Square (algebra)

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Square (algebra)

In algebra

Algebra

Algebra is a branch of mathematics concerning the study of structure , relation , and quantity. Together with geometry, mathematical analysis, combinatorics, and number theory, algebra is one of the main branches of mathematics....
, the square of a number is that number multiplied

Multiplication

Multiplication is the Operation of scaling one number by another. It is one of the four basic operations in elementary arithmetic .Multiplication is defined for Natural number in terms of repeated addition; for example, 4 multiplied by 3 can be calculated by adding 3 copies of 4 together:...
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

Real number

In mathematics, the real numbers may be described informally in several different ways. The real numbers include both rational numbers, such as 42 and −23/129, and irrational numbers, such as pi and the square root of two; or, a real number can be given by an infinite decimal representation, such as 2.4871773339...., where the digits co...
, the value of x² is equal to the area

Area

Area is a quantity expressing the two-dimensional size of a defined part of a surface, typically a region bounded by a closed curve. The term surface area refers to the total area of the exposed surface of a 3-dimensional solid, such as the sum of the areas of the exposed sides of a polyhedron....
of a square

Square (geometry)

In Euclidean geometry, a square is a regular polygon with four equal sides and four equal angles . A square with vertices ABCD would be denoted ....
of edge length x.

A positive integer

Integer

The integers are natural numbers including 0 and their negative and non-negative numberss . They are numbers that can be written without a fractional or decimal component, and fall within the set ....
that is the square of some other integer, for example 25 which is 5², is known as a square number

Square number

In mathematics, a square number, sometimes also called a perfect square, is an integer that can be written as the square of some other integer; in other words, it is the product of some integer with itself....
, 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 + ...

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Encyclopedia

In algebra

Algebra

Multiplication

Real number

Area

Square (geometry)

In Euclidean geometry, a square is a regular polygon with four equal sides and four equal angles . A square with vertices ABCD would be denoted ....
of edge length x.

A positive integer

Integer

Square number

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 is . The first terms of this series (the Square pyramidal number

Square pyramidal number

In mathematics, a pyramid number, or square pyramidal number, is a figurate number that represents a pyramid with a base and four sides. These numbers can be expressed in a formula as...
s) 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

Real

Real most often refers to reality, the state of things as they actually exist.Real may also refer to:...
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

Square root

In mathematics, a square root of a number x is a number r such that r2 = x, or, in other words, a number r whose square is x....
can be taken of a negative number within the system of real number

Real number

Complex number

In mathematics, the complex numbers are an extension of the real numbers obtained by adjoining an imaginary unit, denoted i, which satisfies:...
s, beginning with the imaginary unit

Imaginary unit

In mathematics, physics, and engineering, the imaginary unit is denoted by or the Latin or the Greek iota . It allows the real number system, to be extended to the complex number system, Its precise definition is dependent upon the particular method of extension....
i, which by convention is one of the square roots of −1.

Squaring is also useful in statistics

Statistics

Statistics is a Mathematics pertaining to the collection, analysis, interpretation or explanation, and presentation of data. It also provides tools for prediction and forecasting based on data....
in determining the standard deviation

Standard deviation

In statistics, standard deviation is a simple measure of the variability or statistical dispersion of a data set. A low standard deviation indicates that all of the data points are very close to the same value , while high standard deviation indicates that the data are ?spread out? over a large range of values....
of a set of values. The deviation of each value from the mean of the set is defined as the difference . These deviations are squared, then a mean is taken of the new set of numbers (each of which is positive). This mean is the variance

Variance

In probability theory and statistics, the variance of a random variable, probability distribution, or sample is one measure of statistical dispersion, averaging the squared distance of its possible values from the expected value ....
, and its square root is the standard deviation. In finance

Finance

The field of finance refers to the concepts of time, money and risk and how they are interrelated. Banks are the main facilitators of funding through the provision of credit, although private equity, mutual funds, hedge funds, and other organizations have become important....
, the volatility

Volatility (finance)

Volatility most frequently refers to the standard deviation of the continuously compounded returns of a financial instrument with a specific time horizon....
of a financial instrument is the standard deviation of its values.

Square (algebra)

Uses

See also