Difference between revisions of "Square root"

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The '''square root''' of a number ''X'' is the number that, multiplied by itself, results in ''X''.  The symbol for the square root of ''X'' is <math>\sqrt{X}</math>.  Many computer languages and spreadsheet programs use "''sqr(X)''" or "''sqrt(X)''"to express this, as do people when mathematical symbols are not conveniently available.
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The '''square root''' of a number ''x'' is the number that, multiplied by itself, results in ''x''.  The symbol for the square root of ''x'' is <math>\sqrt{x}</math>.
  
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==Two square roots for each number==
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All numbers have two square roots, one positive and the other negative:
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:<math>3 \cdot 3 = 9</math> and <math>-3 \cdot -3 = 9</math>
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As such, the technically correct way of writing is
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:<math>\sqrt{9} = \pm 3</math>
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When speaking of "''the'' square root of ''x''", people usually refer to the positive square root. For example:
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:<math>\sqrt{9} = 3</math>
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All numbers have two square roots, although they don't have to be distinct. For example, the square root of zero has two values: +0 and -0.
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==Irrational numbers as square roots of whole numbers==
 
Some square roots are relatively simple [[whole number]]s, for example, 3 the square root of 9. Others are less amenable to expression, such as the square root of 2 (=1.414...), which has been proven to be an [[irrational number]].
 
Some square roots are relatively simple [[whole number]]s, for example, 3 the square root of 9. Others are less amenable to expression, such as the square root of 2 (=1.414...), which has been proven to be an [[irrational number]].
  
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Negative numbers have square roots that lie outside the [[real number]]s: Multiplying a real number by itself always results in a positive number. The square roots of negative numbers involve what are called [[imaginary number]]s, constructed via use the square root of -1, which is labelled ''i''.  Thus, the square root of -9 is 3''i''.
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==Square roots of negative numbers==
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Negative numbers have square roots that lie outside the [[real number]]s: Multiplying a real number by itself always results in a positive number. The square roots of negative numbers involve what are called [[imaginary number]]s:
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:<math>\sqrt{-1} = i</math> which for example leads to <math>\sqrt{-9} = 3i</math>
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The statements in [[#Two square roots for each number|"Two square roots for each number"]] also apply to negative numbers. Thus the technically correct way of writing the above example would be:
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:<math>\sqrt{-9} = \pm 3i</math>
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==Alternate expressions and notations==
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Many computer languages and spreadsheet programs use "''sqr(x)''" or "''sqrt(x)''" to describe (or calculate) the square root of ''x''. This is also the common notation used when mathematical symbols are not conveniently available.
  
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Technically, all numbers have two square roots, one positive and the other negative, since squaring the negative number results in a positive number. Thus, sqr(9)=plus or minus 3, written +/-3; likewise <math>\sqrt{-9} = \pm3i</math>.
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The square root of a number can also be denoted as <math>x^{\frac{1}{2}}</math>. This can readily be seen by the rule of adding powers when multiplying:
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:<math>x^{\frac{1}{2}} \cdot x^{\frac{1}{2}} = x^{\frac{1}{2}+\frac{1}{2}} = x^{1} = x</math>
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If superscript is not available, the common way of writing this is '''x^(1/2)'''.
  
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The square root of a number can also be denoted as <math>X^{\frac{1}{2}}</math>, or ''X''^(1/2) if superscript is not available.  This can readily be seen by the rule of adding powers when multiplying: <math>X^{\frac{1}{2}} \cdot X^{\frac{1}{2}} = X^{\frac{1}{2}+\frac{1}{2}} = X^{1} = X</math>
 
 
[[category:mathematics]]
 
[[category:mathematics]]

Revision as of 22:29, April 12, 2007

The square root of a number x is the number that, multiplied by itself, results in x. The symbol for the square root of x is <math>\sqrt{x}</math>.

Two square roots for each number

All numbers have two square roots, one positive and the other negative:

<math>3 \cdot 3 = 9</math> and <math>-3 \cdot -3 = 9</math>

As such, the technically correct way of writing is

<math>\sqrt{9} = \pm 3</math>

When speaking of "the square root of x", people usually refer to the positive square root. For example:

<math>\sqrt{9} = 3</math>

All numbers have two square roots, although they don't have to be distinct. For example, the square root of zero has two values: +0 and -0.

Irrational numbers as square roots of whole numbers

Some square roots are relatively simple whole numbers, for example, 3 the square root of 9. Others are less amenable to expression, such as the square root of 2 (=1.414...), which has been proven to be an irrational number.

Square roots of negative numbers

Negative numbers have square roots that lie outside the real numbers: Multiplying a real number by itself always results in a positive number. The square roots of negative numbers involve what are called imaginary numbers:

<math>\sqrt{-1} = i</math> which for example leads to <math>\sqrt{-9} = 3i</math>

The statements in "Two square roots for each number" also apply to negative numbers. Thus the technically correct way of writing the above example would be:

<math>\sqrt{-9} = \pm 3i</math>

Alternate expressions and notations

Many computer languages and spreadsheet programs use "sqr(x)" or "sqrt(x)" to describe (or calculate) the square root of x. This is also the common notation used when mathematical symbols are not conveniently available.

The square root of a number can also be denoted as <math>x^{\frac{1}{2}}</math>. This can readily be seen by the rule of adding powers when multiplying:

<math>x^{\frac{1}{2}} \cdot x^{\frac{1}{2}} = x^{\frac{1}{2}+\frac{1}{2}} = x^{1} = x</math>

If superscript is not available, the common way of writing this is x^(1/2).

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