# Difference between revisions of "Imaginary number"

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An '''imaginary number''' is a [[complex number]] whose [[real part]] is equal to 0. In the [[complex plane]], these numbers lie on the [[imaginary axis]]. They are sometimes also refered to as ''pure imaginary numbers'', | An '''imaginary number''' is a [[complex number]] whose [[real part]] is equal to 0. In the [[complex plane]], these numbers lie on the [[imaginary axis]]. They are sometimes also refered to as ''pure imaginary numbers'', | ||

− | The | + | The number <math>z</math> is imaginary if and only if <math>z = i \textrm{Im}(z)</math>, where <math>i = \sqrt{-1}</math> is the [[imaginary unit]] and <math>\textrm{Im}</math> is the [[imaginary part]] [[function]]. |

− | The imaginary unit is <math>i</math>, which is <math>\sqrt{-1}</math>. <math>i</math> can be treated as a variable with the distinctive property that <math>i^2= -1</math>. Powers of <math>i</math> can be equal to only 4 possibilities: <math>-i, i, 1, -1</math> because the values cycle. This means that <math>i^1=i</math>, <math>i^2=-1</math>, <math>i^3=-i</math>, <math>i^4=1</math>, <math>i^5=i....</math>. | + | The imaginary unit is <math>i</math>, which is <math>\sqrt{-1}</math>. <math>i</math> can be treated as a variable with the distinctive property that <math>i^2= -1</math>. Powers of <math>i</math> can be equal to only 4 possibilities: <math>-i, i, 1, -1</math> because the values cycle. This means that <math>i^1=i</math>, <math>i^2=-1</math>, <math>i^3=-i</math>, <math>i^4=1</math>, <math>i^5=i....</math>. |

− | Multiplying imaginary/complex numbers is not so much different from multiplying variables. In fact, the easiest way to multiply imaginary numbers is to treat <math>i</math> as a variable and then check to see if there is either <math>i^2</math> or <math>i^4</math> in which case you switch it its real value accordingly. | + | Multiplying imaginary/complex numbers is not so much different from multiplying variables. In fact, the easiest way to multiply imaginary numbers is to treat <math>i</math> as a variable and then check to see if there is either <math>i^2</math> or <math>i^4</math> in which case you switch it its real value accordingly. |

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+ | Dividing imaginary/complex numbers is also somewhat different than normal variables. The first thing you want to do in a fraction with complex or imaginary numbers in the numerator or denominator is to make the denominator real. To do this you'll want to multiply both sides by the denominator's [[Complex conjugate]]. This will make the denominator and possibly the numerator real. From there you must simplify the fraction, and you'll either get a complex or real number. | ||

+ | It is a common mistake to say that <math>\sqrt{1}</math> is <math>i</math>, rather than <math>\pm i</math>, which allows for many mathematical fallacies to be made such as <math>1=2</math>. Another mistake that is made often is that <math>\sqrt{-4}= 4i</math>, but is really <math>\sqrt{-4}=2i</math> since it is under a radical. | ||

==See Also== | ==See Also== |

## Revision as of 11:10, 7 April 2011

An **imaginary number** is a complex number whose real part is equal to 0. In the complex plane, these numbers lie on the imaginary axis. They are sometimes also refered to as *pure imaginary numbers*,

The number is imaginary if and only if , where is the imaginary unit and is the imaginary part function.

The imaginary unit is , which is . can be treated as a variable with the distinctive property that . Powers of can be equal to only 4 possibilities: because the values cycle. This means that , , , , .

Multiplying imaginary/complex numbers is not so much different from multiplying variables. In fact, the easiest way to multiply imaginary numbers is to treat as a variable and then check to see if there is either or in which case you switch it its real value accordingly.

Dividing imaginary/complex numbers is also somewhat different than normal variables. The first thing you want to do in a fraction with complex or imaginary numbers in the numerator or denominator is to make the denominator real. To do this you'll want to multiply both sides by the denominator's Complex conjugate. This will make the denominator and possibly the numerator real. From there you must simplify the fraction, and you'll either get a complex or real number.

It is a common mistake to say that is , rather than , which allows for many mathematical fallacies to be made such as . Another mistake that is made often is that , but is really since it is under a radical.

## See Also

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