Question

Question 1: What are the characteristics of a complex number?

Question 2: What is the relationship between a complex number and its conjugate?

Question 3: Describe the usefulness of the conjugate and its effect on other complex numbers.

Answer 1

Complex numbers are made up of a real and an imaginary part and written in the form:
\[a+bi\]Where a and b are real numbers and i is an imaginary number.

Complex conjugates are a pair of real numbers, both having the same real part, but with imaginary parts of equal magnitude and opposite signs, for example:
\[3+4i\]\[3-4i\]Are complex conjugates.

Pairs of conjugates are significant because the imaginary unit i is qualitatively indistinct from its additive and multiplicative inverse -i, as they both satisfy the definition for the imaginary unit:
\[x^2=-1\]Thus in most "natural" settings, if a complex number provides a solution to a problem, so does its conjugate, such as is the case for complex solutions of the quadratic formula with real coefficients.