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Complex Numbers and Quadratic Equations - The Modulus and the Conjugate of a Complex Number

Grade 11CBSE

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

🔑Concepts

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The Conjugate of a Complex Number: For any complex number z=a+ibz = a + ib, its conjugate is denoted by zˉ\bar{z} and is defined as a−iba - ib. Visually, in the Argand plane, zˉ\bar{z} is the mirror image or reflection of the point P(a,b)P(a, b) about the real axis (horizontal x-axis), resulting in the point (a,−b)(a, -b).

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The Modulus of a Complex Number: The modulus of z=a+ibz = a + ib, denoted by ∣z∣|z|, is the non-negative real number a2+b2\sqrt{a^2 + b^2}. Geometrically, it represents the absolute distance of the point zz from the origin (0,0)(0, 0) in the complex plane, similar to how the hypotenuse of a right-angled triangle is calculated using the Pythagorean theorem.

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Relationship between Modulus and Conjugate: A fundamental identity connects these two concepts: zzˉ=∣z∣2z \bar{z} = |z|^2. This shows that the product of a complex number and its conjugate is always a non-negative real number equal to the square of its distance from the origin.

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Multiplicative Inverse: The reciprocal or multiplicative inverse of a non-zero complex number zz is given by z−1=zˉ∣z∣2z^{-1} = \frac{\bar{z}}{|z|^2}. This formula allows us to express the inverse in the standard form a+iba + ib by using the conjugate to rationalize the denominator.

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Distributive Properties of Conjugates: The conjugate operation distributes over all basic arithmetic operations. Specifically, the conjugate of a sum, difference, product, or quotient of complex numbers is equal to the sum, difference, product, or quotient of their individual conjugates (e.g., z1+z2‾=zˉ1+zˉ2\overline{z_1 + z_2} = \bar{z}_1 + \bar{z}_2).

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Multiplicative Properties of Modulus: The modulus of a product of complex numbers is the product of their moduli, ∣z1z2∣=∣z1∣∣z2∣|z_1 z_2| = |z_1| |z_2|. Similarly, for quotients, ∣z1z2∣=∣z1∣∣z2∣|\frac{z_1}{z_2}| = \frac{|z_1|}{|z_2|} (provided z2≠0z_2 \neq 0). Visually, multiplying complex numbers scales their distances from the origin multiplicatively.

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Triangle Inequality: For any two complex numbers z1z_1 and z2z_2, the property ∣z1+z2∣≤∣z1∣+∣z2∣|z_1 + z_2| \leq |z_1| + |z_2| holds true. Geometrically, this represents the fact that in a triangle formed by the origin and the points representing z1z_1 and z1+z2z_1 + z_2, the length of one side is always less than or equal to the sum of the lengths of the other two sides.

📐Formulae

If z=a+ibz = a + ib, then zˉ=a−ib\bar{z} = a - ib

∣z∣=a2+b2|z| = \sqrt{a^2 + b^2}

zzˉ=∣z∣2z \bar{z} = |z|^2

z−1=1z=zˉ∣z∣2=a−iba2+b2z^{-1} = \frac{1}{z} = \frac{\bar{z}}{|z|^2} = \frac{a - ib}{a^2 + b^2}

z1±z2‾=zˉ1±zˉ2\overline{z_1 \pm z_2} = \bar{z}_1 \pm \bar{z}_2

z1z2‾=zˉ1zˉ2\overline{z_1 z_2} = \bar{z}_1 \bar{z}_2

(z1z2)‾=zˉ1zˉ2\overline{\left(\frac{z_1}{z_2}\right)} = \frac{\bar{z}_1}{\bar{z}_2}

∣z1z2∣=∣z1∣∣z2∣|z_1 z_2| = |z_1| |z_2| and ∣z1z2∣=∣z1∣∣z2∣|\frac{z_1}{z_2}| = \frac{|z_1|}{|z_2|}

💡Examples

Problem 1:

Find the multiplicative inverse of the complex number z=4−3iz = 4 - 3i.

Solution:

Step 1: Identify a=4a = 4 and b=−3b = -3. Step 2: Find the conjugate zˉ=4+3i\bar{z} = 4 + 3i. Step 3: Calculate the square of the modulus ∣z∣2=a2+b2=42+(−3)2=16+9=25|z|^2 = a^2 + b^2 = 4^2 + (-3)^2 = 16 + 9 = 25. Step 4: Use the formula z−1=zˉ∣z∣2z^{-1} = \frac{\bar{z}}{|z|^2}. z−1=4+3i25=425+325iz^{-1} = \frac{4 + 3i}{25} = \frac{4}{25} + \frac{3}{25}i.

Explanation:

To find the multiplicative inverse, we use the relationship between the conjugate and the square of the modulus to clear the imaginary part from the denominator.

Problem 2:

Given z1=1+iz_1 = 1 + i and z2=3+4iz_2 = 3 + 4i, verify the property ∣z1z2∣=∣z1∣∣z2∣|z_1 z_2| = |z_1| |z_2|.

Solution:

Step 1: Calculate z1z2=(1+i)(3+4i)=3+4i+3i+4i2=3+7i−4=−1+7iz_1 z_2 = (1 + i)(3 + 4i) = 3 + 4i + 3i + 4i^2 = 3 + 7i - 4 = -1 + 7i. Step 2: Calculate ∣z1z2∣=(−1)2+72=1+49=50=52|z_1 z_2| = \sqrt{(-1)^2 + 7^2} = \sqrt{1 + 49} = \sqrt{50} = 5\sqrt{2}. Step 3: Calculate ∣z1∣=12+12=2|z_1| = \sqrt{1^2 + 1^2} = \sqrt{2}. Step 4: Calculate ∣z2∣=32+42=9+16=25=5|z_2| = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5. Step 5: Multiply ∣z1∣|z_1| and ∣z2∣|z_2|: ∣z1∣∣z2∣=2×5=52|z_1| |z_2| = \sqrt{2} \times 5 = 5\sqrt{2}. Since 52=525\sqrt{2} = 5\sqrt{2}, the property is verified.

Explanation:

This example demonstrates that the modulus of the product of two complex numbers is equivalent to the product of their individual moduli.