Convert the rectangular form of the complex number
2−2i
into polar form. Show all work and label the modulus and argument.
We start by finding the modulus (magnitude) of the complex number:
|2-2i| = √(2^2 + (-2)^2) = √8 = 2√2
Next, we find the argument (angle) of the complex number:
tanθ = Im/Re = (-2)/2 = -1
Since the real part is positive and the imaginary part is negative, the angle θ is in the fourth quadrant. We can find the angle by taking the arctan of -1 and adding 360 degrees:
θ = arctan(-1) + 360° = 315°
Therefore, the polar form of 2-2i is:
2√2 cis(315°)
where cis(theta) is a shorthand notation for cos(theta) + i sin(theta).
redo it but show how to find the nodulous (steps) and argument (steps)
Certainly!
To convert 2-2i from rectangular form to polar form, we need to find its modulus (magnitude) and argument (angle).
Modulus: The modulus of a complex number is the distance from the origin to the point that represents the number in the complex plane. In this case, we can use the Pythagorean theorem to find the modulus:
|2 - 2i| = sqrt( (2^2) + (-2)^2 ) = sqrt(8)
To simplify the root, we can factor out a 2:
|2 - 2i| = 2 * sqrt(2)
So the modulus of 2-2i is 2 sqrt(2).
Argument: The argument of a complex number is the angle it makes with the positive real axis in the complex plane. To find the argument, we want to determine the arctan of the ratio of the imaginary part to the real part.
theta = arctan( -2/2 )
= arctan( -1 )
This gives us an angle in the fourth quadrant (where x is positive and y is negative), so we need to add 360 degrees to get the standard angle measure:
theta = arctan( -1 ) + 360
= 315 degrees
So the argument of 2-2i in polar form is 315 degrees (or pi/4 radians).
Putting it all together, we can write 2-2i in polar form as:
2 sqrt(2) cis(315 degrees)
or
2 sqrt(2) cis( pi/4 )
To convert a complex number from rectangular form (a+bi) to polar form (r∠θ), we need to find the modulus (r) and the argument (θ) of the complex number.
In rectangular form, the given complex number is 2 - 2i.
Step 1: Find the modulus (r):
The modulus of a complex number is the distance from the origin to the point representing the complex number in the complex plane. It can be found using the formula:
r = √(a^2 + b^2)
In this case, a = 2 and b = -2. Plugging in these values, we have:
r = √(2^2 + (-2)^2)
= √(4 + 4)
= √8
Simplifying further, we have:
r = √(4 * 2)
= √4 * √2
= 2√2
So, the modulus (r) of the given complex number is 2√2.
Step 2: Find the argument (θ):
The argument of a complex number is the angle formed by the positive real axis and the line connecting the origin and the point representing the complex number in the complex plane. It can be found using the formula:
θ = tan^(-1)(b / a)
In this case, a = 2 and b = -2. Plugging in these values, we have:
θ = tan^(-1)((-2) / 2)
= tan^(-1)(-1)
= -π/4
So, the argument (θ) of the given complex number is -π/4.
Therefore, the polar form of the complex number 2 - 2i is 2√2 ∠ -π/4.