Geometrical Meaning of Argument and Modulus - I: Complex Number

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1. Arg(z) = pi/3

z = 0 is excluded

since z = 0 is the only complex number whose argument is not defined

2. Arg(z) > pi/3

Since principal argument belongs to (-pi, pi], therefore Argument cannot be greater than pi.

therefore Arg(z) > pi/3 =>) 

z = 0 is excluded

3. |Arg(z)| < pi/4

=> -pi/4 < Arg(z) < pi/4

z = 0 is excluded

4.

=> Arg(z) = pi

since, principal argument belongs to , therefore Argument cannot be greater than pi.

z = 0 is excluded

5. 

-pi/3 is included

z = 0 is excluded

6. Complex number  , equal only when z is purely real number or '0'




IMP
7. Arg(|z|) = 0

Since |z| is a Positive Real number, therefore it lies on right side of x - axis i.e. on positive Real axis and x-axis has '0' argument because it makes 0 radian angle from x-axis

For z = 0 , argument does not exist

Therefore equation is satisfied for all values of z except z = 0

8. 

Any Positive Real number has argument 0 and any negative real number has argument pi

Modulus of Any complex number except '0' is a Positive Real number therefore Argument of mod of any complex number is '0'


Since, 

therefore, 



This is always true because argument of modulus of any complex number is 0.

Therefore LHS and RHS both are zero for all z except i and -i because at '0' argument does not exist

Therefore equation is true for all value of z except i and -i




9. Im(z) > 1

If z = x + iy

then, y > 1

10. Im(z - 2) > 1

same region as of point 9 

z = x + iy

=> z -2 = (x - 2) + iy

=> y > 1




11. Im(z - i) > 1

=> y - 1 > 1

or, y > 2

since z - i = x + i (y - 2)




12. -1 < Im(z + i) < 1

=> -1 < y + 1 < 1

=> -2 < y < 0

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