# Dictionary Definition

squared adj : having been made square

# User Contributed Dictionary

## English

### Pronunciation

### Verb

squared- past of square

# Extensive Definition

In algebra, the square of a number
is that number multiplied by itself. To
square a quantity is to multiply it by itself. Its notation is a
superscripted "2"; a
number x squared is written as x2. Thus:

A positive integer that is the square of
some other integer, for example 25 which is 52, is known as a
square
number, or more simply a square.

It is often also useful to note that the square
of any number can be represented as the sum (for 0≤n)

- 1 + 1 + 2 + 2 + ... + (n − 1) + (n − 1) + n.

For instance, the square of 4 or 42 is equal
to

- 1 + 1 + 2 + 2 + 3 + 3 + 4 = 16.

This is the result of adding a column and row of
thickness 1 to the square graph of three (like a tic tac toe
board). You add three to the side and four to the top to get four
squared. This can also be useful for finding the square of a large
number quickly. For instance, the square of

- 522 = 502 + 50 + 51 + 51 + 52 = 2500 + 204 = 2704.

In addition, it can be seen that another
equivalent sum may be used to represent the square of a number. The
square of a number N is the sum of the first N odd numbers. The
square of 1 is 1; the square of 2 is

- 1 + 3 = 4;

the square of 7 is

- 1 + 3 + 5 + 7 + 9 + 11 + 13 = 49.

and so on. This, of course is the same as the
previous sum method but with every two numbers following the
initial number added to each other:

- 1 + ( 1 + 2 ) + ( 2 + 3 ) + ( 3 + 4 ) + ... = 1 + 3 + 5 + 7 + ...

The general term of the series
1^2+2^2+3^2+4^2+...+n^2 is n(n+1)(2n+1)/6. The first terms of this
series (the Square
pyramidal numbers) are :

0, 1, 5, 14, 30, 55, 91, 140, 204, 285, 385, 506,
650, 819, 1015, 1240, 1496, 1785, 2109, 2470, 2870, 3311, 3795,
4324, 4900, 5525, 6201...

## Uses

Since the product of two real negative numbers is positive, and the product of two real positive numbers is also positive, it follows that no square number is negative. This has important consequences. It follows, in particular, that no square root can be taken of a negative number within the system of real numbers. This leaves a gap in the real number system that mathematicians fill by postulating imaginary numbers, beginning with the imaginary unit i, which by convention is one of the square roots of −1.Squaring is also useful for statisticians in
determining the standard
deviation of a population or sample
from its mean. Each datum
is subtracted from the mean, and the result is squared. Then an
average is taken of the new set of numbers (each of which is
positive). This average is the variance, and its square root
is the standard deviation -- in finance, the volatility.

squared in Bosnian: Kvadrat (algebra)

squared in German: Quadrat (Arithmetik)

squared in Esperanto: Kvadrato (algebro)

squared in Estonian: Ruut (algebra)

squared in French: Carré (algèbre)

squared in Italian: Quadrato (algebra)

squared in Dutch: Kwadraat

squared in Polish: ²

squared in Portuguese: Quadrado
(aritmética)

squared in Russian: Квадрат (алгебра)

squared in Simple English: Square
(mathematics)

squared in Finnish: Neliö (algebra)

squared in Swedish: Kvadrat (aritmetik)

squared in Chinese: 平方