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Imaginary Concepts -- Complex Numbers

Module by: Kenny M. Felder. E-mail the author

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Summary: This module introduces the concept of complex numbers in Algebra.

Note: Your browser may not currently support MathML. See our browser support page for additional details. You can always view the correct math in the PDF version.

A “complex number” is the sum of two parts: a real number by itself, and a real number multiplied by ii. It can therefore be written as a+bia+bi, where aa and bibi are real numbers.

The first part, aa, is referred to as the real part. The second part, bibi, is referred to as the imaginary part.

Table 1
Examples of complex numbers a+bia+bi (aa is the “real part”; bibi is the “imaginary part”)
3+2i3+2i a=3a=3, b=2b=2
ππ a=πa=π, b=0b=0(no imaginary part: a “pure real number”)
-i-i a=0a=0, b=-1b=-1 (no real part: a “pure imaginary number”)

Some numbers are not obviously in the form a+bia+bi. However, any number can be put in this form.

Table 2
Example 1: Putting a fraction into a+bia+bi form (ii in the numerator)
3−4i53−4i5 size 12{ { {3 - 4i} over {5} } } {} is a valid complex number. But it is not in the form a+bia+bi, and we cannot immediately see what the real and imaginary parts are.
To see the parts, we rewrite it like this:
3−4i53−4i5 size 12{ { {3 - 4i} over {5} } } {}== 3535 size 12{ { {3} over {5} } } {}– 45i45 size 12{ { {4} over {5} } } {}i
Why does that work? It’s just the ordinary rules of fractions, applied backward. (Try multiplying and then subtracting on the right to confirm this.) But now we have a form we can use:
3−4i53−4i5 size 12{ { {3 - 4i} over {5} } } {} a=a= 3535 size 12{ { {3} over {5} } } {}, b=– 45b=–45 size 12{ { {4} over {5} } } {}
So we see that fractions are very easy to break up, if the ii is in the numerator. An ii in the denominator is a bit trickier to deal with.
Table 3
Example 2: Putting a fraction into a+bia+bi form (ii in the denominator)
1i1i size 12{ { {1} over {i} } } {} == 1⋅ii⋅i1⋅ii⋅i size 12{ { {1 cdot i} over {i cdot i} } } {} Multiplying the top and bottom of a fraction by the same number never changes the value of the fraction: it just rewrites it in a different form.
== i−1i−1 size 12{ { {i} over { - 1} } } {} Because i•i i•i is i2i2, or –1.
=-i=-i This is not a property of ii, but of –1. Similarly,5−15−1 size 12{ { {5} over { - 1} } } {} = –5=–5.
1i1i size 12{ { {1} over {i} } } {}: a=0a=0, b=-1b=-1 since we rewrote it as -i-i, or 0-1i0-1i

Finally, what if the denominator is a more complicated complex number? The trick in this case is similar to the trick we used for rationalizing the denominator: we multiply by a quantity known as the complex conjugate of the denominator.

Definition of Complex Conjugate

The complex conjugate of the number a+bia+bi is a-bia-bi. In words, you leave the real part alone, and change the sign of the imaginary part.

Here is how we can use the “complex conjugate” to simplify a fraction.

Table 4
Example: Using the Complex Conjugate to put a fraction into a+bia+bi form
53−4i53−4i size 12{ { {5} over {3 - 4i} } } {} The fraction: a complex number not currently in the form a + b i a+bi
== 5(3+4i)(3−4i)(3+4i)5(3+4i)(3−4i)(3+4i) size 12{ { {5` \( 3+4i \) } over { \( 3 - 4i \) \( 3+4i \) } } } {} Multiply the top and bottom by the complex conjugate of the denominator
== 15+20i32−(4i)215+20i32−(4i)2 size 12{ { {"15"+"20"i} over {3 rSup { size 8{2} } - \( 4i \) rSup { size 8{2} } } } } {} Remember, (x+a)(x–a)=x2–a2(x+a)(x–a)=x2–a2
== 15+20i9+1615+20i9+16 size 12{ { {"15"+"20"i} over {9+"16"} } } {} (4i)2=42i2=16(–1)=–16(4i)2=42i2=16(–1)=–16, which we are subtracting from 9
== 15+20i2515+20i25 size 12{ { {"15"+"20"i} over {"25"} } } {} Success! The top has ii, but the bottom doesn’t. This is easy to deal with.
== 15251525 size 12{ { {"15"} over {"25"} } } {}++ 20i2520i25 size 12{ { {"20"i} over {"25"} } } {} Break the fraction up, just as we did in a previous example.
== 3535 size 12{ { {3} over {5} } } {}++ 45i45 size 12{ { {4} over {5} } } {}i So we’re there! a=a= 3535 size 12{ { {3} over {5} } } {} and b=45b=45 size 12{ { {4} over {5} } } {}

Any number of any kind can be written as a+bia+bi. The above examples show how to rewrite fractions in this form. In the text, you go through a worksheet designed to rewrite −13−13 size 12{ nroot { size 8{3} } { - 1} } {} as three different complex numbers. Once you understand this exercise, you can rewrite other radicals, such as ii size 12{ sqrt {i} } {}, in a+bia+bi form.

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