\require{AMSmath} \require{eqn-number}

Imaginary Numbers

(last update:   17 Feb 2013)

Imaginary numbers are used throughout mathematics, science, and engineering.

In mathematics an imaginary number is often denoted by using the character "i". In electrical engineering it is usually denoted using the letter "j".

\label{eqn:MathematicsBasedImaginaryFund} i \cdot i = -1
\label{eqn:MathematicsBasedImaginary} i = \sqrt{-1}

For mathematics.

\label{eqn:EEBasedImaginaryFund} j \cdot j = -1
\label{eqn:EEBasedImaginary} j = \sqrt{-1}

For electrical engineering.

The different letter for electrical engineering is used because the letter "i" is often used to denote electrical current. The use of "j" helps to avoid confusion.

Other representations

Historically complex numbers have been represented as a pair of numbers[1],

\label{eqn:ClasicMathBasedImaginary} (a,b)

In this representation the a represents the real part of an imaginary number, and the b represents the imaginary part.

Properties

The inverse is the negative

\label{eqn:InverseIsNegJ} \frac{1}{i} = -i
1 = -i(i)
1 = (-1)(i \cdot i)
1 = (-1)i^2
1 = (-1)(-1)
1 = 1

Association across a difference

If the following is true,

ia - ib = i(a-b)

and,

ia - ib = ia + \frac{1}{i}b

then,

i(a-b) = ia + \frac{1}{i}b

Factoring out an 1/i,

i(a-b) = \frac{1}{i}(i^2 a + b)
i(a-b) = \frac{1}{i}([-1] a + b)
i(a-b) = \frac{1}{i}(-a + b)
i(a-b) = \frac{1}{i}(b -a)
i(a-b) = -j(b -a)

In this way the sense of the difference can be swapped by using the conjugate of the imaginary term.

[1]: "Complex Analysis",Ian Stewart, David Tall, 1983, Cambridge University Press

[2]: "Engineering Circuit Analysis",William H. Hayt, Jr., Jack E. Kemmerly, 1993, McGraw-Hill