IIT JEE , AIEEE Forum - Mathematics , Algebra

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Imaginary axis

real axisThe above diagram is an Argand diagram. Notice that the real numbers are on the x-axis and the imaginary numbers are on the y-axis. Finding imaginary numbers in this plane are as easy as finding points in the real plane. In the form a + bi, a is the real part and b is the imaginary part. Move a units right or left ( depending on + or - ) and b units up or down
(depending on + or -).
Rectangular form: z = a + bi
Polar form: z = r cos 0 + (r sin 0)i (remember a = r cos 0 and b = r sin 0 substitute these in and presto!)
factor out the r and get: z = r(cos 0 + i sin 0)
Math shorthand looks like this: z = r cis 0
The absolute value of z (the distance from any point to the origin) =
_______
| z | = / a² + b² (good old Pythagorus again)

Example problems:

1) Express 3 cis 50^o in rectangular form.
    Solution: using the fact that a = r cos 0 and b = r sin 0 implies that
                   a = 3 cos 50 and b= 3 sin 50. Using your calculator gets us
                   a = 1.93 and b = 2.30 with both answers rounded to hundredths.
                Therefore the answer is: 1.93 + 2.30i
2) Express -1 -2i in polar form.
                                                          __________         __
    Solution: Use the fact that r = \/(-1)² + (-2)² r = \/ 5 = 2.24. Now use the fact that the Tan 0 = (y/x) (see page 1 if you forgot!!).   0 = tan^-1(2/1). Using your calculator gives us 63.4^o. Add 180 (why? we are in the 3rd quadrant!!!) 63.4 + 180 = 243.4
                  Therefore, the answer is: 2.24 cis 243.4^o

To Multiply two complex numbers in polar form:
1) Multiply their absolute values
2) Add their polar angles.
In math terms if z₁ = r cis w and z₂ = s cis y then z₁z₂ = rs cis ( w + y)

Example problem:

1) Express (5cis 30^o)(7cis60^o) in polar and rectangular form.
    Solution: Polar form first: multiply the radii and add the angles.
                    Answer in polar form: 35 cis 90^o
        Now change this to get the rectangular form.
Remember, a = r cos 0 and b = r sin 0
so a = 35 cos 90 = 0 and b = 35 sin 90 = 35.
Answer in rectangular form is: 0 + 35i

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