Addition and subtraction of whole numbers. Exercises solved

In this section we are going to see step by step how to add and subtract whole numbers in great detail.

Adding and subtracting whole numbers is a source of many errors and is also a concept of great utility in algebra, so it is very important that you have it very clear. It must be a fundamental pillar in your mathematical base.

We are used to add and subtract natural numbers, or what is the same, only with positive numbers. But with integers, we will have to operate negative numbers and that at first puzzles us a little.

The difference between adding and subtracting whole numbers from when we add and subtract with natural numbers is that we now have negative numbers.

We must have very clear, what are the integers and how they are ordered.

We are going to see it little by little so that you can assimilate concepts.

Sum of positive integers

Adding positive numbers has no major complication. All you have to do is add the numbers, just like adding natural numbers. For example:

Addition and subtraction of integers

What you are really doing is adding the absolute values of the numbers and then adding the plus sign to the result, or rather, leaving it the same because the plus sign is not written.

Keep this concept because it is key to understand the next section.

Negative integer subtraction

Subtracting negative numbers is a very confusing concept, because it is very difficult for us to understand that the result is greater in absolute value than the numbers we are subtracting.

To understand it better, let’s make this subtraction of negative numbers:

Addition and subtraction of whole numbers solved exercises

To subtract negative numbers, we have to follow the same procedure as to add positive numbers: we have to add their absolute values and then add the minus sign to the result. That is, we perform 8+3+1=12 and add the minus sign:

addition and subtraction of integers with parentesis exercises solved

Subtracting negative numbers will result in an even more negative number, or what is the same with the largest absolute value. It is the same concept as adding positive numbers. The result is another, more positive number.

Remember that in the previous lesson, negative numbers grew in absolute value the further to the left they were on the number line and positive numbers grew in absolute value the further to the right of the line we are.

Before we continue, let’s make it clear why it is necessary to enclose negative numbers in parentheses when we operate with them and how to act with these parentheses.

Parenthesis in integers

In general, negative numbers are represented with the minus sign in front:

addition and subtraction exercises of positive and negative integer numbers
As they already have a sign, when we perform operations with them, if we write them like this, as it is, without adding anything else, it would create confusion to have two consecutive signs, as for example in this sum:

how integers are added

Therefore, it is necessary to enclose them in parentheses, to distinguish between the sign of the operation and the sign of the negative number:

sum and subtract integers

On the other hand, we could also enclose positive numbers in parentheses:

how to add and subtract integers

But since normally the positive sign is not written, it is not necessary to enclose them in parentheses, since we would not have the problem of having two consecutive signs:

how to add and subtract integers

How are parentheses of integers removed?

To solve an operation with integers enclosed in parentheses, first remove them, as we will see later.

The parentheses of integers can be removed when they are preceded by a plus or minus sign. We have two signs and we have to keep one of them.

To do this, you must use these rules:

  • When an integer enclosed in parentheses is preceded by a plus sign, the parentheses are removed and the integer sign is retained. For example:

sum and subtract integers

The sign we have left is the same as the one we had inside the parenthesis.

If we do not have any sign in front of the parenthesis, it is equivalent to being preceded by a plus sign and the sign is kept. In other words, we can remove parentheses directly:

how to add and subtract integers

  • When an integer enclosed in parentheses is preceded by a minus sign, the parentheses are removed and the integer sign is changed. For example:

how whole numbers are added and subtracted

Now, we have a number with the opposite sign to the original sign.

Do you realize what happened? The result of eliminating the parenthesis is to obtain a number with only one sign. You will see it clearer when we apply it in the addition and subtraction, do not worry.

Now we are going to continue with the additions and subtractions of integers.

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Add and subtract integers

We are now going to learn how to solve additions and subtractions where positive and negative numbers appear. In operations with integers, negative numbers will be enclosed in parentheses.

We are going to see how it is solved step by step with an example:

sum of integers

First of all, we have to remove the parentheses, taking into account the sign in front of the parentheses, as we have just seen:

as integers are added and subtracted

So the operation is as follows:

how to add integers

Now, on the one hand, we add the positive numbers and on the other hand, we subtract the negative numbers:

exercises of resolved integers

And we have left:

subtract from integer numbers

Finally, we subtract the absolute values of the positive number and the negative number and write the sign of the greater (the greater is 16, then the sign of the result is positive, which is not written):

subtract integers with parentheses

And we have arrived at the result.

Let’s see another example and solve it a little faster:

exercises of resolved integers

First, we remove parentheses:

Addition and subtraction of integer positive and negative numbers resolved exercises

Now we add positive numbers, subtract negative numbers, and we’re left:

subtract from integer numbers

Now we subtract absolute values and put the sign of the greater (in this case is greater the absolute value of -6 and therefore the result is negative):

add integers

And we would have the solution.
The fact that the negative number is larger in absolute value than the positive number, is a concept that usually causes much confusion, since our mind tends to make a subtraction as we have always known and is blocked.

For example, we block ourselves when performing this operation:

how to subtract integers

To solve it, we must follow the step of subtracting the absolute values and putting the sign of the major:

subtract integers

However, this step is performed automatically when the positive number is in front and is greater than the negative number:

subtract positive and negative integers

But in reality, we have subtracted the absolute values and placed the greater sign in the result, without realizing it.

It also causes a lot of confusion when a positive number is subtracted from another negative of greater absolute value:

additions and subtractions of integers

And in this case, it’s not automatic even if we turn it around:

sums with integers

To arrive at the result, we only have to subtract the absolute values and put the greater sign:

addition and subtraction with signed numbers

Always subtract the absolute values and put the sign of the major
.

General procedure for adding and subtracting integers

To finish, let’s summarize the general procedure for adding and subtracting integers:

  1. Remove parenthesis taking into account the sign in front.
  2. Add positive numbers and subtract negative numbers
    • On the one hand we add the absolute value of the negative numbers and then we add the minus sign
    • On the other hand we add the absolute value of the positive numbers and then we add the plus sign
  3. Subtract absolute values and put the major sign

Integer exercises

Solve the following operations:

how to add negative numbers

Solution:

exercises solved whole numbers

How to add negative and positive numbers

how to resolve integer numbers

restas con numeros integos

integer numbers sum and subtract

exercises with integer numbers

sumas y subtas con numeros inos

subtract with integer numbers

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