Sum

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

• Addition ($2+4+6=12$)
• Summation ($\sum _{k=1}^{3}k=1+2+3=6$)
• Computerization:

Sum = 0

For I = M to N

Sum = Sum + X(I)

Next I (in Visual BASIC)

There are types of summing, chiefly:

• additive summation ("adding")
• divisive summation ("dividing")
• factorial summation ("taking the factorial")
• fractional summation ("as a fraction")
• multiplicative summation or product summing
• percentile summation ("percent of" / "per cent. of" ; 2nd spelling termed archaic)
• root summation ("rooting")
• subtractive summation ("subtracting" or "minusing")

Sigma notation

Sigma notation is a mathematical notation to write long sums in a short way. Sigma notation uses the Greek letter Sigma ($\Sigma$), 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, often denoted by $i$, $j$ or $k$) along with 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

$\sum _{i=1}^{n}0=0$

$\sum _{i=1}^{n}1=n$

$\sum _{i=1}^{n}n=n^{2}$

$\sum _{i=1}^{n}i={\frac {n(n+1)}{2}}$

$\sum _{i=1}^{n}i^{2}={\frac {n(n+1)(2n+1)}{6}}$

$\sum _{i=1}^{n}i^{3}={\frac {n^{2}(n+1)^{2}}{4}}$

$\sum _{i=1}^{\infty }a_{i}=\lim _{t\to \infty }\sum _{i=1}^{t}a_{i}$

Applications

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

$\sum _{i=1}^{4}{\frac {1}{2^{i}}}={\frac {1}{2^{1}}}+{\frac {1}{2^{2}}}+{\frac {1}{2^{3}}}+{\frac {1}{2^{4}}}$

The geometric series of a repeating decimal can be represented in summation. For example:

$\sum _{i=1}^{\infty }{\frac {3}{10^{i}}}=0.333333...={\frac {1}{3}}$

The concept of an integral is a limit of sums, with the area under a curve being defined as:

$\lim _{n\to \infty }\sum _{i=1}^{n}f(x_{i}^{*})\Delta x$

Related pages

• Product (mathematics)

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 Summation at Wikimedia Commons
• Sigma Notation Archived 2015-09-21 at the Wayback Machine on PlanetMath
• Derivation of Polynomials to Express the Sum of Natural Numbers with Exponents Archived 2013-02-18 at the Wayback Machine