Addition is arguably the simplest arithmetic operation beyond simply counting the quantity of something. Indeed, one of the ways children learn to add two small numbers together is to count on from one of the numbers (usually the larger of the two). If the numbers are sufficiently small, this can be done using the fingers. Take the addition of seven and four as an example. Take the number seven as the starting point and count on, using the fingers of one hand to keep track of how much you have added. Incrementing seven by one four times gives eight, nine, ten, and eleven - by which point you have used four fingers, you can stop counting on, and the answer is eleven. Of course this is somewhat laborious, and only works for very small numbers.

The simplest form of addition produces the sum of two numbers (sometimes referred to as the addends). The notation used to represent the addition of two numbers shows the numbers, separated by a plus sign ("+"). For example, the addition of the numbers two and three would be written as: "2 + 3". When three or more numbers are added together, the term summation may be used. Such an operation could be written as: "2 + 3 + 4 + 5 . . . ". One of the reasons why addition may be considered the simplest of the four arithmetic operations is because it is both commutative and associative. The commutative nature of addition means that the addends may appear in any order without affecting the result. For example, the following expressions yield the same result:

2 + 7 = 9

7 + 2 = 9

Note that in the above notation, the result follows an equals sign ("="). Addition is also said to be associative. What does this mean? We can effectively consider the addition of three numbers (the addends) as two separate additions. First, we add together the first two numbers. We then add the result of this addition to the third number. We would get the same result, however, by adding the last two numbers together first, and then adding the first number to the result. For example, the following expressions give the same result (note that the use of brackets indicates which addition operation is carried out first):

(3 + 5) + 9 = 17

3 + (5 + 9) = 17

There are many techniques that allow larger numbers to be added together using mental arithmetic which we will not go into here. It is however generally acknowledged that through practice, children acquire a number of "addition facts" that are either committed to memory or can quickly be derived from other known facts. For example, a child may remember that four plus four equals eight, and reason that four plus five must therefore be one more than eight and rightly conclude that the answer is nine. Over time, many of these derived facts also become committed to memory and can be quickly recalled. Given our decimal system of counting, there seems to be a general consensus that more complex additions are greatly facilitated by the ability to fluently recall a hundred addition facts. This means knowing the result of adding any two numbers, either of which may have any value from one to ten. The addition facts can be represented in tabular form, as shown below. The intersection of each row and column gives the result of adding the number at the start of the row to the number at the top of the column (note that for additions involving zero, the addition of zero to any number simply yields that number).

 0 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10 11 2 3 4 5 6 7 8 9 10 11 12 3 4 5 6 7 8 9 10 11 12 13 4 5 6 7 8 9 10 11 12 13 14 5 6 7 8 9 10 11 12 13 14 15 6 7 8 9 10 11 12 13 14 15 16 7 8 9 10 11 12 13 14 15 16 17 8 9 10 11 12 13 14 15 16 17 18 9 10 11 12 13 14 15 16 17 18 19 10 11 12 13 14 15 16 17 18 19 20

The availability of a lookup-table does not significantly extend the range of additions that can quickly be carried out beyond the simple technique of counting on using fingers, although it may be slightly quicker. Memorisation of the addition facts, together with the use of various techniques for deriving hitherto unknown facts, allows us to solve more complex addition problems. Some people are able to add together a number of relatively large (three or four-digit) numbers using only mental arithmetic. Sadly, I am not one of them. As the numbers become larger, it is necessary to use more sophisticated methods to derive the correct result. Many people would probably argue that, given the availability of cheap and powerful pocket calculators today, it is completely unnecessary to learn these techniques. In order to be able to understand more advanced mathematical concepts however, it is necessary to have a working knowledge of the methods used for carrying out basic arithmetic. These methods have varied, as theories about the best way to teach mathematics have changed. The important thing to remember, however, is that they constitute a basic set of problem solving skills. Familiarity with the basic tools used for arithmetic facilitates the acquisition of the more sophisticated skills required to solve more complex problems.

As mentioned earlier, the addition of two relatively small numbers can often be carried out using mental arithmetic. When adding together several three- or four-digit numbers without the aid of a calculator, it may be necessary to use a pencil (or pen) and paper. One of the most widely used methods of adding together multi-digit numbers is columnar addition. In this method, the digits in each column are added together, and the result of the addition is written beneath each column. The process starts with the right-most column and proceeds from right to left. If the result for any column is greater than ten, the rightmost digit of the result is placed beneath the current column, and the remaining digit (called the carry, because it is carried to the left) is placed on the next line down, beneath the column immediately to the left of the current column. The carry will be added to the result for the column under which it appears. Consider the following example:

0 0 4 8 7
+ 7 9 1 5
+ 0 0 3 6
= 8 4 3 8
0 1 1 1