AskDefine | Define squared

Dictionary Definition

squared adj : having been made square

User Contributed Dictionary

English

Pronunciation

Verb

squared
  1. past of square

Adjective

  1. raised to the second power

Translations

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:
If x is a positive real number, the value of x2 is equal to the area of a square of edge length x.
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
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