In arithmetic, addition is a type of basic mathematical operation. In primary education, students are taught to add numbers in the decimal system, starting with single digits and progressively tackling more difficult problems. In this section, we will learn the addition of two or more numbers.

The addition is a term used to describe to add two or more numbers together. In other words, it adds two or more numbers together. The number that is left and right of the plus symbol is called addend, and the number after the equal sign is called addition or sum. ### Notation

A plus symbol (+) is used to denote the addition. It appears between the two numbers that are called infix notation. The other synonym of addition is added, sum, plus, and total. Sometimes it is also represented by the symbol ∑ (sigma). It is used when we have to add a large number. By using the plus symbol, we can perform addition between different numbers such as integers, real numbers, decimal numbers, complex numbers, etc. Besides this, it is also used in algebra to add vectors and matrices.

For example, in the first basket, there are five apples, and in the second basket, there are 4 apples. If we count the apples of both the basket, we get 9 apples. In arithmetic, we can represent it in the mathematical expression as:

5+4=9

• In addition, the order of addition does not matter. It always gives the same answer.
2+4+6+8=20 or 8+6+4+2=20 or 6+2+8+4=20
• Adding 0 to any number or vice-versa gives the same number as a result.
7+0=7 or 0+7=7
• If we add any number to itself two-times, it is the same as multiplying a number by 2.
8+8=16
It is same as:
8×2=16
• The repeated addition of 1 is the same as counting.
1+1=2,1+1+1=3

The following table helps the children to memorize the sum of two numbers. You can find the sum of two numbers, between 0 to 10. Before moving to the addition, we must be aware of the term carry.

In arithmetic, a carry is a digit that is transferred from the right column to the left column and added to the transferred column.

### Addition of one-digit Numbers

We can find the addition of one-digit numbers with the help of the above table. Suppose we want to add 2 and 3 together. Search 2 in the left-most column, and 3 in the topmost row. In the current row, move down until you reach in front of the selected column. The square contains the addition of the numbers 2 and 3, i.e., 5. Similarly, we can find the sum of any single-digit number.

### Addition of Two-digit Numbers

• Arrange the given numbers in the column for easy understanding.
• Add the ones place digits together, transfer the carry, if any. It gives the unit place of the answer.
• Add the tens place digits and carry them from the previous step (if any).
• Write the answer.

Let's implement the above steps in an example.

Example: Add 24 and 32.

Solution: Example: Add 98 and 22.

Solution: ### Addition of Three-digit Numbers

• Arrange the given numbers in the column for easy understanding.
• Add the ones place digits together, transfer the carry, if any. It gives the unit place of the answer.
• Add the tens place digits, and carry from the previous step (if any), transfer the carry, if any. It gives the tens place of the answer.
• Add the hundreds of place digits, and carry from the previous step (if any). It gives the hundreds or thousands or both (depend on the sum) place of the answer.
• Write the answer.

Let's implement the above steps in an example.

Example: Add 367 and 492.

Solution: Example: Add 847 and 564.

Solution: Similarly, we can add four-digit numbers, also.

### Addition of Integers

An integer includes all positive and negative numbers, including 0. A number may have a positive or negative sign. The addition of integers with the sign follows the rules. Generally, we do not represent a positive number with + sign. In the following table, we have summarized the additional rule of positive and negative numbers.

We have taken two numbers a and b as addend and z as sum.

Sign Remarks Examples
(+) + (+) = + Always gives positive result a + b = z
(+) + (-) = - If a>b, the result will be +ive else -ive a + (-b) = z or -z
(-) + (+) = - If a<b, the result will be -ive else +ive (-a) + b = -z or z
(-) + (-) = - Always gives negative result (-b) + (-b) = -z

### Examples

10+20=30
25+(-20)=25-20=5
20+(-25)=20-25=-5
(-22)+(10)=10-22=-12
(-10)+(22)=22-10=12
(-40)+(-20)=-40-20=-60

### Addition of Decimal Numbers

To add two or more decimal numbers, follow the rules given below:

• Write the number in the column form but remember that decimal point must be lined up.
• Make the number of equal lengths, if unequal.
• Add the columns together and put a decimal point in the answer.

Example: Add 56.3457 and 2.4.

Solution: Example: Add 12.02 and 45.11.

Solution: Example: Add 33.89, 0.0073, and 6.

Solution: ### Addition of Rational Number

The rational numbers are the numbers that are in the fraction form . Let's see how to add the rational number.

When the denominator of each fraction is the same:

• Add the numerators and put the resultant in the answer.
• Simplify the fraction if required.

In general, we can say that if are two fractions, the addition of fractions will be: Remember: To simplify a fraction, numerator and denominator must be divisible by the same number.

Example: Find the sum of .

Solution: On simplifying the fraction , we get 2.

Hence, the sum of is 2.

When the denominator of each fraction is unlike (not similar):

• Find the LCM of the denominators because we need to make denominators the same.
• Divide the LCM by the denominators.
• Multiply the resultant in the numerators, respectively, and simplify.
• Add numerators, and get the answer.

In general, we can say that if are two fractions, the addition of fractions will be: Example: Solution:

Let's solve the question according to the above steps.

Find the LCM of the denominators. Divide the LCM by the denominators. Multiply the resultant (from the above step) in the numerators, respectively, and simplify.  ### Addition of Complex Numbers

Complex numbers are added by adding real and imaginary parts separately. In general, we can say that if a+bi and c+di are two complex numbers, then the addition of these numbers will be:

(a+bi)+(c+di)=(a+c)+(b+d)i

Example: Add (6+4i) and (5+3i).

Solution:

In the above example, 6 and 5 are real parts, and 4i and 3i are imaginary parts. So, we will add real parts together and imaginary parts together.

(6+4i)+(5+3i)=(6+5)+(4i+3i)
(6+4i)+(5+3i)=(11+7i)

The sum of (6+4i) and (5+3i) is (11+7i).

Example: Add (12+10i) and (7-9i).

Solution:

(12+10i)+(7-9i)=(12+7)+(10i-9i)
(6+4i)+(5+3i)=(19+i)

The sum of (12+10i) and (7+9i) is (19+i)

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