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Published byKenyon Alvarez Modified over 2 years ago

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An introduction to the complex number system

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Through your time here at COCC, youve existed solely in the real number system, often represented by a number line.

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Just what is a real number, anyway? Can you give me an example of a real number?

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All of those real numbers (and many, many more) fit into place on the number line you know and love so well...

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But...what lies above and below our beloved number line?

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To visualize, mathematicians placed another axis perpendicular to the real axis...

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Look familiar?

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The horizontal axis is still real.... Real axis

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But...what to call the vertical axis? Real axis

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Remember, in MTH 065, whenever you found the square root of a negative number....and just stopped? What did you write for your solution? No Real Number

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Well, the region above and below the real axis is where all of those square roots of negative numbers live. This axis is called the imaginary axis...unfortunately. But why, Sean? Why is it called imaginary? Please tell us!

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The imaginary unit is cleverly called i and is defined like this:

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Real axis Imaginary axis i is placed right here on our new plane......notice that i is not real, so it doesnt touch the real axis.

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Now we have Squaring both sides, we must also have

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Real axis Imaginary axis i 2 goes right here... i 2 is not imaginary... its real!

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Lets continue the pattern...

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Real axis Imaginary axis Where would i 3 go?

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And, one last example...

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Real axis Imaginary axis And so, i 4 rejoins the real realm...

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Real axis Imaginary axis Great...but what about a number thats over here?

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Real axis Imaginary axis Its not on either axis!

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Real axis Imaginary axis This type of number is called complex; it has both a real and imaginary part.

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Real axis Imaginary axis Its real coordinate is – 2 and its imaginary coordinate is 3.

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Real axis Imaginary axis Its written – 2 + 3i.

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Believe it or not...that last complex number is one of the solutions to our previous quadratic equation! Prove it, Rule! We think youre full of it!

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Real axis Imaginary axis x = – 2 3i.

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Lets try two more...

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