Mathematical Investigations III
Name:
Mathematical Investigations III - A View of the World
An Introduction to Complex Numbers
The system of real numbers is not quite algebraically “complete” since not all polynomials have zeroes in the real numbers. For example, the quadratic polynomial has no real number zeroes. In order to be able to solve all polynomial equations, we need to expand our system of numbers.
Let , so . With this definition of the imaginary unit, all polynomials whose coefficients are real numbers have zeroes. For example, the function above has zeroes and , since
and
.
In fact, with this definition, all negative real numbers have square roots. For instance, .
Complex numbers are numbers that can be written as where a and b are real numbers. All real numbers are complex numbers, since the real number can be written as. Therefore, the real numbers are a subset of the complex numbers. Other examples of complex numbers include (which can be written as) and .
Since the definition of complex numbers is based on real numbers (a and b), operations with complex numbers are natural extensions of operations with real numbers and so are fairly intuitive. Perform each of the following operations and write all complex answers in this form.
1. Simplify.
+
2. Simplify.
3. Multiply and simplify.
3. (cont.)
Previously, you learned to rationalize the denominator of by multiplying the numerator and denominator by the conjugate of the denominator, namely. Do that now:
Now apply that same technique to rewrite . Multiply the numerator and denominator by so that its denominator is rationalized.
We call the complex conjugate of .
Complex Conjugates
If is a complex number, the complex conjugate of z is and is denoted .
Remark: We will mostly use the simpler term “conjugate” instead of “complex conjugate”.
Remark: We usually pronounce as “z bar”.
4. Use the conjugate of each denominator to rationalize the following:
5. Solve each of the following equations for z. Express your answer in a + bi form.
(a) (b)
6. Now try your hand at simplifying powers of the imaginary unit, i. Look for a pattern.
i1 = i (wow!)
7. What pattern do you notice about the powers of i?
8. Use the pattern you found to evaluate.
We have seen how to solve linear equations involving complex numbers. We now shift our
attention to quadratic equations.
9. Solve each quadratic equation below by completing the square:
(a) (b) (c)
10. What do you notice about the pairs of solutions to the quadratic equations in Question 9?
11. Solve each of the following equations for z over the complex numbers. The quadratic formula may be useful.
(a) (b)
12. Does your observation from Question 11 hold with the pairs of solutions to the quadratic equations in Question 11? Explain.
13. What was different about the coefficients in the quadratic equations between Question 9 and Question 11?
14. Summarize the observation about the pair of solutions to a quadratic equation: if the coefficients are real numbers and one solution is a non-real complex number, then the solutions are…
15. (Optional problem—come back to problem 15 if you have time after completing the rest of this packet.) Use the quadratic formula to prove your answer to Question 14 about the solutions of the quadratic equation
if a, b, and c are all real numbers.
16. Let . Enter this as y1(x) in your calculator. Then use your calculator to evaluate at the following pairs of points (you may simply type y1(1 + i); the “i” key is 2ND-Catalog on the TI-89):
(a) ,
(b) ,
(c) ,
17. Describe the relationship demonstrated by your answers to the previous Question. Your response should probably have a in it somewhere!
Poly 7.XXX Rev. S11