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A standard number could be represented in the form of words, digits, an expanded form, or a pictorial diagram. Polar representation is another way of presenting a number in terms of modulus ( r ) and arguments (θ). This type of presentation is essential for polar coordinates and is helpful for the graphical representation as it helps to plot the coordinates of the point on the graph. Graphically and theoretically, a polar coordinate could be presented in a rectangular expression or a polar expression.
Rectangular representation is the rectangular expression of a complex number and is given as z = a +bi, where i is the imaginary number and (a, b) is the rectangular coordinates. The polar representation is another form of representation and is the most adaptable type, obtained from the rectangular expression (z=a+ib). The formula which makes polar representation of the number practicable to rectangular form is given as:
z = r cosθ + i r sinθ which then equals to r (cosθ + i sinθ)
Here,
z= complex number
r= absolute value or modulus
i= imaginary number
θ= argument of the number
Graphical Representation of Complex Number
The polar form is the most helpful type for representing a number with the help of polar coordinates. The general formula for a number with polar coordinates (x, y) is z = r cosθ + i r sinθ. This formula is of large scale use for the graphical representation of a number. It makes the process much easier for determining the navigation and position of the coordinate in the graph. Consider an example with a complex number z = x+iy in a two-dimensional coordinate system for better understanding. The figure below is the diagrammatic form of the polar expression with the coordinate named “A,” which makes the angle of θ with the horizontal component and has polar coordinates (x, y).
According to the diagram, the horizontal and vertical axes mentioned are real and imaginary, respectively. The constant “r” represents the length of the vector from the origin (0,0) to point A (x,y). The line joining the origin to point A makes an angle of value θ with the horizontal component, the positive X-axis.
Let the graph’s origin be named O and the x- coordinate of point A be called B. As per the figure, we can see a triangle formation called AOB. Since the segment AB is perpendicular to the X-axis, we could make use of the Pythagorean Theorem and can write,
r2 = x2 + y2
From trigonometric ratios, we know that,
Cos θ = adjacent side of angle θ / Hypotenuse= x/r
Sin θ = opposite side of angle θ / Hypotenuse = y/r
Now, multiplying each side by r,
r x cosθ = x and r x sinθ = y
Substituting the value (magnitude) of x and y in the rectangular form z = x+ iy,
z= x+ iy = r*cos θ + r*sinθ = r*( cos θ +i sin θ )
z= r*( cos θ +i sin θ )
Therefore from the expression mentioned above, we can prove that the polar form of a complex number is given as z= r( cos θ +i sin θ ).
Where, r=|z|=√(x2+y2), x =r *cosθ, y =r*sinθ
And the value of θ could be,
- θ=tan-1(y/x) for x>0……( If z lies in the first or fourth quadrant)
- θ=tan-1(y/x)+π ……..( If z lies in the second quadrant)
- θ=tan-1(y/x)-180° for x<0 ……..( If z lies in the third quadrant)
Conversion of Coordinates from Rectangular Form to Polar Form
As of now, you may have a brief idea about rectangular representation and the polar representation of complex numbers. Both these presentation forms play an essential role in the concept of imaginary numbers in mathematics. Because they can be used as the solutions for many questions and are dependent on each other, this dependency property can be proved ahead in our polar representation of complex number study material as we convert a complex number from its rectangular form to polar form.
The conversion is carried out using the formulas of r = √(a2 + b2) and θ = tan-1(b / a). The argument’s value (r) should be calculated by taking the square root of the sum of squares of the coordinates as mentioned in the rectangular form. For knowing the value of the magnitude (θ), simply take the inverse tangent function for the value “b” divided by the value of “a.” For better understanding, let’s consider an example with the complex number z= -1 + √3i.
Our priority step would be to calculate the modulus value and the argument. Therefore calculating the angle between the line segments.
tan-1(√3 / (-1)) = tan-1(-√3) = -tan-1(√3) = 60 °.
Since the complex number lies in the second quadrant, the value of argument θ will be 180°-60°= 120°. Therefore the value of the argument ( θ ) has its value as 120°. Now calculating the value of “r” or the magnitude of z which is: |z| = √((-1)2 + (√3)2) = √(1+3) = √4 = 2
Therefore the value of modulus is 2. Hence the converted polar representation is
z= 2( cos 60° + i sin 60° )
Products of Polar Form of a Number
Let us discuss the product of polar forms in the study material notes on the Polar representation of Complex numbers. Consider two complex numbers in polar form as z = r1(cos θ1 + i sin θ1), y = r2(cos θ2 + i sin θ2).
Multiplying both the complex numbers,
z*w = r1(cos θ1 + i sin θ1) * r2(cos θ2 + i sin θ2)
= r1r2 [(cos θ1cos θ2 – sin θ1sin θ2) + i (sin θ1cos θ2 + cos θ1sin θ2)]
= r1r2 [cos(θ1 + θ2) + i sin(θ1 + θ2)]
Therefore, the product of two polar forms of a number is given as:
Product = r1r2 [cos(θ1 + θ2) + i sin(θ1 + θ2)]
Conclusion
The concept of polar representation of complex numbers is crucial to learn. It helps solve any mathematical concept related to imaginary numbers. Also, the polar representation could be a valuable tool to simplify the problem and make its solving process more efficient. This polar representation of the complex number study material provided above could help you understand the topic more quickly and efficiently. Studying all issues from the study material notes on polar representation notes on complex numbers can help you achieve the milestone over the concept of complex numbers in mathematics.
