# Addition

## Elementary addition

Addition is the basic operation of the basic operations in arithmetic. If we add two numbers, we basically count the values of both numbers together and present the result as a new number.

In Figure 1 two numbers are added: 6 + 4 = 10, verbally: "six plus four equals ten" or "six added to four is ten" or "six and four makes ten"...

Adding any two natural numbers under ten is a basic skill anyone should be able to perform fluently, almost without thinking.

In Figure 2 addition under ten is ordered in an accessible table.

+ | 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 |

In fact, students start by learning all the additions that result in 10 or less than 10.
Then, for larger sums, they learn to first "make 10" and add what is left.
"Making 10" is first adding the *complement* to 10.
For example, 1 is called the complement to 10 of 9 and 7 is called the *complement* to 10 of 3:

9 + 8 = (9 + 1) + 7 = 10 + 7 = 17

3 + 8 = (3 + 7) + 1 = 10 + 1 = 11

The elements that are added together are called the *terms* of the addition, and the result is called the *sum*.

Addition is *commutative*, meaning that changing the order of the *terms* does not change the *sum*:

3 + 4 = 4 + 3

2 + 3 + 4 = 4 + 2 + 3

What effect does this property have on the table in Figure 2?

Adding zero to a number (or a number to zero) has no effect: 3 + 0 = 3.

## Adding terms > 10

If we master addition under ten *and* understand the principles of the decimal numeral system,
we can try some more difficult additions:

In 50 + 30 both digits 5 and 3 are in the tens position of its number, thus the digits can be added: 5 + 3 = 8 and in the tens position this makes 80.

50 + 30 = (5+3) × 10 = 8 × 10 = 80

60 + 40 = (6+4) × 10 = 10 × 10 = 100.

700 + 500 = (7+5) × 100 = 12 × 100 = 1200.

So, we constantly need to *regroup* ones by ones, ten by tens, hundreds by hundreds, etc:

55 + 4 = 50 + 5 + 4 = 50 + 9 = 59.

46 + 6 = 40 + 6 + 6 = 40 + 12 = 40 + 10 + 2 = 50 + 2 = 52.

219 + 23 = 200 + 10 + 20 + 9 + 3 = 200 + 30 + 12 = 200 + 40 + 2 = 242.

77 + 45 = 70 + 40 + 7 + 5 = 110 + 12 = 100 + 10 + 12 = 100 + 22 = 122.

## Addition algorithm

So far, you should have been able to calculate the given examples fluently in your head
(if not, practice here).
More difficult additions can be executed using pen and paper and the standard *addition algorithm*,
also called *long addition* or *column addition*.

1 | 1 | |

1 | 2 | 7 |

9 | 5 | |

2 | 2 | 2 |

In the above example the addition algorithm is used to calculate 127 + 95 = 222.
The recipe is as follows:

We write the digits of the involved numbers in a table with horizontal rows and vertical columns.
**From right to left** the first column is for the ones positions,
the second column for the tens positions, the third column for the hundreds positions etc.
Then we write the numbers to be added (127 and 95) under each other with their digits in the corresponding columns.
Then we add the digits in each column, starting with the **right** column.
Each time we write the result in the bottom row under the horizontal line:

7 + 5 = 12.
We write the 2 in the bottom row in the ones column and the 1 (= 10)
needs to be "carried" to the tens column,
and in the next step to be added to the other tens digits in this column. Thus:
1 + 2 + 9 = 12 (the first digit "1" is the "carry").
And again we write the 2 in the result row, and carry the 1 (= 100) to the next column.
This example ends with the third column: 1 + 1 = 2 (the first "1" is again the "carry").
Thus: 127 + 95 = 222.
If we want, we can write the "carries" in the corresponding columns in a top row, as done in the above example.

This compact and widely used algorithm always works for all additions of natural numbers, also if we need to add up more than two numbers. Some more examples:

1 | 1 | ||

5 | 9 | 4 | |

6 | 3 | 7 | |

1 | 2 | 3 | 1 |

2 | 1 | 2 | |

1 | 8 | 0 | 9 |

7 | |||

6 | 7 | 7 | |

9 | 3 | ||

7 | 0 | 0 | |

3 | 2 | 8 | 6 |

### Practice application

With the next app you can practice with the standard addition algorithm. Use the algorithm on a piece of paper to add-up the numbers. Click the exercise or click "CHECK" to see if you did it right.

Exercises: