Topic: Review of Complex Numbers

## Question

Compute the Magnitude of the following Complex Numbers= a) $e^2$

b) $e^{2j}$

c) $j$

What properties of the complex magnitude can you use to check your answer?

a) $|e^2|= \sqrt{(e^2)^2}= e^2$ (Clarkjv 18:33, 10 January 2011 (UTC))

b) $|e^{(2j)}| = \sqrt{(e^2)^2} = e^2$ (Clarkjv 18:33, 10 January 2011 (UTC))

c) $|j| = \sqrt{(0^2+1^2)} = 1$(Clarkjv 18:33, 10 January 2011 (UTC))

Instructor's comments: First of all, bravo for being the first to dare to post on the course wiki!!! The answer to a) is correct and the justification is fine, because the number considered is real. However, the approach from a) does not extend to b), because in b) the number is complex. To obtain the magnitude of a complex number, you can multiply it by its complex conjugate and then take the square root of the result. Can somebody please propose a different answer for b)? Please keep the answer above "as is", since it it a very common mistake. The answer and justification of c) are both correct. -pm

b) $|e^{(2j)}| = |(cos(2) + j*sin(2))| = \sqrt{(cos(2))^2 + (sin(2))^2} = \sqrt{1} = 1$ (cmcmican 10:29, 10 January 2011 (UTC))

Instructor's comments: This answer is correct: you used Euler's formula, which is going to be very useful in ECE301. This is an alternative to multiplying the number by its complex conjugate and taking the square root of the result. A quick note though on the symbol $*$: we will be using it to denote the convolution operation later on, so it will be important not to use it to denote multiplication anymore. -pm