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Complex Numbers and Quadratic Equations - Argand Plane and Polar Representation

Grade 11CBSE

Review the key concepts, formulae, and examples before starting your quiz.

🔑Concepts

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The Argand Plane: A two-dimensional coordinate system where the horizontal x-axis represents the real part (Real Axis) and the vertical y-axis represents the imaginary part (Imaginary Axis). Every complex number z=a+biz = a + bi is represented as a unique point P(a,b)P(a, b) on this plane. Visually, this creates a one-to-one correspondence between the set of complex numbers and the points in a plane.

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Modulus of a Complex Number: Geometrically, the modulus ∣z∣=a2+b2|z| = \sqrt{a^2 + b^2} represents the distance between the origin O(0,0)O(0,0) and the point P(a,b)P(a,b). If you draw a line from the origin to the point, the modulus is the length of this line segment, forming the hypotenuse of a right-angled triangle with base aa and height bb.

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Argument (Amplitude): The argument θ\theta of a complex number is the angle that the line segment OPOP makes with the positive direction of the real axis, measured in the counter-clockwise direction. Visually, it describes the rotational position of the point around the origin.

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Principal Argument: To ensure a unique value, the principal argument arg(z)\text{arg}(z) is restricted to the interval (−π,π](-\pi, \pi]. Visually, its value depends on the quadrant: In the 1st quadrant, θ=α\theta = \alpha; in the 2nd quadrant, θ=π−α\theta = \pi - \alpha; in the 3rd quadrant, θ=−(π−α)\theta = -(\pi - \alpha); and in the 4th quadrant, θ=−α\theta = -\alpha, where α=tan⁡−1∣ba∣\alpha = \tan^{-1}|\frac{b}{a}|.

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Polar Representation: A complex number z=a+biz = a + bi can be expressed in terms of its modulus rr and argument θ\theta as z=r(cos⁡θ+isin⁡θ)z = r(\cos \theta + i\sin \theta). In this form, rr is the 'radius' or distance from the center, and θ\theta is the 'angle' of rotation, effectively describing the point using polar coordinates (r,θ)(r, \theta).

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Conjugate on the Argand Plane: The conjugate of z=a+biz = a + bi, denoted by zˉ=a−bi\bar{z} = a - bi, is represented by the point (a,−b)(a, -b). Geometrically, this is a reflection or mirror image of the point zz across the real axis (x-axis).

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Purely Real and Purely Imaginary Numbers: On the Argand plane, purely real numbers (where b=0b=0) lie entirely on the horizontal Real Axis. Purely imaginary numbers (where a=0a=0) lie entirely on the vertical Imaginary Axis. The origin represents the complex number 0+0i0 + 0i.

📐Formulae

Modulus: ∣z∣=r=a2+b2|z| = r = \sqrt{a^2 + b^2}

Argument Calculation: tan⁡α=∣ba∣\tan \alpha = |\frac{b}{a}|, where α\alpha is the acute angle.

Polar Form: z=r(cos⁡θ+isin⁡θ)z = r(\cos \theta + i\sin \theta)

Conversion identities: a=rcos⁡θa = r\cos \theta and b=rsin⁡θb = r\sin \theta

Principal Argument (θ\theta) for z=a+biz = a + bi:

  • Q1 (a>0,b>0a>0, b>0): θ=α\theta = \alpha
  • Q2 (a<0,b>0a<0, b>0): θ=π−α\theta = \pi - \alpha
  • Q3 (a<0,b<0a<0, b<0): θ=−(π−α)\theta = -(\pi - \alpha)
  • Q4 (a>0,b<0a>0, b<0): θ=−α\theta = -\alpha

💡Examples

Problem 1:

Represent the complex number z=1+i3z = 1 + i\sqrt{3} in the polar form.

Solution:

  1. Identify aa and bb: Here a=1a = 1 and b=3b = \sqrt{3}.
  2. Calculate the modulus rr: r=a2+b2=12+(3)2=1+3=4=2r = \sqrt{a^2 + b^2} = \sqrt{1^2 + (\sqrt{3})^2} = \sqrt{1 + 3} = \sqrt{4} = 2
  3. Calculate the acute angle α\alpha: tan⁡α=∣ba∣=∣31∣=3\tan \alpha = |\frac{b}{a}| = |\frac{\sqrt{3}}{1}| = \sqrt{3} Since tan⁡π3=3\tan \frac{\pi}{3} = \sqrt{3}, we have α=π3\alpha = \frac{\pi}{3}.
  4. Determine the quadrant: Since a>0a > 0 and b>0b > 0, the point (1,3)(1, \sqrt{3}) lies in the first quadrant. Therefore, the principal argument θ=α=π3\theta = \alpha = \frac{\pi}{3}.
  5. Write in polar form: z=r(cos⁡θ+isin⁡θ)=2(cos⁡π3+isin⁡π3)z = r(\cos \theta + i\sin \theta) = 2(\cos \frac{\pi}{3} + i\sin \frac{\pi}{3})

Explanation:

We first find the magnitude (modulus) of the vector and then find its direction (argument). Since the point is in the first quadrant, the calculated acute angle is the argument.

Problem 2:

Convert the complex number z=−1−iz = -1 - i into polar form.

Solution:

  1. Identify aa and bb: a=−1,b=−1a = -1, b = -1.
  2. Calculate the modulus rr: r=(−1)2+(−1)2=2r = \sqrt{(-1)^2 + (-1)^2} = \sqrt{2}
  3. Calculate the acute angle α\alpha: tan⁡α=∣−1−1∣=1  ⟹  α=π4\tan \alpha = |\frac{-1}{-1}| = 1 \implies \alpha = \frac{\pi}{4}
  4. Determine the quadrant: Since a<0a < 0 and b<0b < 0, the point lies in the third quadrant. The principal argument is θ=−(π−α)\theta = -(\pi - \alpha).
  5. Calculate θ\theta: θ=−(π−π4)=−3π4\theta = -(\pi - \frac{\pi}{4}) = -\frac{3\pi}{4}
  6. Polar form: z=2(cos⁡(−3π4)+isin⁡(−3π4))z = \sqrt{2}(\cos(-\frac{3\pi}{4}) + i\sin(-\frac{3\pi}{4}))

Explanation:

For points in the third quadrant, the argument is measured clockwise from the negative real axis or calculated as −(π−α)-(\pi - \alpha) to stay within the principal range (−π,π](-\pi, \pi].