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What is a sum?

Sum Definition

A sum is the result of numbers or expressions being added together. It comes from the Latin word "summus" which means "total".  

The first number in a sum is called the augend and the remaining numbers are called the addends.

A sum can be an unsimplified expression like the left side of this equation. Or it can be a simplified version like the right side of this equation. 

A lot of elementary school teachers will say that a sum is "the answer to an addition problem" because the "answer" is a simplified version of the actual sum.

Examples with Numbers

8 + 1 = 9

In this example, 8 + 1 is the sum and 9 is the simplified sum. 

2 + 1 + 7 = 10

In this example, 2 + 1 + 7 is the sum and the simplified sum is 10

Examples with Variables

7x 5y = 12

In this example, 7x + 5y is the sum.

This sum can't be simplified because we don't know what x and y are, but 12 is equal to the sum so you could say "12 is the sum of 7x and 5y." 

13a + 11b - 7c

When you are working with variables, it's usually easiest to interpret any subtraction as adding a negative.

So, in this example, you could say...

13a + 11b - 7c is the sum of the terms 13a11b, and -7c

Technically, you could say that 13a is an augend, 11b is an addend, and -7c is a subtrahend.

However, if any of the variables are negative numbers, that could change an addend into a subtrahend, or vice versa.

So, when we work with variables, we usually call all of the parts of the sum "terms" instead of differentiating between augends, addends, minuends, and subtrahends.

Summation Notation

In calculus and advanced algebra, you may be asked to calculate really long (or even infinite) sums like this...

\(3+7+11+15+19+23+27+31+35...\)

These types of sums are called series and they are often written in summation notation. 

When will I use sums?

You use sums all the time because addition is one of the most basic math concepts. Almost all math problems involve addition in some way.

But sums can also be used in ways besides basic addition. The buttons below will take you to pages that explain how to use advanced sums. 

Using Sums in Algebra

Consecutive Integers

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Factoring with Sum of Squares

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Factoring with Sum of Cubes

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Using Sums in Geometry

Sum of the Interior Angles of a Triangle

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Sum of the Interior Angles of a Polygon

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Sum of the Exterior Angles of a Polygon

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Using Sums in Advanced Algebra & Calculus

Sum of Finite Arithmetic Sequences

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Sum of Finite Geometric Sequences

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Sum of Infinite Geometric Sequences

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Resources

Free Printable Worksheets

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Free Online Practice Problems

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