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.