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Translations of Encyclopedia about Mathematics

 

Addition

Addition is used to add together the total amount of some aggregate or amount. Smaller amounts can be counted using the fingers of the hand. With larger amounts though, it is easier to use a calculator, although we can perform such adding using notations.

The numbers we wish to add together are called addents and the total of the addents are called the sum.

If we would like to add together larger numbers without the use of a calculator, we can use the following procedure:

256

+

518

carry over

1

=

774

Write down the addents one below one another. Then add each column separately, writing the total below. If however the sum of a particular column is greater than 9, such as in our example (6+8=14), we write down the second digit but carry the first digit (in this case 1, marked green above) to the next column to the left, adding it to that column. Adding the second column together we get the number 7, which we write below that column. The far left column also gives us a value of 7, to give us a final figure of 774.

Another example: 957 + 45 + 4970 + 1 + 800 = 6773

957

+

45

+

4970

+

1

+

800

carry over

211

=

6773

It does not make any difference what order you place the various addents in in that the so-called cumulative law still applies (ie- 3+1 and 1+3 give the same result, 4).

However, if using brackets, the law of association applies such that (a + b) + c = a + (b + c). Brackets are used to define the order of operations, where functions within the brackets are computed first. But, because we are only adding, the end result still remains the same.

Example: (3 + 4) + 5 = (7) + 5 = 7 + 5 = 12

Also, (+a) = +a, and (-a) = -a. Generally, the plus sign with positive numbers are left out and understood.

When counting with real numbers, we must be aware of the sign (positive or negative).

When adding to real numbers (in our case a and b), the following generally applies:

1. (+a) + (+b) = + (a + b)

Example

(+2) + (+6) = (+2) + 6 = 2 + 6 = 8 = (2 + 6 ) = +(2 + 6)

2. (-a) + (-b) = - (a + b)

Example

(-2) + (-6) = (-2) - 6 = -2 - 6 = -8 = -(8) = -(2 + 6)

3. (+a) + (-b) = + (a b) = - (b a)

Example

(+6) + (-2) = +6 2 = 6 2

= 4 = +(4) = +(6 2) as with

= 4 = -(-4) = -(2 - 6)

4. (-a) + (+b) = + (b a) = - (a b)

Example

(-6) + (+2) = -6 + 2

= -4 = +(-4) = +(2 6), as with

= -4 = -(4) = -(6 2)

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