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Thought-Process to Discover Knowledge

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mathsComplex NumbersComplex Plane and Polar Form

### Argument of a complex Number

Argument is defined for a complex number represented as a point in Argand plane.

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A Complex number a+i b is represented on complex plane as shown in the figure. What is the angle the line segment OP makes with the real axis?

• tan^(-1)(b/a)
• tan^(-1)(b/a)
• 0

The answer is 'tan^(-1)(b/a)'

The angle between the real axis in positive direction and the line segment OP is called the argument of the complex number. The argument of a complex number is given as text(arg) z.

What does "argument" mean?

• an independent element that plays a role in determining the value of something
• an independent element that plays a role in determining the value of something
• complete absence and avoidance of mentioning of something

The answer is 'an independent element that plays a role in determining the value of something'.
For example the arguments of a function f() = 2xy+3 are the variables x and y. In this case, the arguments define the value of the function.

In complex numbers, the modulus provides the absolute value. In addition to the modulus, to completely specify the complex number, the additional independent element required is the "argument".

For z=a+i b in CC, argument is given by text(arg) z = tan^(-1)(b/a)

What is arg(z) called?

• Pronunciation : Say the answer once

The answer is 'Argument or arg'.

The angle made by the line segment with the real axis is the argument of the complex number.

Argument: For a complex number z=a+i b in CC, the argument of z is given as  text(arg) z = tan^(-1)(b/a)

Find the argument of 1+i

• tan^(-1)1
• tan^(-1)1
• sqrt(2)

The answer is 'tan^(-1)1'

What are the possible values of tan^(-1)1?

• only one possible value pi/4
• Two possible values pi/4 and (3pi)/4
• (n pi)/4 where n = 1,3,5,cdots
• (n pi)/4 where n = 1,3,5,cdots

The answer is '(n pi)/4 where n = 1,3,5,cdots'.

The possible values of tan^(-1)1 are (n pi)/4 where n = 1,3,5,cdots.

What are the possible values of tan^(-1)1, if the opposite side and adjacent sides are given as 1?

• Answer is same : (n pi)/4 where n = 1,3,5,cdots
• Answer is different : (n pi)/4 where n = 1,5,9,cdots
• Answer is different : (n pi)/4 where n = 1,5,9,cdots

'Answer is different : (n pi)/4 where n = 1,5,9,cdots'

Note that: tan^(-1)1 = (n pi)/4 where n = 1,3,5,cdots but

tan^(-1)  (text(opposite-side)=1)/(text(adjacent-side)=1) = (n pi)/4 where n = 1,5,9,cdots and

tan^(-1)  (text(opposite-side)=-1)/(text(adjacent-side)=-1) = (n pi)/4 where n = 3,7,11,cdots

The argument of a complex number can be given in the form 2n pi + theta where n=0,1,2.... The argument corresponding to n=0 is called the principal argument. The range of principal argument is -pi <= theta <= pi.

Principal argument of a complex number is in the range -pi to pi.

Principal Argument: For a complex number z=a+i b in CC, the principal argument of z is given as  text(arg) z = tan^(-1)(b/a), where -pi <= text(arg) z <= pi

Given the complex numbers z_1, z_2, z_3, z_4 as shown in figure. The value of argument of complex numbers are
•  0 < text(arg) z_1 < pi/2
•  pi/2 < text(arg) z_2 < pi
•  0< text(arg) z_3 <-pi/2
•  -pi/2< text(arg) z_4 <-pi

Solved Exercise Problem:

Find the principal argument of 1-i

• pi/4
• -pi/4
• -pi/4
• +-pi/4

The answer is '-pi/4'

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