In mathematics, addition, represented by the symbol , is an operation which combines two mathematical objects together into another mathematical object of the same type, called the sum. Addition can occur with simple objects such as numbers, and more complex objects such as vectors and matrices.

Arithmetic 

In arithmetic, addition is finding the total of two or more numbers. The sign for addition is "+", and the name for the sign "+" is "plus". Another name for the total is the sum.

Counting examples 
For example, there are objects in two groups. The objects are small circles: "o". One group has five of these objects. The other group has three of these objects. To find the total number of objects in both groups, the objects can be counted. Another way to find the number of objects in both groups is to add the numbers in each group.

Adding

Another method is to add the numbers of objects in group A and group B, since they are already counted. In symbols:

There are rules for adding numbers that people learn. There are also rules for adding numbers that are built into machines (binary adder). The rule says that:

In another counting example, Sally and Bill have 2 children. Sally and Bill get 3 more children. Sally and Bill have added three children to their two children, and now have five children.

Vertical Addition

The animation above demonstrates the addition of seven hundred and eighty six and four hundred and sixty seven, the problems digits have been separated into units, tens and hundreds (place value).

First, the units 6 and 7 are added together to make 13, so 1 ten and 3 units, with the 3 written below and the 1 ten carried to the tens column. Next, in the tens column, the 1, 8 and 6 are added together to make 15 tens, so 1 hundred and 5 tens, with the 5 written below and the 1 hundred carried to the hundreds column. Finally, in the hundreds column, 1, 7 and 4 are added together to make 12 hundreds, so 1 thousand and 2 hundreds, with the 2 written below and the 1 thousand carried to the thousand column. The final answer is thus one thousand two hundred and fifty three.

A measurement example
Tom wants to know the distance between his house and Sally's house. Bob's house is 300 meters east of Tom's house. Sally's house is 120 meters east of Bob's house:

Tom's house      300 meters      Bob's house      120 meters      Sally's house

The distance from Tom's house to Sally's house can be found by adding the distances already measured. The distance from Tom's house to Bob's house, added to the distance from Bob's house to Sally's house, is the same as the distance from Tom's house to Sally's house. That is, three hundred (300) meters plus 120 meters.

Hence Sally's house is 420 meters to the east of Tom's house.

Properties

Commutativity

Addition is commutative, meaning that one can change the order of the numbers in a sum, but still get the same result. For example:

Associativity 
Addition is also associative, which means that when three or more numbers are added together, the order of operations does not change the result.

For any three numbers a, b, and c, it is true that . For example,  and , which means that .

Addition as increase 
Addition can also mean to make bigger.

Example of addition as increase 
 For example, Tom has a house. Tom puts a new room on the house. This new room is called an addition.
 John is making food. To make the food taste better, John puts salt in the food. That is, John adds salt to the food. The salt is a food additive or addition to the food.

Addition table

Related pages 
 Identity element
 Order of operations

References

Other websites 
 AAA Math: Addition

Basic English 850 words
Hyperoperations