Class 11 Mathematics (041) · Chapter 4 NotesComplex Numbers and Quadratic Equations

Complex numbers are an extension of the real number system that includes solutions to equations like x² + 1 = 0, which have no real roots. A complex number is of the form a + ib, where a and b are real numbers, and i is the imaginary unit with the property i² = -1. The real part of the complex number is a, and the imaginary part is b. For example, 2 + 3i is a complex number where 2 is the real part and 3 is the imaginary part.

• Complex numbers extend the real number system
• Forma + ib, where a and b are real numbers
• i is the imaginary unit with i² = -1
• a is the real part, b is the imaginary part

The imaginary unit i is defined by the property i² = -1. This unit allows us to represent the square roots of negative numbers. For example, √-4 can be written as 2i because (2i)² = 4i² = 4(-1) = -4. The powers of i cycle every four exponents: i¹ = i, i² = -1, i³ = -i, i⁴ = 1, and then the cycle repeats. This cyclic nature is useful in simplifying expressions involving powers of i.

• i² = -1
• Allows representation of square roots of negative numbers
• Powers of i cycle every four exponentsi, -1, -i, 1

Complex numbers can be added, subtracted, multiplied, and divided. Addition and subtraction are performed by combining the real and imaginary parts separately. For example, (2 + 3i) + (1 + 4i) = (2 + 1) + (3i + 4i) = 3 + 7i. Multiplication involves using the distributive property and the fact that i² = -1. Division of complex numbers involves multiplying the numerator and the denominator by the conjugate of the denominator to eliminate the imaginary part in the denominator.

• Addition(a + ib) + (c + id) = (a + c) + i(b + d)
• Subtraction(a + ib) - (c + id) = (a - c) + i(b - d)
• Multiplication(a + ib)(c + id) = (ac - bd) + i(ad + bc)
• Division(a + ib)/(c + id) = [(a + ib)(c - id)]/(c² + d²)

The modulus of a complex number z = a + ib is given by |z| = √(a² + b²). The modulus represents the distance of the point (a, b) from the origin in the Argand plane. The conjugate of a complex number z = a + ib is z = a - ib. The conjugate has important properties, such as zz = |z|² and z + z = 2Re(z). The conjugate is useful in various operations, including division and finding the inverse of a complex number.

• Modulus|z| = √(a² + b²)
• Conjugatez = a - ib
• Propertieszz = |z|², z + z = 2Re(z)
• Useful in division and finding the inverse

The Argand plane is a graphical representation of complex numbers where the x-axis represents the real part and the y-axis represents the imaginary part. Each complex number z = a + ib corresponds to a point (a, b) in the Argand plane. The polar representation of a complex number involves expressing it in terms of its modulus and argument (angle). The argument θ is the angle that the line from the origin to the point (a, b) makes with the positive x-axis. This representation is useful in multiplying and dividing complex numbers using trigonometric identities.

• Graphical representation of complex numbers
• x-axisreal part, y-axis: imaginary part
• Polar representationz = r(cosθ + i sinθ)
• r = modulus, θ = argument