ML Aggarwal Class 12 Maths Solutions Section A Chapter 14 Complex Numbers

Chapter 14: Complex Numbers

14.1 Complex Numbers

A complex number is a number of the form \( a + ib \) where \( a \) and \( b \) are real numbers and \( i = \sqrt{-1} \). It can also be written as an ordered pair \( (a, b) \). The set of all complex numbers is denoted by \( C = \{z : z = a + ib, a, b \in \mathbb{R}\} \).

Real and Imaginary Parts: For a complex number \( z = a + ib \), the real part is \( \text{Re}(z) = a \) and the imaginary part is \( \text{Im}(z) = b \). Note that the imaginary part is itself a real number.

Types of Complex Numbers:

• If \( b = 0 \), then \( z = a \) is a real number.
• If \( a = 0 \) and \( b \neq 0 \), then \( z = ib \) is a purely imaginary number.
• If \( b \neq 0 \), then \( z = a + ib \) is a non-real complex number.

Equality of Complex Numbers: Two complex numbers \( z_1 = a + ib \) and \( z_2 = c + id \) are equal if and only if \( a = c \) and \( b = d \).

14.2 Algebra of Complex Numbers

14.2.1 Addition: For \( z_1 = a + ib \) and \( z_2 = c + id \), the sum is \( z_1 + z_2 = (a + c) + i(b + d) \).

Properties:

• Commutative: \( z_1 + z_2 = z_2 + z_1 \)
• Associative: \( (z_1 + z_2) + z_3 = z_1 + (z_2 + z_3) \)
• Additive Identity: \( z + 0 = z \)
• Additive Inverse: For \( z = a + ib \), the negative is \( -z = -a - ib \) such that \( z + (-z) = 0 \)

14.2.2 Subtraction: For \( z_1 = a + ib \) and \( z_2 = c + id \), the difference is \( z_1 - z_2 = (a - c) + i(b - d) \).

14.2.3 Multiplication: For \( z_1 = a + ib \) and \( z_2 = c + id \), the product is \( z_1 z_2 = (ac - bd) + i(ad + bc) \).

Properties:

• Commutative: \( z_1 z_2 = z_2 z_1 \)
• Associative: \( (z_1 z_2) z_3 = z_1 (z_2 z_3) \)
• Multiplicative Identity: \( z \cdot 1 = z \)
• Multiplicative Inverse: For \( z = a + ib \neq 0 \), the inverse is \( z^{-1} = \frac{a}{a^2 + b^2} - i\frac{b}{a^2 + b^2} \) such that \( z \cdot z^{-1} = 1 \)
• Distributive: \( z_1(z_2 + z_3) = z_1 z_2 + z_1 z_3 \)

14.2.4 Division: For \( z_1 = a + ib \) and \( z_2 = c + id \neq 0 \), the quotient is
\( \frac{z_1}{z_2} = \frac{ac + bd}{c^2 + d^2} + i\frac{bc - ad}{c^2 + d^2} \)

14.2.5 Integral Powers: For any complex number \( z \), positive integral powers are \( z^1 = z, z^2 = z \cdot z, z^3 = z^2 \cdot z \), and so on. For non-zero \( z \), negative powers are \( z^{-1} = \frac{1}{z}, z^{-2} = \frac{1}{z^2} \), and so on. Zero power gives \( z^0 = 1 \).

14.2.6 Powers of \( i \): The integral powers of \( i \) follow a pattern:

• \( i^0 = 1, i^1 = i, i^2 = -1, i^3 = -i \)
• \( i^4 = 1, i^5 = i, i^6 = -1, i^7 = -i \)
• For any integer \( k \): \( i^{4k} = 1, i^{4k+1} = i, i^{4k+2} = -1, i^{4k+3} = -i \)

14.2.7 Modulus of a Complex Number: For \( z = a + ib \), the modulus (or absolute value) is \( |z| = \sqrt{a^2 + b^2} \). Note that \( |z| \geq 0 \).

Properties of Modulus:

• \( |-z| = |z| \)
• \( |z| = 0 \) if and only if \( z = 0 \)
• \( |z_1 z_2| = |z_1| |z_2| \)
• \( \left|\frac{z_1}{z_2}\right| = \frac{|z_1|}{|z_2|} \) (provided \( z_2 \neq 0 \))

14.2.8 Conjugate of a Complex Number: For \( z = a + ib \), the conjugate is \( \overline{z} = a - ib \).

Properties of Conjugate:

• \( \overline{(\overline{z})} = z \)
• \( \overline{z_1 + z_2} = \overline{z_1} + \overline{z_2} \)
• \( \overline{z_1 - z_2} = \overline{z_1} - \overline{z_2} \)
• \( \overline{z_1 z_2} = \overline{z_1} \cdot \overline{z_2} \)
• \( \overline{\left(\frac{z_1}{z_2}\right)} = \frac{\overline{z_1}}{\overline{z_2}} \) (provided \( z_2 \neq 0 \))
• \( |\overline{z}| = |z| \)
• \( z \overline{z} = |z|^2 \)
• \( z^{-1} = \frac{\overline{z}}{|z|^2} \) (provided \( z \neq 0 \))

14.2 Representations of Complex Numbers

A complex number can be shown in two ways - through a geometric picture or through trigonometric form.

14.2.1 Geometric Representation (Argand Diagram): Every complex number \( z = x + iy \) can be shown as a point \( P(x, y) \) in a coordinate plane. The x-axis is called the real axis, and the y-axis is called the imaginary axis. This plane is called the complex plane, and the picture is called an Argand diagram.

Key Geometric Ideas:

• The distance from the origin O to point P is \( |z| = \sqrt{x^2 + y^2} \).
• If \( z_1 = x_1 + iy_1 \) and \( z_2 = x_2 + iy_2 \) are two complex numbers with corresponding points \( P_1(x_1, y_1) \) and \( P_2(x_2, y_2) \), then the distance between the two points is \( |P_1P_2| = \sqrt{(x_1 - x_2)^2 + (y_1 - y_2)^2} = |z_1 - z_2| \).
• If a point Q divides the segment \( P_1P_2 \) in the ratio \( m : n \), then Q has coordinates \( \left(\frac{mx_2 + nx_1}{m + n}, \frac{my_2 + ny_1}{m + n}\right) \).

Question. If complex numbers z₁, z₂, z₃ are the vertices A, B, C respectively of an isosceles right angled triangle at C, then show that (z₁ - z₂)² = 2(z₁ - z₃)(z₃ - z₂).
Answer: Begin by placing C at the origin. Since the triangle is isosceles and right-angled at C with BC = CA = k, we can set z₁ = k, z₂ = ik, and z₃ = 0. Now compute the left side: (z₁ - z₂)² = (k - ik)² = k²(1 - i)² = k²(1 - 2i - 1) = -2k²i. Next, compute the right side: 2(z₁ - z₃)(z₃ - z₂) = 2(k - 0)(0 - ik) = 2k(-ik) = -2k²i. Since both sides equal -2k²i, the statement is proven.
In simple words: Position C at the origin and use the equal side lengths to place the other two vertices. Then expand both sides of the equation and verify they match.

Exam Tip: When working with geometric conditions on complex numbers, always use the hint provided - placing a key vertex at the origin simplifies the algebra significantly and makes the proof straightforward.

Question. If ω (≠ 1) is a cube root of unity, then find the conjugate of 2ω - 3i.
Answer: The cube roots of unity satisfy ω³ = 1 and include the complex roots ω = \( \frac{-1 + i\sqrt{3}}{2} \) and ω = \( \frac{-1 - i\sqrt{3}}{2} \). These are complex conjugates of each other. If ω = \( \frac{-1 + i\sqrt{3}}{2} \), then the conjugate of ω is \( \frac{-1 - i\sqrt{3}}{2} \). Therefore, 2ω - 3i = 2 · \( \frac{-1 + i\sqrt{3}}{2} \) - 3i = -1 + i√3 - 3i = -1 + i(√3 - 3). The conjugate of this expression is -1 - i(√3 - 3) = -1 - i√3 + 3i = -1 + i(3 - √3).
In simple words: Take the conjugate of each part separately: the real part stays the same, but the imaginary part changes sign.

Exam Tip: Remember that for a cube root of unity ω ≠ 1, the value is always complex, not real. Use the standard forms and their conjugate relationship to simplify the final answer.

Question. If z is a complex number, prove that (z - 1)(z̄ - 1) = |z - 1|².
Answer: Let z = x + iy, where x and y are real numbers. Then z̄ = x - iy. Now, z - 1 = (x - 1) + iy. The product (z - 1)(z̄ - 1) becomes [(x - 1) + iy][(x - 1) - iy] = (x - 1)² - (iy)² = (x - 1)² + y². This is exactly the definition of |z - 1|², which equals (x - 1)² + y². Therefore, (z - 1)(z̄ - 1) = |z - 1|².
In simple words: When you multiply a complex number by its conjugate, you always get the square of its distance from the origin (or from any shifted point like 1).

Exam Tip: This is a key property: for any complex number w, the product w · w̄ always equals |w|². Use this whenever you need to convert a product involving conjugates into a modulus squared.