Question

Explain, in complete sentences and in your own words, the answers to the following questions relating to complex numbers.

What are the characteristics of a complex number? (1 point)
What is the relationship between a complex number and its conjugate? (2 point)
Describe the usefulness of the conjugate and its effect on other complex numbers. (2 points)

Answer 1

A complex number is composed of a real part and an imaginary part.
The real part is a real number. The imaginary part is a real number times i.

Answer 2

1. It is 2-dimensional as opposed to 1-dimensional.

2. The conjugate of a complex number is the complex number with the imaginary part changed from positive to negative (or vice versa). On an Argand diagram, it appears that the complex number has been reflected in the x axis.

3. The conjugate can be added to the original number and then halved (i.e. (1/2)(z+z*) ) to extract the real part of the variable. This can be modified slightly to extract the imaginary part instead (i.e. (1/2)(z-z*) ).

Answer 3

The conjugate of a complex number has an opposite imaginary part. For example, the number a+bi has conjugate a-bi.

Answer 4

Complex number is a number which can be put in the form a + bi.

Answer 5

Sum of a number and its conjugate is 2 times the real part.
Product of a number and its conjugate is the sum of the squares of the real and imaginary parts, and is the square of the amplitude or "length" of the number.