square (algebra)

Information about square (algebra)

For other uses, see square.


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

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 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... Square pyramidal numbers

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.

See also

Square may mean:

Entertainment

  • Square (album), an album by Buck 65
  • Square (band), a musical trio from Lincoln, Nebraska
  • Square Enix, a video game company
  • Square Co.

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Algebra is a branch of mathematics concerning the study of structure, relation and quantity. The name is derived from the treatise written by the Arabic[1] mathematician, astronomer, astrologer and geographer,
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Multiplication is the mathematical operation of adding together multiple copies of the same number. For example, four multiplied by three is twelve, since three sets of four make twelve:


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subscript or superscript is a number, figure, symbol, or indicator that appears smaller than the normal line of type and is set slightly below or above it – subscripts appear below the baseline, while superscripts are above.
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In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2.4871773339…. The real numbers include both rational numbers, such as 42 and −23/129, and irrational numbers, such as π and
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Area is a physical quantity expressing the size of a part of a surface. The term Surface area is the summation of the areas of the exposed sides of an object.

Units

Units for measuring surface area include:
square metre = SI derived unit

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In plane (Euclidean) geometry, a square is circle with four sides.

Classification

A square is a regular quadrilateral. Likewise it is also a special case of a rhombus, kite, parallelogram, and trapezoid.
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The integers (from the Latin integer, which means with untouched integrity, whole, entire) are the set of numbers including the whole numbers (0, 1, 2, 3, …) and their negatives (0, −1, −2, −3, …).
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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.
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Real may refer to:
  • Reality, something that exists
  • Real (galley), the flagship of Don Juan de Austria in the Battle of Lepanto in 1571
  • Real (bicycle manufacturer), a bicycle manufacturer
  • Real

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A negative number is a number that is less than zero, such as −3. A positive number is a number that is greater than zero, such as 3. Zero itself is neither positive nor negative.
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A negative number is a number that is less than zero, such as −3. A positive number is a number that is greater than zero, such as 3. Zero itself is neither positive nor negative.
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In mathematics, a square root of a number x is a number r such that , or in words, a number r whose square (the result of multiplying the number by itself) is x.
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In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2.4871773339…. The real numbers include both rational numbers, such as 42 and −23/129, and irrational numbers, such as π and
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imaginary number (or purely imaginary number) is a complex number whose square is a negative real number. Imaginary numbers were defined in 1572 by Rafael Bombelli. At the time, such numbers were thought not to exist, much as zero and the negative numbers were regarded by
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In mathematics, the imaginary unit (or sometimes the Latin or the Greek iota, see below) allows the real number system to be extended to the complex number system . Its precise definition is dependent upon the particular method of extension.
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In probability and statistics, the standard deviation of a probability distribution, random variable, or population or multiset of values is a measure of the spread of its values. It is usually denoted with the letter σ (lower case sigma).
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sample is a subset of a population. Typically, the population is very large, making a census or a complete enumeration of all the values in the population impractical or impossible. The sample represents a subset of manageable size.
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In statistics, mean has two related meanings:
  • the arithmetic mean (and is distinguished from the geometric mean or harmonic mean).
  • the expected value of a random variable, which is also called the population mean.

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variance of a random variable (or somewhat more precisely, of a probability distribution) is one measure of statistical dispersion, averaging the squared distance of its possible values from the expected value.
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Volatility is the measure of the state of instability.
  • Volatility (finance) frequently refers to the standard deviation of the change in value of a financial instrument with a specific time horizon.

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In mathematics, a square root of a number x is a number r such that , or in words, a number r whose square (the result of multiplying the number by itself) is x.
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cube of a number n is its third power — the result of multiplying it by itself three times:

n3 = n × n × n.

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In mathematics, the Pythagorean theorem (AmE) or Pythagoras' theorem (BrE) is a relation in Euclidean geometry among the three sides of a right triangle. The theorem is named after the Greek mathematician Pythagoras, who by tradition is credited with its discovery and
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Exponentiation is a mathematical operation, written an, involving two numbers, the base a and the exponent n.
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