# Addition

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:

4 | 8 | 7 | |

7 | 9 | 1 | 5 |

3 | 6 | ||

8 | 4 | 3 | 8 |

1 | 1 | 1 |

Adding together the digits in the first (rightmost) column gives the answer 18 (7 + 5 + 6 = 18). The 8 goes beneath the first column, while the 1 goes beneath the second column (but lower down, in order to leave space for the result of the next addition). Adding together the digits in the second column gives the answer 12 (8 + 1 + 3 = 12), but we need to add the 1 that was carried from the first column, which gives us 13. The 3 goes in the answer beneath the second column, while the 1 is carried beneath column three. The addition of the digits in column thee gives 13 (4 + 9), plus the carried 1 gives 14. The 4 goes in the answer beneath column three, and the 1 will be carried beneath column four (the leftmost column). In this final column, we only have a single digit (7). This is added to the 1 carried beneath column four to give 8 in the answer beneath that column. Since there are no further columns to be added or digits carried over to the fifth column, we have completed the addition and the answer is 8,438 (you can check the result with a calculator if you like).

The addition of *real numbers* (i.e. fractional numbers with a floating point) can also be carried out using columnar addition in exactly the same way. The only additional point to remember (no pun intended) is that the decimal point for each number to be added, and for the answer, must appear in the same column. Examine the following example to see how this works (as previously, you can check the accuracy of the result using a calculator).

1 | 2 | 9 | . | 7 | 6 | 5 | ||

4 | 8 | 7 | 6 | . | 2 | 3 | ||

9 | 7 | . | 5 | 3 | 6 | 4 | ||

5 | 1 | 0 | 3 | . | 5 | 3 | 1 | 4 |

1 | 2 | 2 | 1 | 1 | 1 |

There are a few points to remember about addition. First of all, when it comes to adding things together we are not restricted to numbers alone. We can add together quantities such as length, area, volume and so on. It is important to make sure, however, that the quantities being added together are expressed in the same units. You cannot, for example, add lengths expressed in a mixture of metres and centimetres. The expression "2m + 57cm" cannot be evaluated. The values must be expressed as either centimetres or metres. The *International System of Units* (usually referred to simply as *SI units*) specifies the *metre* as the standard unit of length, so we often use the metre as the unit for expressing length when measuring things like the size of a room or a football pitch. A value of 57 centimetres would therefore be expressed as 0.57 metres, since 1 metre = 100 centimetres.

Another point to remember is that not all arithmetic operations are associative, so when we are dealing with complex expressions that include other operations (i.e. subtraction, multiplication or division) as well as addition, we must first determine the order in which the various parts of the expression should be evaluated. We will be looking at this issue in some detail on another page. Finally, note that you will at some point encounter situations where one or more of the addends in an expression is *negative*. For now, it is sufficient to remember that the sum of two numbers with a different *polarity* (i.e. one positive, one negative) is equal to the difference between them. The polarity of the result will be the same as that of the addend with the greatest *absolute value*. If the addends both have the same absolute value, the result will be zero.