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An introduction to the complex number system

An introduction to the complex number system. Through your time here at COCC, you’ve existed solely in the real number system, often represented by a number line . Just what is a “real” number, anyway? Can you give me an example of a real number? .

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An introduction to the complex number system

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  1. An introduction to the complex number system

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

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

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

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

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

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

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

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

  10. Look familiar?

  11. The horizontal axis is still “real”.... Real axis

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

  13. 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”

  14. 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!”

  15. The “imaginary unit” is cleverly called i and is defined like this:

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

  17. Now we have Squaring both sides, we must also have

  18. Imaginary axis i2 goes right here... Real axis i2 is not imaginary... it’s real!

  19. Let’s continue the pattern...

  20. Imaginary axis Where would i3 go? Real axis

  21. And, one last example...

  22. Imaginary axis And so, i4 rejoins the real realm... Real axis

  23. Imaginary axis Real axis Great...but what about a number that’s over here?

  24. Imaginary axis Real axis It’s not on either axis!

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

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

  27. Imaginary axis Real axis It’s written – 2 + 3i.

  28. Believe it or not...that last complex number is one of the solutions to our previous quadratic equation! “Prove it, Rule! We think you’re full of it!”

  29. Imaginary axis x = – 2  3i. Real axis

  30. Let’s try two more...

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