# Sum

A sum is the result of adding two or more numbers or terms. In the addition problem below,

15 + 9 = 24

the 15 and 9, the numbers being added, are referred to as addends. The result, 24, is the sum of 15 and 9. We can also use the term "sum" as a verb. For example, "sum 15 and 19" refers to the action of adding 15 and 19, which again gives us a sum of 24.

In the above example, we added only two natural numbers, but it is possible to sum many different types of numbers, as well as expressions. In early mathematics, sums most typically include only the operation of addition, but this doesn't have to be the case.

15 - 9 = 4

Even though we subtracted 9, we can actually look at this problem as:

15 + (-9) = 4

with the use of negative numbers, and can still consider 4 a sum.

Also, although we only added 2 numbers in both examples above, we can add as many numbers as we want. In algebra, summation notation is used when we need to add many numbers that follow a specific pattern, so that we don't have to write out all the terms in the summation. The symbol uses the greek letter sigma, which looks like:

## Addition using simpler sums

When trying to find the sum of multiple numbers, or larger numbers, it is helpful to remember the properties of addition. In particular, the commutative and associative properties of addition allow us to move numbers around and group them in ways that allows us to more easily add them.

The commutative property of addition says that 1 + 3 = 3 + 1 = 4. The order in which we add doesn't matter.

The associative property of addition says that (1 + 2) + 10 = 1 + (2 + 10) = 13, so it doesn't matter whether we add 1 and 2 first, or 2 and 10 first. The result is the same.

It can sometimes be helpful, especially when first learning addition, to break numbers apart into numbers we may be more comfortable with.

Example

27 + 41 = ?

If we aren't yet comfortable with addition using columns and carrying, we could break each digit in the above problem up into a sum of smaller numbers, then rearrange and group them using properties of addition.

27 = 20 + 7 = 10 + 10 + 5 + 2

41 = 20 + 20 + 1 = 10 + 10 + 10 + 10 + 1

Now that we have broken up the problem into 10s, 5s, 2s, and 1s, this should make it easier to find the sum:

10 + 10 + 10 + 10 + 10 + 10 = 60

5 + 2 + 1 = 8

60 + 8 = 68