**Introduction**

Addition is one of the most important arithmetic operations used in mathematics. It is the method of calculating the total of two or more numbers to know the sum of the numbers. We use addition not just while studying mathematics but in solving our day to day problems as well. Before we learn how to add numbers let us understand the definition of addition.

**What is Addition?**

Addition, in mathematics, can be defined as the process of combining two or more numbers together to make a new total or sum. The numbers to be added together are called **addends** and the result thus obtained is called the **sum**.

**Addition of a Single Digit Number with a Single Digit Number**

We know that the single digit numbers are 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9. This is the order in which we count the numbers where 2 comes after 1, 3 comes after 2 and so on**. A single digit has the place value of a unit in the place value system of the numbers.**

**Addition by Forward Counting**

Let us consider a situation where you have 3 chocolates. Your brother gives you 4 more chocolates. How many chocolates do you have now?

In order to add 4 chocolates to the 3 chocolates that you already have, starting from 3, you will move 4 places to the right of the counting of numbers. So, 4 points after 3, in the number system is 7. Hence, 3 + 4 = 7

Let us take another example.

Let us add 2 to 4. Again if we want to add 2 to 4 we will have to move 2 numbers to the right of 4. So, moving 2 points ahead of 4, we will get, 6 which means that 4 + 2 = 6.

Now, that we have understood how to add a single digit number to another single digit number, let us learn how to add a single digit number to a two digit number.

**Addition of a Single Digit Number with a Two Digit Number**

In the counting of numbers, two digit numbers start from 10. Then we have 11, 12, 13, 14, 15, ….. and so on. It is important to note here that two digit number “ab” can be written in the form 10 x a + b, where a has the ten’s place value and b has the unit value in the number system.

**Addition with No Carrying Forward**

For adding a Single Digit with a Two Digit Number, the digit at the one’s place value of first number will be added to the digit at the one’s place of the second number.

Let us understand it using an example.

Suppose we want to add 5 and 13.

5 – 5 is at the unit’s place in the number.

13 – 1 is the ten’s place while 3 is the unit’s place.

So, we add the unit’s digit of both the number as shown below –

Here we can see that 5 + 3 = 8 < 10, so the result obtained was a single digit number at the unit’s place. But if we were to add the numbers 5 + 16, we would have got 5 + 6 = 11, then how would we have written 11 in the answer. This is where the concept of carrying forward comes into force.

**Addition with Carrying forward**

Let us consider two numbers 5 and 16. How will we add these two numbers? Let us find out.

5 = 0 tens + 5 ones

16 = 1 tens + 6 ones.

Therefore,

5 + 16 =

0 tens + 5 ones

+1 tens + 6 ones

——————–

1 tens + 11 ones

——————–

Now, 11 ones = 1 tens + 1 ones

Therefore, we have 5 + 16 = 1 tens +1 tens + 1 ones = 2 tens + 1 ones.

Hence, 5 + 16 = 21

**Addition of a Two Digit Number with a Two Digit Number**

Let us now learn how to add two numbers of two digits. We have just discussed that in a two digit number, we have two digits, one at the ten’s place and the other at the one’s place. Let us see how to add these numbers.

**Addition with No Carrying Forward**

Consider two numbers 12 and 16. Suppose we want to add these two numbers. How will we find their sum? Let us find out.

Recall that,

12 = 1 tens + 2 ones

16 = 1 tens + 6 ones

Therefore,

The sum of 12 and 16 will be given by adding the respective one’s digits with each other and the ten’s digits with each other. We will have,

12 +16 = =

1 tens + 2 ones

+ 1 tens + 6 ones

———————

2 tens + 8 ones

———————

**Hence, 12 + 16 = 28**

**Addition with Carrying forward**

Here we can see that 2 + 6 = 8 < 10, so the result obtained was a single digit number at the unit’s place. But if we were to add the numbers such as 8 + 9, we would have got 8 + 9 = 17, then how would we have written 17 in the answer. This is the same situation, we discussed while understanding two to add a single digit and a two digit number. Here as well, we will proceed in the same manner, i.e. by carrying forward.

Let us consider two numbers 28 and 16. How will we add these two numbers? Let us find out.

We will follow the following steps to add these numbers.

- We know that 28 = 2 tens + 8 ones and 16 = 1 tens + 6 ones
- Now, 8 ones + 6 ones = 14 ones. As the sum of the digits at the one’s place exceeds 9, you must carry ones into tens.
- 14 ones = 10 ones + 4 ones = 1 tens + 4 ones
- Write 4 under ones column and carry
- Add the tens
- 1 ten ( that was carried over ) + 2 tens + 1 ten = 4 tens.
- Thus 28 + 16 = 44

**Addition Using a Number Line**

Let us now understand the concept of adding one or digit numbers using the number line.

We move to the right on our number line for adding two numbers. The following example shall help you understand how to add two numbers to a number line.

For example, you want to add 3 + 2

**Steps involved – **

1. First draw a number line having both negative as well as positive numbers.

2. Mark the first number 3 on this number line.

3. Move 2 points right as we have to add the numbers.

4. The number we have our point on is our answer, i.e. 5

Hence 3 + 2 = 5

**Addition Using the Hundred Grid**

Another way of adding two digits numbers is by using the hundred grid. A hundred grid is where you mark numbers from 1 to 100 in the following manner –

Let us use the above grid to add two numbers. For this let us have two numbers 57 and 16 and we want to find their sum.

We shall use the following steps to add these two numbers using the hundred grid –

- Mark the bigger number on the hundred grid. In this case, the bigger number is 57. So we ill mark it on the number grid. We will get,

- If the number to be added is more than 9, break it into tens and ones. In our case, we have the number 16. So, breaking it into tens and ones, we have, 16 = 1 tens + 6 ones.
- Jump as many tens as in the second number. The number 16 has 1 tens, so we will jump 1 tens to reach the number 67 and mark it on the grid as shown below –

- Now, we will move forward as many ones as in the second number. 16 has 6 ones, so, we will move 6 numbers forward from 67 to reach 73 and show it on the grid as below –

- The number that we have reached is our answer. So, 57 + 16 = 73

**The Addition Table**

The understanding of the addition table is very important for better learning of the addition of small numbers. The addition table is a table that shows the results of the addition of a single digits number with another single number.

How do we use the above table to add two single digit numbers? Let us find out.

Suppose we want to add the number 6 and 9.

In order to use this table, of the two given numbers, we will mark one number of the horizontal ( rows ) top of the table and the other in the vertical (columns ) headings of the table. For instance, we will mark 6 on the vertical side and 8 on the side of the row to get –

Now we will check the intersection of these two numbers on the matrix table. In this case, we can see that they intersect at the number 14. Hence, 8 + 16 = 14

**Some Important Results**

- Zero added to any number gives the same number, for example, 0 + 7 = 7.
- When you add 1 to a number, the answer is the number “ just after” example, 5 + 1 = 6.
- Changing order of numbers while adding does not change the answer, for example 4 + 3 = 3 + 4 = 7.
- “ + “ is the sign used for addition.
- 5 + 3 = 8 is read as “ five plus three is equal to eight”.
- The answer of the addition is called the sum.

**Key Notes and Summary**

- Addition, in mathematics, can be defined as the process of combining two or more numbers together to make a new total or sum. The numbers to be added together are called
**addends**and the result thus obtained is called the**sum**. - A single digit has the place value of a unit in the place value system of the numbers.
- For adding a Single Digit with a Two Digit Number, the digit at the one’s place value of first number will be added to the digit at the one’s place of the second number.
- The addition table is a table that shows the results of the addition of a single digits number with another single number.

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