kidzsearch.com > wiki Explore:images videos games

# Sum

The **sum** of two numbers is what we get when we add the two numbers together. This operation is called **summation**. There are a number of ways of writing sums, with the most common being:

- Addition ([math]2+4+6 = 12[/math])
- Summation ([math]\sum_{k=1}^3 k = 1+2+3=6[/math])
- Computerization:

- Sum = 0
- For I = M to N
- Sum = Sum + X(I)
- Next I (in Visual BASIC)

## Sigma notation

**Sigma notation** is a mathematical notation to write long sums in a short way. Sigma notation uses the Greek letter Sigma, (**∑**), and takes upper and lower **bounds** which tell us where the sum begins and where it ends. The lower bound usually has a variable (called the **index**) given a value, such as "i=2". This tells us that the summation begins at 2, and goes up by 1 until it reaches the number on the top.

## Properties

- [math]\sum_{i=1}^n 0 = 0[/math]

- [math]\sum_{i=1}^n 1 = n[/math]

- [math]\sum_{i=1}^n n = n^2[/math]

- [math]\sum_{i=1}^n i = \frac{n(n+1)}{2}[/math]

- [math]\sum_{i=1}^n i^2 = \frac{n(n+1)(2n+1)}{6}[/math]

- [math]\sum_{i=1}^n i^3 = \frac{n^2 (n+1)^2}{4}[/math]

## Applications

Sums are used to represent series and sequences. For example,

- [math]\sum_{i=1}^4 \frac{1}{2^i} = \frac{1}{2^1} + \frac{1}{2^2} + \frac{1}{2^3} + \frac{1}{2^4}[/math]

The geometric series of a repeating decimal can be represented in summation,

- [math]\sum_{i=1}^\infty \frac{3}{10^i} = 0.333333... = \frac{1}{3}[/math]

The concept of an integral is a limit of sums. The area under a shape being defined as:

- [math]\lim_{n \to \infty} \sum_{i=1}^n f(x_i^*)\Delta x[/math]

## Further reading

- Nicholas J. Higham, "The accuracy of floating point summation",
*SIAM J. Scientific Computing***14**(4), 783–799 (1993).

## Other websites

- Media related to Sum at Wikimedia Commons
- Sigma Notation on PlanetMath
- Derivation of Polynomials to Express the Sum of Natural Numbers with Exponents