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1-10 Introduction to Complex Numbers What is a complex number?

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To see a complex number we have to first see where it shows up Solve both of these Uhoh…….what do I do here?

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Um, no solution???? does not have a real answer. It has an imaginary answer. To define a complex number we have to create a new variable. This new variable is i

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Definition: Note: i is the representation for, not a simplification of So, following this definition:

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And it cycles…. Do you see a pattern yet?

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What is that pattern? We are looking at the remainder when the power is divided by 4. Why? Every doesnt matter. It is what remains after all of the are taken out. Try it with

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Hints to deal with i 1. Find all is at the beginning of a problem. 2. Treat all is like variables, with all rules of exponents holding. 3. Reduce the power of i at the end by the rules we just learned..

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Examples

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OK, so what is a complex number? A complex number has two parts – a real part and an imaginary part. A complex number comes in the form a + bi real imaginary

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And just so you know… All real numbers are complex 3 = 3 + 0i All imaginary numbers are complex 7i = 0 + 7i Again, treat the i as a variable and you will have no problems.

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Lets try these 4 problems.

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More Practice

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Complex Number System Reals Rationals (fractions, decimals) Integers (…, -1, -2, 0, 1, 2, …) Whole (0, 1, 2, …) Natural (1, 2, …) Irrationals.

Complex Number System Reals Rationals (fractions, decimals) Integers (…, -1, -2, 0, 1, 2, …) Whole (0, 1, 2, …) Natural (1, 2, …) Irrationals.

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