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4.4 The Modulus and the Conjugate of a Complex Number

🎓 Class 11 Mathematics CBSE Theory Ch 4 — Complex Numbers and Quadratic Equations ⏱ ~15 min
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This MCQ module is based on: 4.4 The Modulus and the Conjugate of a Complex Number

This mathematics assessment will be based on: 4.4 The Modulus and the Conjugate of a Complex Number
Targeting Class 11 level in Algebra, with Advanced difficulty.

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4.4 The Modulus and the Conjugate of a Complex Number

Modulus
For \(z=a+ib\), the modulus is the non-negative real number \[\boxed{\;|z|=\sqrt{a^2+b^2}\;}\] The modulus is zero iff \(a=b=0\), i.e. iff \(z=0\).
Conjugate
The conjugate of \(z=a+ib\) is \[\boxed{\;\bar z=a-ib\;}\] The conjugate flips the sign of the imaginary part. \(\bar z\) is sometimes written \(z^*\).

Algebraic identities (proof exercises)

For complex numbers \(z, z_1, z_2\), the following hold:

Modulus
\(|z_1z_2|=|z_1|\,|z_2|\)
\(\left|\dfrac{z_1}{z_2}\right|=\dfrac{|z_1|}{|z_2|}\) (\(z_2\ne 0\))
\(z\bar z=|z|^2\)
Conjugate
\(\overline{z_1+z_2}=\bar z_1+\bar z_2\)
\(\overline{z_1z_2}=\bar z_1\,\bar z_2\)
\(\overline{(\bar z)}=z,\quad z+\bar z=2\,\text{Re}(z),\quad z-\bar z=2i\,\text{Im}(z)\)

4.5 Argand Plane and Polar Representation

The Swiss bookkeeper Jean-Robert Argand? (1806) noticed that complex numbers correspond naturally to points in a 2-dimensional plane: the complex number \(z=x+iy\) is identified with the point \(P(x,y)\). The plane equipped with this identification is the Argand plane; the x-axis is called the real axis, the y-axis the imaginary axis.

X (real) Y (imag) O 1 2 −1 −2 1 2 3 4 −1 A(2,4) = 2+4i B(−2,3) = −2+3i C(0,1) = i D(2,−1) = 2−i E(−1,−2) = −1−2i
Fig 4.1: Complex numbers plotted as points in the Argand plane.

Modulus = distance from origin

The modulus \(|z|=\sqrt{x^2+y^2}\) is exactly the distance \(OP\) between the origin and the representing point \(P(x,y)\).

X Y O(0,0) P(x, y) |z|=√(x²+y²) x y
Fig 4.2: |z| is the length of OP — Pythagoras' theorem applied to coordinates (x, y).

Conjugate = mirror image across the real axis

Reflecting \(P(x,y)\) across the x-axis gives \(Q(x,-y)\), which is exactly the point representing \(\bar z\):

X Y O P(x, y) → z Q(x, −y) → z̄
Fig 4.3: P represents z, Q represents z̄. They are mirror images across the real axis.

Polar form

Pick \(z=x+iy\ne 0\) in the Argand plane. Let \(r=|z|=\sqrt{x^2+y^2}\) and let \(\theta\) be the angle the line \(OP\) makes with the positive real axis (measured anti-clockwise). Then

\[x=r\cos\theta,\qquad y=r\sin\theta,\]

so

Polar form
\[\boxed{\;z=r(\cos\theta+i\sin\theta)\;}\] where \(r=|z|\ge 0\) and \(\theta\) is called the argument \(\arg z\). Any \(\theta\) differing by a multiple of \(2\pi\) is also a valid argument; the unique value in \((-\pi,\pi]\) is the principal argument \(\text{Arg}(z)\).

To compute \(\theta\): use \(\tan\theta=y/x\) and pick the quadrant from the signs of \(x\) and \(y\).

Quadrant rule for the argument
Quadrant of (x, y)I (x>0, y>0)II (x<0, y>0)III (x<0, y<0)IV (x>0, y<0)
Arg z range(0, π/2)(π/2, π)(−π, −π/2)(−π/2, 0)
Interactive: Argand Plane & Polar Form

Drag the red point. The simulation displays the rectangular form, modulus, and principal argument live; the dashed line shows |z| measured from the origin.

Re Im
z = 3 + 2i  |  |z| = 3.606  |  Arg(z) = 33.69° = 0.5880 rad
Polar form: z = 3.606 (cos 0.588 + i sin 0.588)

Worked Examples

Example 6. Find the conjugate and modulus of \(z=2-3i\), and the multiplicative inverse.
\(\bar z=2+3i\). \(|z|^2=4+9=13\), so \(|z|=\sqrt{13}\). The inverse is \[z^{-1}=\dfrac{\bar z}{|z|^2}=\dfrac{2+3i}{13}=\dfrac{2}{13}+\dfrac{3}{13}i.\]
Example 7. Express \(z=-\sqrt 3+i\) in polar form.
\(r=|z|=\sqrt{3+1}=2\). For Arg: \((x,y)=(-\sqrt 3, 1)\) lies in Q2 (x<0, y>0). \(\tan\alpha=|y/x|=1/\sqrt 3\), so the reference angle is \(\pi/6\). In Q2, \(\theta=\pi-\pi/6=5\pi/6\). Hence \[z=2\!\left(\cos\dfrac{5\pi}{6}+i\sin\dfrac{5\pi}{6}\right).\]
Example 8. Express \(z=1-i\) in polar form.
\(r=\sqrt{1+1}=\sqrt 2\). \((x,y)=(1,-1)\) is in Q4. \(\tan\alpha=1\), reference angle \(\pi/4\). In Q4, \(\theta=-\pi/4\). So \[z=\sqrt 2\!\left(\cos\!\left(-\dfrac{\pi}{4}\right)+i\sin\!\left(-\dfrac{\pi}{4}\right)\right).\]
Example 9. Find the conjugate of \(\dfrac{(3-2i)(2+3i)}{(1+2i)(2-i)}\).
Use \(\overline{z_1/z_2}=\bar z_1/\bar z_2\) and \(\overline{z_1z_2}=\bar z_1\bar z_2\). So required conjugate is \[\dfrac{(3+2i)(2-3i)}{(1-2i)(2+i)}.\] Compute numerator: \((3+2i)(2-3i)=6-9i+4i-6i^2=12-5i\). Denominator: \((1-2i)(2+i)=2+i-4i-2i^2=4-3i\). Ratio: \[\dfrac{12-5i}{4-3i}\cdot\dfrac{4+3i}{4+3i}=\dfrac{48+36i-20i-15i^2}{16+9}=\dfrac{63+16i}{25}=\dfrac{63}{25}+\dfrac{16}{25}i.\]
Activity: Plot Polar Forms by Hand
L3 Apply
Materials: Graph paper, ruler, protractor.
Predict: Will multiplying \(1+i\) by itself rotate or scale? By how much?
  1. Plot \(z_1=1+i\) on graph paper. Measure \(|z_1|\) with a ruler and Arg with a protractor. Confirm \(|z_1|=\sqrt 2\) and Arg \(=\pi/4\).
  2. Compute \(z_1^2=(1+i)^2=1+2i+i^2=2i\). Plot it. Measure modulus and argument.
  3. You should find \(|z_1^2|=2\) (modulus squared) and Arg = \(\pi/2\) (argument doubled). The pattern: multiplication = scale moduli + add arguments.
  4. Confirm with \(z_1^3\) and \(z_1^4\).
Multiplication of complex numbers in polar form: \(r_1(\cos\theta_1+i\sin\theta_1)\cdot r_2(\cos\theta_2+i\sin\theta_2)=r_1r_2(\cos(\theta_1+\theta_2)+i\sin(\theta_1+\theta_2))\). Moduli multiply; arguments add. This is why repeated multiplication of \(z\) traces an exponential spiral.

Competency-Based Questions

Scenario: A 2D rotation of a vector by angle \(\alpha\) (anti-clockwise) is implemented by multiplying its complex representation by \(e^{i\alpha}=\cos\alpha+i\sin\alpha\). A graphics engine uses this to rotate a point P at \(z=3+4i\) by 60° about the origin.
Q1. The modulus of \(z=4-3i\) is:
L3 Apply
  • (a) 1
  • (b) 5
  • (c) 7
  • (d) 25
Answer: (b) 5. \(|z|=\sqrt{16+9}=5\).
Q2. (Fill the blank) The conjugate of \(z=-1+i\) lies in quadrant ____ of the Argand plane.
L2 Understand
Answer: \(\bar z=-1-i\), which is in Q3 (x<0, y<0).
Q3. The polar form of \(z=-1-i\sqrt 3\) is:
L3 Apply
Answer: \(r=\sqrt{1+3}=2\). \((x,y)=(-1,-\sqrt 3)\) is in Q3. Reference angle: \(\tan^{-1}(\sqrt 3)=\pi/3\). In Q3, principal Arg \(=-\pi+\pi/3=-2\pi/3\). So \(z=2(\cos(-2\pi/3)+i\sin(-2\pi/3))\).
Q4. (True/False) "\(|z_1+z_2|\le|z_1|+|z_2|\) for all complex \(z_1,z_2\), with equality only if they have the same argument." Justify geometrically.
L5 Evaluate
True. This is the triangle inequality. Geometrically, \(z_1, z_2\) are two vectors from the origin; \(z_1+z_2\) is the parallelogram diagonal. The diagonal length is at most the sum of the side lengths, with equality only when the two sides are parallel — i.e. same argument.
Q5. Apply the rotation: rotate \(z=3+4i\) by 60° about the origin (multiply by \(\cos 60°+i\sin 60°\)). Find the new complex number.
L4 Analyse
Solution: \(\cos 60°+i\sin 60°=\dfrac{1}{2}+\dfrac{\sqrt 3}{2}i\). Multiply: \((3+4i)(\tfrac{1}{2}+\tfrac{\sqrt 3}{2}i)=\tfrac{3}{2}+\tfrac{3\sqrt 3}{2}i+2i+2\sqrt 3 i^2=\left(\tfrac{3}{2}-2\sqrt 3\right)+\left(\tfrac{3\sqrt 3}{2}+2\right)i\). Modulus is preserved: \(|z|=5\) before and after.

Assertion–Reason Questions

Assertion (A): \(z\bar z=|z|^2\) for every complex number \(z\).
Reason (R): \((a+ib)(a-ib)=a^2+b^2\).
(a) Both true, R explains A.
(b) Both true, R doesn't explain A.
(c) A true, R false.
(d) A false, R true.
Answer: (a). R is the algebraic step that gives \(a^2+b^2=|z|^2\), the very content of A.
Assertion (A): The principal argument of \(-1\) is \(\pi\).
Reason (R): Arg z lies in \((-\pi, \pi]\).
(a) Both true, R explains A.
(b) Both true, R doesn't explain A.
(c) A true, R false.
(d) A false, R true.
Answer: (a). \(-1\) is on the negative real axis. The angle from the positive real axis is \(\pi\), and π lies in \((-\pi,\pi]\) by the rule (R).
Assertion (A): Multiplying any complex number by \(i\) rotates it by 90° anti-clockwise.
Reason (R): \(i=\cos(\pi/2)+i\sin(\pi/2)\), so multiplication by \(i\) adds \(\pi/2\) to the argument.
(a) Both true, R explains A.
(b) Both true, R doesn't explain A.
(c) A true, R false.
(d) A false, R true.
Answer: (a). Polar-form multiplication adds arguments; \(|i|=1\) so modulus is preserved. R is exactly the geometric reason for A.

Frequently Asked Questions

What is the modulus of a complex number?
The modulus of z = a + ib is |z| = √(a² + b²). Geometrically, it is the distance of the point (a, b) from the origin in the Argand plane.
What is the conjugate of a complex number?
The conjugate of z = a + ib is z̄ = a − ib. Geometrically it is the reflection of z across the real axis. Note z·z̄ = |z|² and z + z̄ = 2 Re(z).
What is the Argand plane?
The Argand plane is the coordinate plane in which a complex number z = x + iy is identified with the point P(x, y). The x-axis is the real axis; the y-axis is the imaginary axis.
What is the polar form of a complex number?
z = r(cos θ + i sin θ), where r = |z| ≥ 0 and θ = arg(z) is the angle the line OP makes with the positive real axis. The principal argument satisfies −π < θ ≤ π.
How do you convert a + ib to polar form?
Compute r = √(a² + b²); find θ from cos θ = a/r and sin θ = b/r (or tan θ = b/a, choosing the correct quadrant). Then z = r(cos θ + i sin θ).
What is the principal argument?
The unique value θ of arg(z) lying in the interval (−π, π], denoted Arg(z). Other valid arguments differ from Arg(z) by integer multiples of 2π.
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