a natural number or an integer because natural numbers are itself integers

There are three possibilities when adding integers. They are:

• Addition between two positive integers
• Addition of two negative integers
• Addition between a positive and a negative integer.

The addition of two positive integers result in a positive answer. For example, +4 + (+3) = +7. Positive integers are never written with a positive sign and for this case, the answer is just 7.

When a positive and a negative integer are added, the numbers are subtracted without signs and the answer is assigned the sign of the larger integer. For example, to add 10 + (-15) = -5, the larger number in this case is 15 without the sign. Therefore subtract 15 and 10 to get 5 and assign the answer the sign of 15 which is -5.

When adding negative integers, the numbers are added and the sum assumes the sign of the original integers. For example, – 5 + (-1) = – 6.

## How to Subtract Integers?

Like addition, there are also three possibilities of subtraction of integers:

• Subtraction of two positive integers
• Subtraction of two negative integers
• Subtraction of a positive and a negative integer.

For the ease of subtraction, problems involving subtraction of integers can be modeled into the following transformation:

• The subtraction sign is converted into addition sign
• Take the inverse of the integer that comes after addition the sign.

For example, to subtract (-6) – (8) using the above transformation:

Step 1:

Covert the subtraction sign to an addition sign

Step 2:

Take the inverse of the integer that comes after the addition sign. The inverse of 8 is -8.

Add the integers and assign the sign of the larger integer

Example 1

Convert the subtraction sign to addition sign

Subtract and put the sign of larger integer

Example 2

Since both integers are negative;

Add the integers together and put the sign of the original integers to the result.

There is only one rule that you have to remember when subtracting integers! Basically, you are going to change the subtraction problem to an addition problem.

## Number 1 Rule for Subtracting Integers

When SUBTRACTING integers remember to ADD the OPPOSITE.

TIP: For subtracting integers only, remember the phrase:
“Keep – change – change”

What does that phrase mean?

Keep – Change – Change is a phrase that will help you “add the opposite” by changing the subtraction problem to an addition problem.

• Keep the first number exactly the same.
• Change the subtraction sign to an addition sign.
• Change the sign of the last number to the opposite sign. If the number was positive change it to negative OR if it was negative, change it to positive.

Let’s take a look at a few examples to help you better understand this process.

## Example 1 – Subtracting Integers Using Keep-Change-Change

Here’s an example for the problem: 12 – (-6) = ?

Keep 12 exactly the same. Change the subtraction sign to an addition sign. Change the -6 to a positive 6. Then add and you have your answer! Notice how we rewrote this subtracting problem as an addition problem, and then utilized our addition rules!

If you rewrite every subtraction problem as an addition problem, then you will only have to remember one set of rules.

Let’s take a look at another example using the keep-change-change rule.

## Example 2 – Subtracting Integers by Rewriting as Addition As you can see, if you rewrite you subtracting problems as addition problems, you will be able to easily find the difference using your addition rules.

Using the Keep-Change-Change rule is a great way to remember how to rewrite the subtraction problem as an addition problem.

If you are continuing your study of integer rules, be sure to check out the multiplication and division rules for integers.

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Wondering how to perform subtraction on integers, don’t worry this page will help you to understand everything that you need to know. This page will explain the rules, procedure, and also few examples. Subtraction represents the value that is been removed from the existing value. For example, if a basket has 4 fruits and now if we remove two fruits from the basket then the basket will be left with only two fruits, this shows subtraction.

### Rules to Subtract Integers | How to Subtract Integers?

If you are familiar with the concept of adding two integers, then I’m sure you can perform subtraction very easily.

While subtracting any integer we need to follow a few simple rules. That is first we need to convert the given subtraction problem into an addition problem. This means we have to convert the subtraction sign into an addition sign.

1. Take the first integer and change its operation from subtraction to addition
2. Take the second number and get its opposite sign
3. Then finally proceed with the regular addition process

### Subtracting Integers Possibilities

Whenever we are adding two integers, we will come across the below-mentioned possibilities.

• Subtracting two positive integers
• Subtracting two negative integers.
• Subtracting one positive integer and one negative integer.

### Subtracting Integers Examples

Example 1: Subtracting two positive integers

Solution:

Initially, we have to change the operation

That means subtraction becomes an addition

Now we have to change the sign of the second integer

Which gives us 5 – 8 = 5 + (-8)
= -3.

Example 2: Subtracting one positive integer and one negative integer

Solution:

First of all, we have to change the operation

That means subtraction becomes an addition.

Now we have to change the sign of the second integer

Which gives us 15 – (-4) = 15 + (4)
= 19

Example 3: Subtracting one positive integer and one negative integer

Solution:

First, we have to change the operation

That means subtraction becomes an addition

Now we have to change the sign of the second integer

Which gives us (-32) – (7) = (-32) + (-7)
= -39

Example 4: Subtracting two negative integers.

Solution:

First, we have to change the operation

That means subtraction becomes an addition

Now we have to change the sign of the second integer

Which gives us (-21) – (-3) = (-21) + (3)
= -18

### FAQs on Subtraction of Integers

1. Can we subtract two integers which are having different signs?

Yes, we can subtract two integers having different signs.

2. What is the result for the subtraction of (5) – (-5)?

On subtracting (5) – (-5) we will get the result 10.

3. How do you subtract an integer without a number line?

We can subtract an integer using its absolute value.

4. What happens when you subtract a negative sign from a positive sign?

We can use the number line as a model to help us visualize adding and subtracting of signed integers. Just think of addition and subtraction as directions on the number line. There are also several rules and properties that define how to perform these basic operations.

To add integers having the same sign, keep the same sign and add the absolute value of each number.

To add integers with different signs, keep the sign of the number with the largest absolute value and subtract the smallest absolute value from the largest.

Subtract an integer by adding its opposite.

Watch out! The negative of a negative is the opposite positive number. That is, for real numbers,

Here’s how to add two positive integers:

If you start at positive four on the number line and move seven units to the right, you end up at positive eleven. Also, these integers have the same sign, so you can just keep the sign and add their absolute values, to get the same answer, positive eleven.

Here’s how to add two negative integers:

If you start at negative four on the number line and move eight units to the left, you end up at negative twelve. Also, these integers have the same sign, so you can just keep the negative sign and add their absolute values, to get the same answer, negative twelve.

Here’s how to add a positive integer to a negative integer:

If you start at negative three on the real number line and move six units to the right, you end up at positive three. Also, these integers have different signs,

so keep the sign from the integer having the greatest absolute value and subtract the smallest absolute value from the largest.

Subtract three from six and keep the positive sign, again giving positive three.

Here’s how to add a negative integer to a positive integer:

If you start at positive five on the real number line and move eight units to the left, you end up at negative three. Also, these integers have different signs, so keep the sign from the integer having the greatest absolute value and subtract the smallest absolute value from the largest, or subtract five from eight and keep the negative sign, again giving negative three.

To subtract a number, add its opposite:

Because they give the same result, you can see that subtracting eight from five is equivalent to adding negative eight to positive five. The answer is – 3.

To subtract a number, add its opposite:

Because they give the same result, you can see that subtracting negative six from negative three is equivalent to adding positive six to negative three. The answer is 3.

If you know how to add integers, I’m sure that you can also subtract integers. The key step is to transform an integer subtraction problem into an integer addition problem. The process is very simple. Here’s how:

## Steps on How to Subtract Integers

Step 1 : Transform the subtraction of integers problem into the addition of integers problem. Here’s how:

• First, keep the first number (known as the minuend).
• Second, change the operation from subtraction to addition.
• Third, get the opposite sign of the second number (known as the subtrahend)
• Finally, proceed with the regular addition of integers.

Step 2 : Proceed with the regular addition of the integers.

Note that you will eventually add integers. So for your convenience, here’s the quick summary of the rules on how to add integers.

• Case 1: Adding two integers having the same sign

Add their absolute values then keep the common sign.

• Case 2: Adding two integers with different signs

Subtract their absolute values (larger absolute value minus smaller absolute value) then take the sign of the number with the larger absolute value.

## Examples of Integer Subtractions

Example 1: Subtract the integers below.

We will need to transform the problem from subtraction to addition. To do that, we keep the first number which is –13 , change the operation from subtraction to addition, then switch the sign of + 4 to – 4.

The final step is to proceed with regular addition. Add their absolute values. Then we determine the sign of the final answer. Since we are adding integers with the same sign, we will keep the common sign which in this case is negative.

Example 2: Subtract the integers below.

Just like before, convert a subtraction problem to an addition problem. Positive 9 remains, switch the operation from “minus” to “plus” then get the opposite sign of the subtrahend (second number) from negative to positive.

Now, let’s add them. We are adding two positive integers so we expect the answer to be positive as well because the common sign is positive.

Example 3: Find the difference of the two integers.

I hope you are already getting the hang of it. Let’s make this an addition of integers problem first then proceed with regular addition of integers with different signs.

So we subtract their absolute values first then get the sign of the number with larger absolute value.

Subtracting the absolute values, we have 24 minus 19 which gives us +5. But the final answer is – 5 because the sign comes from – 24.

You might think of integers as just ordinary numbers, like 3, -12, 17, 0, 7000, or -582. Integers are also called whole numbers because they aren’t divided into parts of numbers, like fractions and decimals are. Read this article to learn everything you need to know about adding and subtracting integers, or skip to the section you need help with.

## Steps

### Adding and Subtracting Positive Integers with a Number Line

1. Understand what a number line is. Number lines turn basic math into something real and physical that you can see in front of you. By just using a few marks and some common sense, we can use them like calculators to add and subtract numbers.
• Your math book might call this point the origin, since it’s where numbers originate, or start from.
• Don’t worry about making the spacing perfect – as long as you’re close enough that you can tell what it’s supposed to mean, the number line will work.
• The left side is the side at the beginning of a sentence.
• The example image shows a number line from -6 to 6.
• An integer is just another way of saying a “whole number”. Fractions like 1/2 (one half) are only part of a number, so they are not integers. Same with a decimal like 0.25 (zero point two five); decimals are not integers.
• Do you think this is too easy? If you’ve done any addition at all, you probably know the answer to 1+2. That’s good: if you know the answer it will be easier to understand how number lines work. Then you can use a number line for more difficult addition problems, or to prepare you for more difficult math like algebra.

### Adding and Subtracting Negative Numbers with a Number Line

1. Learn what a number line is. If you don’t know how to make a number line, go back to Adding and Subtracting Positive Numbers with a Number Line to learn how.
• For example, let’s add 1 and -4. In the standard, familiar number writing you’re used to, this is just:
1 + (-4)
On a number line, we start at 1, move 4 spaces left, and end up at -3.
• For example, consider -4. When we add -4 to 1, it decreases 1 by 4. We can “say this in math” by writing

We’d write this on a number line, as starting with our pointer at 1, then adding a move 4 spaces to the left (in other words, adding a -4). Since it’s an equation, one thing equals another – so the reverse works too:

• On a number line, we start with our pointer at 5, decrease by 8, and arrive with our pointer at -3.

Now we’ll only be moving 7 left, so we have

• We already know that 5 – 8 = -3, so let’s take 5 – 8 out of our equation now and put in -3:
5 – (8 – 1) = 5 – 7 = -3 + 1
• We already know what 5 – (8 – 1) is — it’s going one space less than 5 – 8. Our equation can show the fact that 5 – 8 gives us -3, and going one space short gives us -2. Our equation can be written like this now:

We can express this as a simple, more general rule for writing math:

first number plus a second number = first number minus (negative second number)
Or, in more simple terms like you’ve probably heard in a math class:

A number line is a visual representation of numbers on a straight line.

## Answer: Integers can be added and subtracted without a number line in different ways depending upon the sign of the numbers.

There are some rules for doing these operations. Before we start learning these methods of integer operations, we need to remember a few things. If there is no sign in front of a number, it means that the number is positive.

## Explanation:

Case 1: Signs are the same

If the signs are the same, add and keep the same sign.

• (+) + (+) = add the numbers and the answer is positive

Example : 2 + 5 = 7

• (‐) + (‐) = add the numbers without the negative sign and place the negative sign before the answer.

Example : (-5) + (-4) = -9

Case 2: Signs are different

If the signs are different, subtract the numbers and use the sign of the larger number

• (+) + (‐) = subtract the numbers and take the sign of the bigger number

Example: 7 + (-3) = 4

• (‐) + (+) = subtract the numbers and take the sign of the bigger number

Example: (-9) + 6 = -3

### Subtracting Integers Rule:

To subtract a number from another number, the sign of the number (which is to be subtracted) should be changed and then this number with the changed sign should be added to the first number.

• (+) – (+) = Change the sign of the number to be subtracted and add them up. The result takes the sign of the greater number

Example: (+6) – (+2) = (+6) + (-2) = 6 – 2 = 4

• (-) – (-) = Change the sign of the number to be subtracted and add them up. The result takes the sign of the greater number

Example: (-9) – (-6) = (-9) + (+6) = -9 + 6 = -3

• (+) – (-) = Change the sign of the number to be subtracted and add them up. Result is always positive

Example: (+5) – (-3) = (+5) +(+3) = 5 + 3 = 8

• (-) – (+) = Change the sign of the number to be subtracted and add them up. Result is always negative

Example: (-7) – (+2) = -7 – 2 = -9

Thus, we see the rules to add and subtract integers without a number line