Square Roots of Negative Numbers & Graphing of Same

Apr 2010
61
13
Prescott, Arizona
1. At present I am working with Imag. and Complex Numbers. I want to determine the square roots of the first 100 negaive numbers. So far have done first 20. Would someone "in the know" please check and make sure I have them done correctly --

Sq. Rt. Of Equals
-1 i
-2 i x sq.rt. of 2
-3 i x sq.rt. of 3
-4 2i and -2i
-5 i x sq.rt. of 5
-6 i x sq.rt. of 6
-7 i x sq.rt. of 7
-8 2i x sq.rt. of 2
-9 3i and -3i
-10 i x sq.rt. of 10
-11 i x sq.rt. of 11
-12 2i x sq.rt. of 3
-13 i x sq.rt. of 13
-14 i x sq.rt. of 14
-15 i x sq.rt. of 15
-16 4i or -4i
-17 i x sq. rt. 17
-18 3i x sq.rt. of 2
-19 i x sq.rt. of 19
-20 2i x sq.rt. of 5

2. When I am finished with all 100 negative numbers I want to somehow graph them. Can someone give me the method to use? Complex plane? Polar coordinates? Something else?
 

Prove It

MHF Helper
Aug 2008
12,883
4,999
1. At present I am working with Imag. and Complex Numbers. I want to determine the square roots of the first 100 negaive numbers. So far have done first 20. Would someone "in the know" please check and make sure I have them done correctly --

Sq. Rt. Of Equals
-1 i
-2 i x sq.rt. of 2
-3 i x sq.rt. of 3
-4 2i and -2i
-5 i x sq.rt. of 5
-6 i x sq.rt. of 6
-7 i x sq.rt. of 7
-8 2i x sq.rt. of 2
-9 3i and -3i
-10 i x sq.rt. of 10
-11 i x sq.rt. of 11
-12 2i x sq.rt. of 3
-13 i x sq.rt. of 13
-14 i x sq.rt. of 14
-15 i x sq.rt. of 15
-16 4i or -4i
-17 i x sq. rt. 17
-18 3i x sq.rt. of 2
-19 i x sq.rt. of 19
-20 2i x sq.rt. of 5

2. When I am finished with all 100 negative numbers I want to somehow graph them. Can someone give me the method to use? Complex plane? Polar coordinates? Something else?
In general, if \(\displaystyle a > 0\) and \(\displaystyle a \in \mathbf{R}\)

then \(\displaystyle \sqrt{-a} = i\sqrt{a}\).


To graph them, you use a compass and straight-edge.

Draw an Argand Diagram and draw the line \(\displaystyle \textrm{Re}\,{z} = 1\)

Plot the point \(\displaystyle (1, 1)\) and join it to the origin. Now you have a \(\displaystyle 1, 1, \sqrt{2}\) right angle triangle. If you position your compass with its centre at the origin and tip at \(\displaystyle (1, 1)\) and draw the arc to the imaginary axis, you will have plotted \(\displaystyle i\sqrt{2}\).

Now draw a horizontal line from that point. If you join its intersection with \(\displaystyle \textrm{Re}\,{z}\) to the origin, you will have a \(\displaystyle 1, \sqrt{2}, \sqrt{3}\) right angle triangle. By drawing the arc to the imaginary axis, you will now have plotted \(\displaystyle i\sqrt{3}\).

Draw a horizontal line from that point. Join its intersection with \(\displaystyle \textrm{Re}\,{z}\) to the origin and you will have a \(\displaystyle 1, \sqrt{3}, \sqrt{4}\) right angle triangle. Draw the arc to the imaginary axis and you will have plotted \(\displaystyle i\sqrt{4} = 2i\).

Follow this process for as long as you like and you will have plotted the square roots of negative integers.
 
Apr 2010
61
13
Prescott, Arizona
"Argand Diagram" ?

Is this what I would call a "Complex Plane" where vertical axis consists of both positive and negative imaginary numbers and horizontal axis consists of integers? Or, is it something else? If it is something else, where can I find an example of an "Argand Diagram?

Thanks for your help.
 
Apr 2008
748
159
Is this what I would call a "Complex Plane" where vertical axis consists of both positive and negative imaginary numbers and horizontal axis consists of integers? Or, is it something else? If it is something else, where can I find an example of an "Argand Diagram?

Thanks for your help.
Yes this is what you would call a Complex Plane.

A good example of an Argand Diagram can be found on Wikipedia (link below), or just get onto Google or Youtube and do some searches.

Complex plane - Wikipedia, the free encyclopedia
 
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