In this lesson let’s learn about what integers are and how adding and subtracting integers work.

An integer is always a whole number. It cannot be a fraction or a decimal. Also, integers can be positive or negative. Zero is also an integer.

*Examples of integers*

-1, 0, 5, -19, 456, -2587, 97 102, -116 708

*Examples of non-integers*

A non-integer is any number which is not a whole number. So, they can be either fractions or decimals.

3/5, 2.015, – 7/12, 0.00125, -14.133

A number line is the best way to understand positive and negative integers and also, zero (0) in its position.

A set of integers, denoted by Z, can be shown as follows;

**Z = {…, -2, -1, 0, 1, 2, … }**

In mathematics we have to deal with these positive and negative integers a lot. In your primary grades mostly you have to work with positive integers. However, when you start your secondary education, you will have to work with both positive and negative integers very often, especially, adding, subtracting, multiplying and dividing them.

Adding integers include adding both positive and negative integers. It can be adding only positive integers, or both positive and negative integers, or only negative integers.

You know adding only positive integers very well. In fact, you are a master in it.

*2 + 6 = 8*

*56 + 98 = 154*

*456 + 123 = 579*

*645 869 + 98 521 = 744 390*

Yeah…., it’s that simple.

So, now we only have to be careful with adding both positive and negative integers and only negative integers, because there are some adding and subtracting integers rules that you have to keep in your mind while performing these calculations.

*****Remember these rules are valid for both integers and non-integers, which are whole numbers, fractions, decimals etc.*

In this lesson we only work with integers, or whole numbers.

__Rule No. 1:__

When adding two integers, if the signs match we will add the given integers together and keep the sum with the same sign.

__Examples:__

**4 + 2**

**= 6**

Both are positive integers. So, we add the positive integers and the answer is also a positive integer, but we don’t usually show the positive sign. If there’s no sign before an integer it is always a positive integer or number. This is what you have been doing all this time.

Now, let’s follow the same rule with negative integers as well.

**– 4 + (- 2)**

**= -6**

Both are negative integers. So, we add the negative integers and the answer is also a negative integer.

Let’s see what’s happening here;

You have to **add -4 and -2.**

So, this is how we show it;

**– 4 + (- 2)**

**– 4 + (- 2)**

The brackets mean to multiply. So it says to multiply + and -.

**= – 4 – 2**

When we multiply + and – it is always – . Now apply the rule. Add the integers with the same sign.

**= – 6**

The answer is kept with the same sign.

__Explanation:__

__Step 1:__

When the positive sign, and the negative sign of the integer 2 get multiplied with each other it gives a negative sign.

**– 4 + (- 2)**

**= – 4 – 2**

__Step 2:__

Then add the integers with the same sign and keep the answer with the same sign.

**= – 4 – 2**

**= – 6**

This is how when you add two negative integers the answer comes with a negative sign.

****Remember;
When there are no brackets, parentheses, exponents or any other signs in between the integers, always add the integers if the integers have the same sign and the answer should also be kept with same sign.*

**4 + 5 = 9**

*OR*

**– 4 -5 = -9**

__Rule No. 2:__

When adding two integers with the signs that don’t match, one positive and one negative number, we will subtract the integers as if they were all positive and keep the answer with the larger integer’s sign.

Now let’s see what’s happening in these examples;

You have to **add 2 and -7.**

This how we show it;

**2 + (- 7)**

There’s no sign in front of 2 means it is a positive integer.

**2 + (- 7)**

The brackets mean to multiply. So it says to multiply + and -.

**= + 2 – 7**

When we multiply + and – it is always – . Then apply the rule. Two integers with different signs. Subtract the small integer from the large integer.

**= – 5**

The answer is kept with the larger integer’s sign.

__Explanation:__

__Step 1:__

When the positive sign, and the negative sign of the integer 7 get multiplied with each other it gives a negative sign.

**2 + (- 7)**

**= 2 – 7**

__Step 2:__

Then apply the rule. Two integers with different signs, one positive integer and one negative integer. Subtract the small integer from the large integer and keep the answer with the larger integer’s sign.

**= 2 – 7**

**= – 5**

This is how when you add integers with different signs, the answer comes with the larger integer’s sign.

Let’s take another example.

**Add -2 and +7.**

This how we show it;

**– 2 + 7**

There’s no sign in front of 2 means it is a positive integer.

**– 2 + 7**

Two integers with different signs. Subtract the small integer from the large integer.

**= 5**

The answer is kept with the larger integer’s sign.

__Explanation:__

There are two integers with signs that don’t match, one positive integer and one negative integer. Subtract the small integer from the large integer and keep the answer with the larger integer’s sign.

**– 2 + 7**

**= 5**

****Remember;
The rules of adding positive and negative integers can also be applied for adding positive and negative non-integers, such as fractions, decimals etc.*

Subtracting integers include subtracting both positive and negative integers. It can be subtracting only positive integers, or both positive and negative integers, or only negative integers.

You know subtracting the following very well.

*6 – 2 = 4*

*98 – 56 = 42*

*456 – 123 = 333*

*645869 – 98521 = 547 348*

Easy-peasy…! Right?

Likewise, we will have to subtract both positive and negative integers or only positive integers or only negative integers. However, still you must follow some rules when subtracting positive and negative integers.

__Rule No. 1:__

This is similar to adding integers with different signs.

Now, let’s look into how we get these answers.

Look at the following example:

**6 – 8 = ?**

In primary grades, this is something impossible to do, because primary graders have no idea about negative numbers.

If we show 6 – 8 =? In another way;

**6 – (+8) = ?**

It looks like subtracting two positive numbers.

Let’s see what’s happening here;

**6 – (+8)**

The brackets mean to multiply. So it says to multiply – and +.

**= 6 – 8**

When we multiply – and + it is always – . Two integers with different signs. Subtract the small integer from the large integer.

**= – 2**

The answer is kept with the larger integer’s sign.

Let’s take another example.

**Subtract -6 from -8.**

**-6 – (-8)**

The brackets mean to multiply. So it says to multiply – and -.

**= -6 + 8**

When we multiply – and – it is always +. Now you have two integers with different signs. Subtract the small integer from the large integer.

**= 2**

The answer is kept with the larger integer’s sign.

__Rule No. 2:__

This is similar to adding integers with the same sign.

So, how did we get these answers?

Look at the following example:

**6 – (-8) = ?**

Let’s see how to get the answer here;

**6 – (-8)**

The brackets mean to multiply. So it says to multiply – and -.

**= 6 + 8**

When we multiply – and – it is always +. Now you have two integers with the same sign. This is a simple positive integer addition.

**= 14**

The answer is also positive.

Let’s take another example.

**Subtract -6 from 8.**

**-6 – 8 **

We know the rule is to add the integers with the same sign.

**-6 – 8 = – 14**

Keep the answer with the same sign.

****Remember;
The rules of subtracting positive and negative integers can also be applied for subtracting positive and negative non-integers, such as fractions, decimals etc.*