In the real number system, there is no solution to the equation x^2=-1x
2
=−1x, squared, equals, minus, 1. In this lesson, we will study a new number system in which the equation does have a solution.
The backbone of this new number system is the number iii, also known as the imaginary unit.
i^2=-1i
2
=−1i, squared, equals, minus, 1
\sqrt{-1}=i
−1
=isquare root of, minus, 1, end square root, equals, i
By taking multiples of this imaginary unit, we can create infinitely many more new numbers, like 3i3i3, i, i\sqrt{5}i
5
i, square root of, 5, end square root, and -12i−12iminus, 12, i. These are examples of imaginary numbers.
However, we can go even further than that and add real numbers and imaginary numbers, for example 2+7i2+7i2, plus, 7, i and 3-\sqrt{2}i3−
2
i3, minus, square root of, 2, end square root, i. These combinations are called complex numbers.
Defining complex numbers
A complex number is any number that can be written as \greenD{a}+\blueD{b}ia+bistart color #1fab54, a, end color #1fab54, plus, start color #11accd, b, end color #11accd, i, where iii is the imaginary unit and \greenD{a}astart color #1fab54, a, end color #1fab54 and \blueD{b}bstart color #11accd, b, end color #11accd are real numbers.
\begin{array}{ccc} \Large\greenD a&\Large+&\Large \blueD bi \\ \uparrow&&\uparrow\phantom{i} \\ \text{Real}&&\text{Imaginary} \\ \text{part}&&\text{part} \end{array}
a
↑
Real
part
+
bi
↑i
Imaginary
part
\greenD aastart color #1fab54, a, end color #1fab54 is called the \greenD{\text{real}}realstart color #1fab54, start text, r, e, a, l, end text, end color #1fab54 part of the number, and \blueD bbstart color #11accd, b, end color #11accd is called the \blueD{\text{imaginary}}imaginarystart color #11accd, start text, i, m, a, g, i, n, a, r, y, end text, end color #11accd part of the number.
The table below shows examples of complex numbers, with the real and imaginary parts identified. Some people find it easier to identify the real and imaginary parts if the number is written in standard form.
Complex NumberStandard Form \greenD a+\blueD b ia+bistart color #1fab54, a, end color #1fab54, plus, start color #11accd, b, end color #11accd, iDescription of parts
7i-27i−27, i, minus, 2\greenD {-2}+\blueD 7i−2+7istart color #1fab54, minus, 2, end color #1fab54, plus, start color #11accd, 7, end color #11accd, iThe real part is \greenD{-2}−2start color #1fab54, minus, 2, end color #1fab54 and the imaginary part is \blueD 77start color #11accd, 7, end color #11accd.
4-3i4−3i4, minus, 3, i\greenD 4 + (\blueD{-3})i4+(−3)istart color #1fab54, 4, end color #1fab54, plus, left parenthesis, start color #11accd, minus, 3, end color #11accd, right parenthesis, iThe real part is \greenD{4}4start color #1fab54, 4, end color #1fab54 and the imaginary part is \blueD{-3}−3start color #11accd, minus, 3, end color #11accd
9i9i9, i\greenD 0+\blueD9i0+9istart color #1fab54, 0, end color #1fab54, plus, start color #11accd, 9, end color #11accd, iThe real part is \greenD{0}0start color #1fab54, 0, end color #1fab54 and the imaginary part is \blueD 99start color #11accd, 9, end color #11accd
-2−2minus, 2\greenD {-2}+\blueD0i−2+0istart color #1fab54, minus, 2, end color #1fab54, plus, start color #11accd, 0, end color #11accd, iThe real part is \greenD{-2}−2start color #1fab54, minus, 2, end color #1fab54 and the imaginary part is \blueD 00start color #11accd, 0, end color #11accd
Step-by-step explanation:
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