an introduction to the complex number system. through your time here at cocc, youve existed solely...
TRANSCRIPT
![Page 1: An introduction to the complex number system. Through your time here at COCC, youve existed solely in the real number system, often represented by a number](https://reader035.vdocuments.net/reader035/viewer/2022070306/5519d1a25503468b0c8b4826/html5/thumbnails/1.jpg)
An introduction to the complex number system
![Page 2: An introduction to the complex number system. Through your time here at COCC, youve existed solely in the real number system, often represented by a number](https://reader035.vdocuments.net/reader035/viewer/2022070306/5519d1a25503468b0c8b4826/html5/thumbnails/2.jpg)
Through your time here at COCC, you’ve existed solely in the real
number system, often represented by a number line.
![Page 3: An introduction to the complex number system. Through your time here at COCC, youve existed solely in the real number system, often represented by a number](https://reader035.vdocuments.net/reader035/viewer/2022070306/5519d1a25503468b0c8b4826/html5/thumbnails/3.jpg)
Just what is a “real” number, anyway? Can you give me an example of a real number?
![Page 4: An introduction to the complex number system. Through your time here at COCC, youve existed solely in the real number system, often represented by a number](https://reader035.vdocuments.net/reader035/viewer/2022070306/5519d1a25503468b0c8b4826/html5/thumbnails/4.jpg)
Just what is a “real” number, anyway? Can you give me an example of a real number?
![Page 5: An introduction to the complex number system. Through your time here at COCC, youve existed solely in the real number system, often represented by a number](https://reader035.vdocuments.net/reader035/viewer/2022070306/5519d1a25503468b0c8b4826/html5/thumbnails/5.jpg)
Just what is a “real” number, anyway? Can you give me an example of a real number?
![Page 6: An introduction to the complex number system. Through your time here at COCC, youve existed solely in the real number system, often represented by a number](https://reader035.vdocuments.net/reader035/viewer/2022070306/5519d1a25503468b0c8b4826/html5/thumbnails/6.jpg)
Just what is a “real” number, anyway? Can you give me an example of a real number?
![Page 7: An introduction to the complex number system. Through your time here at COCC, youve existed solely in the real number system, often represented by a number](https://reader035.vdocuments.net/reader035/viewer/2022070306/5519d1a25503468b0c8b4826/html5/thumbnails/7.jpg)
All of those real numbers (and many, many more) fit into place on the
number line you know and love so well...
![Page 8: An introduction to the complex number system. Through your time here at COCC, youve existed solely in the real number system, often represented by a number](https://reader035.vdocuments.net/reader035/viewer/2022070306/5519d1a25503468b0c8b4826/html5/thumbnails/8.jpg)
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...
![Page 10: An introduction to the complex number system. Through your time here at COCC, youve existed solely in the real number system, often represented by a number](https://reader035.vdocuments.net/reader035/viewer/2022070306/5519d1a25503468b0c8b4826/html5/thumbnails/10.jpg)
Look familiar?
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The horizontal axis is still
“real”....
Real axis
![Page 12: An introduction to the complex number system. Through your time here at COCC, youve existed solely in the real number system, often represented by a number](https://reader035.vdocuments.net/reader035/viewer/2022070306/5519d1a25503468b0c8b4826/html5/thumbnails/12.jpg)
But...what to call the
vertical axis?
Real axis
![Page 13: An introduction to the complex number system. Through your time here at COCC, youve existed solely in the real number system, often represented by a number](https://reader035.vdocuments.net/reader035/viewer/2022070306/5519d1a25503468b0c8b4826/html5/thumbnails/13.jpg)
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”
![Page 14: An introduction to the complex number system. Through your time here at COCC, youve existed solely in the real number system, often represented by a number](https://reader035.vdocuments.net/reader035/viewer/2022070306/5519d1a25503468b0c8b4826/html5/thumbnails/14.jpg)
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:
1i
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Real axis
Imaginary axisi is placed right here on our new plane...
...notice that i is not real, so it doesn’t touch the real axis.
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Now we have
1i Squaring both sides, we must
also have
2 1i
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Real axis
Imaginary axisi2 goes right here...
i2 is not imaginary...it’s real!
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Let’s continue the pattern...
1i 2 1i 3 2
( 1)
i i i
i
i
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Real axis
Imaginary axisWhere would i3 go?
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And, one last example...
4 2 2
( 1) ( 1)
1
i i i
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Real axis
Imaginary axis
And so, i4 rejoins the real realm...
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Real axis
Imaginary axis
Great...but what about a number that’s over here?
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Real axis
Imaginary axis
It’s not oneither 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
It’s 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!
22( 2) 18x
“Prove it, Rule! We think you’re full of it!”
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Real axis
Imaginary axisx = – 2 3i.
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Let’s try two more...
2 25 6x x
(3 4) 5x x
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