Complex Number

Pronunciation: /kəmˈplɛks nʌm bər/ ?

A complex number is a number that contains two parts: a real part and an 'imaginary' part.[1] The real part is any real number. The imaginary part is a real number multiplied by the imaginary unit written as the lower-case letter 'i'. The imaginary unit i represents squareroot(-1).

Complex numbers are typically written in the form a+bi. where a is the real part and bi is the imaginary part. a+bi can also be written a+b*squareroot(-1) or a+squareroot(-b^2)

An imaginary number is a complex number with no real part, such as squareroot(-5) or 6.2i. This is sometimes called a pure imaginary number.

check mark Understanding Check

Classifying Numbers

For each number, decide which class or classes to which each number belongs. Click 'yes' or 'no' under each class of numbers. A number may belong to more than one class.

NumberComplexRealPure Imaginary
6-6i yes no yes no yes no
5 yes no yes no yes no
2i yes no yes no yes no
-3+4x yes no yes no yes no
Table 1: Classifying numbers

Complex Plane

Complex plane with 4+6i, -2+2i, -6, -6-4i, -4i, 6-2i, 8+4i plotted
Figure 1: Complex plane

Complex numbers can be represented as points on the complex plane.[2] Like the Cartesian plane, the complex plane has two axes: the real axis and the imaginary axis. The real axis is the horizontal axis. The real axis is also the real number line, put into the complex plane. The vertical axis is the imaginary axis. The complex plane is also called an Argand diagram.

In figure 1, find the point 4+6i, which is red. Note that it corresponds to 4 on the real axis and 6 on the imaginary axis. Each point is plotted the same way. If a number has no imaginary part, it is a real number, and is found on the real axis. In figure 1, -6 is a real number on the real axis. If a number is an imaginary number, it is found on the imaginary axis. The number -4i in figure 1 is an example of an imaginary number on the imaginary axis.

Powers of i

Table of powers of i
PowerValue
i1i
i2i·i = -1
i3i·(-1) = -i
i4i·(-i) = -i2 = -(-1) = 1
i5i·1 = i (See i1)
i6Click for answer
i7Click for answer
i8Click for answer
Table 1: Powers of i
When simplifying expressions containing the imaginary constant i, it is important to know the values of the integral powers of i. To understand this, remember the definition of i: i=square root(-1). Since the square root function is the inverse of the square function, (square root(a))^2=a. So, (square root(-1))^2=(i)^2=-1. Table 1 gives the values of the first five integral powers of i. Can you figure out the next three? Come up with your answer, then click on 'Click for answer' to check your answer.

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Manipulative 1: Powers of i. Created with GeoGebra.

Adding Complex Numbers

When adding complex numbers, add corresponding parts. Add the real parts together and the imaginary parts together:
a+bi + c+di = (a+c)+(b+d)i

Here are some examples:
3+2i + 1-3i = (3+1)+(2-3)i = 4-1i = 4-i
4+1i + -2+2i = (4-2)+(1+2)i = 2+3i

Multiplying Complex Numbers

To multiply complex numbers, cross multiply the parts:
×(a+bi)
×(c+di)
×(a·c + bc
×(a·c + a·di + bdi
+ac + (ad + bc)i + bdi2

It is important to remember that i2 = -1. Substitute -1 for i2 in the equation:
= ac + (ad + bc)i + bd·(-1)
= ac + (ad + bc)i - bd·
= ac - bd + (ad + bc)i

Here is an example:
(1-3i)·(-6+2i)
= 1·(-6)+1·2i+(-3)·(-6)i+(-32i2
= -6+2i+18i-6i2
= -6+20i-6i2

Substitute -1 for i2 in the equation:

-6+20i-6i2
= -6+20i-6(-1)
= -6+20i+6
= -6+6+20i
= 0+20i
= 20i

Properties of Complex Numbers

PropertyValueExampleDescription
Additive Identity 0+0i 3-4i + 0+0i = 3+0 + (-4+o)i = 3-4i The additive identity is the number that, when added to any complex number, gives the original number unchanged. The additive identity for complex numbers is 0+0i.
Multiplicative Identity 1+0i (-1+4i)*(1+0i) =(-1*1+-1*0i+4i*1+4i*0i) =-1+0+4i+0i =-1+4i The multiplicative identity is the number that, when multiplied by any complex number, gives the original number unchanged. The multiplicative identity for complex numbers is 1+0i.
Additive Inverse -(a+bi) = -a-bi 3-2i + -3+2i = 3+(-3) + -2i+2i = 0+0i The additive inverse of a complex number is the number that when added to the original number gives the additive identity. The additive identity of an arbitrary complex number a+bi is -(a+bi) = -a-bi.
Multiplicative Inverse (a+bi)^(-1)=1/(a+bi)=1/(a+bi)*(a-bi)/(a-bi)=(a-bi)/(a^2+b^2)=(a/(a^2+b^2)-bi/(a^2+b^2) (2-3i)^(-1)=1/(2-3i)=1/(2-3i)*(2+3i)/(2+3i)=(2+3i)/(2^2+(-3)^2)=(2/(4+9)+3i/(4+9)=2/13+3/13i The multiplicative inverse of a complex number is the number that when multiplied by the original number gives the multiplicative identity. The multiplicative inverse of an arbitrary complex number a+bi is
a/(a^2+b^2)-b/(a^2+b^2)i.
Associate Property of Addition (a+bi)+[(c+di)+(e+fi)]=[(a+bi)+(c+di)]+(e+fi) (2-3i) + [(5 + 2i) + (-3 - 6i)]= [(2-3i) + (5+2i)] + (-3-6i) implies (2-3i) + [2-4i]= [7-i] + (-3-6i) implies 4-7i=4-7i The associative property of addition states that the order in which complex numbers are added does not matter.
Commutative Property of Addition (a+bi)+(c+di)=(c+di)+(a+bi) (2-4i)+(-3-i)=(-1-5i)=(-3-i)+(2-4i) The commutative property of addition states that addends of complex numbers can be switched without changing the result.
Associate Property of Multiplication (a+bi)*[(c+di)*(e+fi)]=[(a+bi)*(c+di)]*(e+fi) (5+2i)*[(-3-6i)*(2-3i)]=[(5+2i)*(-3-6i)]*(2-3i) implies (5+2i)*(-24-3i)=(-3-36i)*(2-3i) implies -114-63i=-114-63i. The associative property of multiplication states that the order in which complex numbers are added does not matter.
Commutative Property of Multiplication (a+bi)*(c+di)=(c+di)*(a+bi) (1+2i)*(-2+i)=(-2+i)*(1+2i) implies -4-3i=-4-3i The commutative property of multiplication states that multiplicands of complex numbers can be switched without changing the result.
Table 1: Properties of complex numbers.

Addition

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Manipulative 2: Addition of complex numbers. Created with GeoGebra.

Multiplication

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Manipulative 3: Multiplication of complex numbers. Created with GeoGebra.

Additive Inverse

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Manipulative 4: Additive inverse of a complex number. Created with GeoGebra.

Multiplicative Inverse

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Manipulative 5: Multiplicative inverse of a complex number. Created with GeoGebra.

Cite this article as:


Complex Number. 2010-01-08. All Math Words Encyclopedia. Life is a Story Problem LLC. http://www.allmathwords.org/en/c/complexnumber.html.

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Revision History


2010-01-08: Added "References" (McAdams, David.)
2008-12-26: Changed equations from html to images (McAdams, David.)
2008-07-17: Added addition, multiplication, and properties of complex numbers (McAdams, David.)
2008-06-09: Added paragraph on imaginary numbers. Changed images to equations (McAdams, David.)
2008-03-25: Changed More Information to current standard (McAdams, David.)
2007-07-12: Initial version (McAdams, David.)

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