The concept of addition helps us to find the total number of objects. Two objects or numbers are said to be added to obtain a larger number called the SUM.

There are many ways to add numbers or objects.

We can use our fingers to perform addition when the numbers that we need to add are less than or equal to 5.
For example, to add 5 and 3 you can open all 5 fingers in one hand and three fingers in the other hand. The sum of 5 and 3 shows the total number of fingers opened on both the hands, so, 5 + 3 = 8.

Example 1: How many burgers are there?

Solution: There are 6 + 4 = 10 burgers.

Example 2: How many cubes are there?

Solution: There are 5 + 2 = 7 cubes.

In the above two examples, you can observe that there are two numbers or objects that are added. They are referred to as “addends” and each addend is separated by the “+” sign; equal “=” symbol implies that the sum of the addends and the result obtained are equal.

Addition can be defined as a process of summing up together. The process involves the addition of two or more than two numbers; the order of numbers arranged does not affect the result.

An addition number sentence or addition sentence is used to express the process mathematically. This representation includes two or more addends, plus signs, an equal symbol, and the sum of addends.

For example, the sum of 9 and 3 is 12, i.e., 9 + 3 = 12. When you interchange the positions of addends, i.e., 3 + 9 the end result remains the same, 3 + 9 = 12. It is read as “three plus nine equals twelve”. Representation 9 + 3 = 12 is an addition sentence.

Example 3: Count the number of objects in each set. Add them and write the total number of objects in the given box.

Solution: Count the number of objects in each set of objects. For example, set 1 has 4 ice creams, set 2 contains 3 ice creams, and set 3 contains 4 ice creams. So, in total, we have 4 + 3 + 4 = 11 ice creams. The same procedure should be followed to find the remaining objects too.

## Addition using Number Line

The concept of adding small whole numbers is easy to understand using a number line. Let’s use an example to understand the process of addition using a number line, as we are already familiar with the notion.

Example 4: Find the sum of 3 and 6 using the number line.

Solution: To understand the process of addition using the number line, follow the steps shown.

Step 1: Draw a number line starting from 0 to 15.

Step 2: Mark the point where one of the two addends is located on the number line or you can also move from 0 the number of steps equal to the chosen addend. Let us choose 3. So, either mark 3 on the number line or move three steps from 0 to 3.

Step 3: To add the other number to the first, move the number of steps in the positive direction equal to the second addend given. Here, it is 6. To add 6 to 3, move 6 steps in the positive direction starting from 3. After 6 steps, the number obtained is 9. Hence, the sum 3 + 6 = 9.

Example 5: Find the sum of – 4 and –3 using the number line.

Solution: To understand the process of addition of two negative numbers using the number line, follow the steps below.

Step 1: Draw a number line starting from -10 to 10. (There is no restriction to take the same numbers on the number line. We can take any numbers that help us to find the answer.)

Step 2: Mark the point where one of the two addends is located on the number line or you can also move from 0, the number of steps equal to the chosen addend. Let us choose – 4. So, either mark – 4 on the number line or move four steps in the negative direction from 0 to – 4.

Step 3: To add the other number to the first, move the number of steps in the negative direction (because the second addend is a negative number) equal to the second addend given. Here, it is –3. To add –3 to –4, move 3 steps in the negative direction starting from –4. After 3 steps, the number obtained is –7. Hence, the sum (–4) + (–3) = –7.

The same procedure is followed to add more than two numbers (positive or negative) using the number line.

Another way of making addition fun is number bond. Number bonds are defined as a pair of numbers that can be combined together to form another number. Number bonds are represented in the pictorial form as shown below.

Let us see some of the examples to know the number bonds function.

The pair of numbers inside the number bonds can differ. For example, 24 can have pair of numbers 20 and 4. Depending upon one number in the pair, the value of the other number is decided.

Try it yourself!

I. Find the missing numbers in number bonds?

II. Write the addition number sentence for the given jumps.

Note: The order of addends does not affect the sum. For example, 5 + 3 = 8. When you change the position of addends, 3 + 5, the sum remains the same. Hence, 5 + 3 = 3 + 5 = 8.

## Addition using Compensation Method

We add numbers using this strategy by making one of the numbers as close to its tens or hundreds as possible. By compensating one number (removing a number) and adding to the other, the number can be converted to tens or hundreds.

In other words, the compensation method is a process of removing some number from one addend and adding the same number to the other addend. Let’s look at a few examples.

Example 6: Add 29 and 17.

Solution: The nearest tens to 29 is 30. Let us take 1 from 17 and add it to 29. Thus, 29 become 29 + 1 =30 and 17 become 17 – 1 = 16.
30 + 16 = 46

Example 7: Add 234 and 43.

Solution: The nearest tens to 234 is 230. Let us take 4 from 234 and add it to 43. Thus, 234 become 234 – 4 = 230 and 43 become 43 + 4 = 47.
230 + 47 = 277

## Addition using Decomposition Method

Decomposing means to separate or divide apart. One type of this strategy is to divide the number to make addition easier. The other type is to divide the number into place values and then add the respective place value numbers together.

Example 8: Add 347 and 259.

Solution: To add the given numbers, decompose the number by their place values and then add them.

347 = 300 + 40 + 7

259 = 200 + 50 + 9

Add the respective place value numbers.

347 + 259 = (300 + 200) + (40 + 50) + (7 + 9)

347 + 259 = 500 + 90 + 16 = 606

## Observations on addition

Certain observations can be made about the type of number we choose or the type of sum we obtain while performing addition:

• When two natural numbers or two whole numbers are added together, the result is a natural or whole number.
For example, consider numbers 10 and 8, the sum 10 + 8 = 18.
• A positive integer is formed by adding two positive numbers.
For example, consider numbers 5 and 7 that are positive integers. The sum 5 + 7 = 13 is also a positive integer.
• A negative integer is formed by adding two negative integers.
For example, consider numbers –6 and –4 that are negative integers. The sum (–6) + (–4) = –10 which is a negative integer.
• Subtract the numbers and use the sign of the greatest integer number when adding positive and negative integers.
•

## Properties of Addition

Three properties that satisfy the addition of numbers are

1. Commutative Property
2. Associative Property
3. Identity Property

Commutative Property
The sum is not affected by changing the sequence of addends.
Example: 1 + 8 = 8 + 1

Associative Property
The sum is not affected by changing the order of the addends.
Example: (4 + 2) + 5 = 4 + (2 + 5)

Identity Property
The sum of 0 and any other number is always the number itself.
Example: 0 + 5 = 5

Activity 1
Draw a line to match the equations in both columns.

## Addition by Grouping and Regrouping Numbers

When there are three or more numbers with higher values, the grouping method makes addition easier. The sum is unaffected by the sequence in which the numbers are added while grouping.

Example 9: Add 3478, 2469, and 1302.

Solution: Let us group the numbers as (3478 + 2469) + 1302.

First, add numbers within the parenthesis that are 3478 + 2469.

Hence, (3478 + 2469) + 1302 = 7249.

A method used in regrouping is similar to that of decomposition. In this method, the numbers are rearranged in place values to perform the addition. Numbers are rearranged as ones, tens, hundreds, and so on. This method is used to carry the addition process for any sized (number of digits) numbers.

Example 10: Add 134 and 86 using the regrouping method of addition.

Solution: Regroup the numbers 134 and 86.

134 = 1 hundreds + 3 tens + 4 ones = 100 + 30 + 4

86 = 8 tens + 6 ones = 80 + 6

Now, let us add them by grouping their place values.

100 + (30 + 80) + (4 + 6) = 100 + (110) + (10)

100 + (100 + 10) + 10 = 2 hundreds + 2 tens = 200 + 20 = 220

134 + 86 = 220

It is possible to add small numbers horizontally. As the values of numbers increase, we use a vertically adding process. To make the addition process easier, we use the place value concept.

Let us consider examples to understand adding numbers vertically.

Example 11: Find the sum of 53 and 25.

Solution: To find the sum, first we write the given numbers using place values as follows:

Add the numbers in the place values and write the sum below them. The sum of digits in ones place is 3 + 5 = 8; the sum of digits in tens place is 5 + 2 = 7. Therefore, the sum of 53 and 25 is 78, which can be written as 53 + 25 = 78.

Example 12: Find the sum of 276 and 94.

Solution: The numbers can be written in place value as shown

Add the numbers in the place values and write the sum below them. Adding 6 and 4 we get 10, we have to write only 0 under ones place and carry 1 to the next place value digit. The number in thousands place which is 7 becomes 7 + 1 = 8. The new digit in tens place of 278 while adding 94 is 8.

Now, add 8 and 9 in tens place, 8 + 9 = 17. Write 7 under tens place value and carry 1 to the next place value digit. Thus, the new digit in hundreds place will be 2 + 1 = 3.

Hence, the required sum of 276 and 94 is 276 + 94 = 370.

Please note that the same procedure can be used for higher numbers also.

## Addition with stories

In a real-life, we use addition in many circumstances. For example; amount paid for groceries, the weight of vegetables and fruits, distance traveled, bank balance, and so on.

Follow the steps to solve word problems.

1. Identify the Problem.
2. Gather Information.
3. Model it mathematically.
4. Solve the problem.
5. Verify the answer.

Let us solve some of our real-life applications of addition.

Example 13: Robin deposited \$12348 in a bank in January, \$24780 in February, and \$19457 in March. What is the amount deposited by him in the bank in these 3 months?

Solution: The data given is written in the table as shown.

 Month Amount deposited (\$) January 12348 February 24780 March 19457

Add the numbers using any of the methods learned. We use the vertical addition method.

Example 14: A library has 2549 books of science, 3470 books of history, 6047 books of philosophy, and 2093 books of economics. How many books are there in the library?

Solution: The data given is written in the table as shown.

Add the numbers 2549, 3470, 6047, and 2093.

## Math Facts

• The sum of the first 100 numbers is 5050.
• The Abacus is faster than a calculator in adding and subtracting numbers.
• See how adding 10 to the numbers result.
• ## Mathemagician

Johann Friedrich Carl Gauss, a German mathematician, is considered as one of the greatest mathematicians for his contributions.

His contributions were in the fields of number theory, geometry, probability theory, geodesy, planetary astronomy, the theory of functions, and potential theory.

In 1792, Gauss made the important discovery that a regular polygon with 17 sides may be formed using only a ruler and compass. Its significance rests not in the result, but in the proof, which was based on a thorough examination of polynomial factorization and opened the way for further Galois theories. His 1797 Ph.D. thesis proved the fundamental theorem of algebra: every polynomial equation with real or complex coefficients has the same number of roots (solutions) as its degree (the highest power of the variable).

Know more about other concepts of Number Operations on Multiplication , and Multiplication Tables.

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