Скачать или смотреть LECTURE 3 COMPLEX NUMBERS CHAPTER 7 CLASS 11

Introduction to Complex Numbers: - In mathematics, complex numbers extend the concept of real numbers by incorporating the square root of negative numbers. The introduction of complex numbers arises from the need to solve equations like x^2 + 1 = 0, which has no real solution.
2. Imaginary Numbers: - The square root of negative numbers less than or equal to zero. x = square root of -1 = i (iota)
3. Square Root of a Negative Number: - If a is a positive real number, then, square root of -a = i * square root of a
4. Powers of i: - The powers of i repeat in a cycle of four.
i^1 = i, i^2 = -1, i^3 = -i, i^4 = 1.
5. Definition of a Complex Number: - A complex number is of the form z = a + ib, where –
a is the real part denoted as Rez,
b is the imaginary part denoted as Imz,
i is the imaginary unit, and i^2 = -1.
6. Formulas and Key Concepts: -
(i) Equality of Complex Numbers: - Two complex numbers z1 = a + ib and z2 = c + id are equal if and only if a = c and b = d.
(ii) Addition and Subtraction of Complex Numbers: - For two complex numbers z1 = a + ib and z2 = c + id –
Addition: - z1 + z2 = (a + c) + i(b + d)
Subtraction: - z1 - z2 = (a - c) + i(b - d)
(iii) Multiplication of Complex Numbers: - For two complex numbers z1 = a + ib and z2 = c + id –
z1z2 = (ac - bd) + i(ad + bc)
(iv) Conjugate of a Complex Number: - The conjugate of a complex number z = a + ib is denoted by overline {z} and is given by:
overline{z} = a – ib.
(v) Modulus of a Complex Number: - The modulus (or absolute value) of a complex number z = a + ib is denoted by |z| and is given by:
|z| = sqrt{a^2 + b^2}
(vi) Division of Complex Numbers: - For two complex numbers z1 = a + ib and z2 = c + id, the division z1/z2 is:
z1/z2 = (ac + bd) + i(bc - ad)/(c^2 + d^2).
7. Polar Form of a Complex Number: - Any complex number z = a + ib can also be represented in polar form.
z = r(costheta + isintheta) where –
r = |z| = sqrt{a^2 + b^2} is the modulus of z,
theta = arg(z) is the argument (or angle) of z.
8. De Moivre’s Theorem: - For a complex number z = r(costheta + isintheta) and any integer n: -
z^n = r^n[(cos (ntheta) + i sin (ntheta)]
Here’s an easy step-by-step approach to solve complex number problems: -

1. Understand the Problem Type: -
Identify the type of problem you’re dealing with:
Is it about simplifying expressions?
Are you required to find the modulus or argument?
Is it about performing addition, subtraction, multiplication, or division of complex numbers?
2. Basic Operations (Addition, Subtraction, Multiplication, Division)
(i) Addition and Subtraction: - Treat complex numbers like binomials (real and imaginary parts separately).
Example: - (3 + 2i) + (1 + 4i) = (3 + 1) + (2i + 4i) = 4 + 6i
(ii) Multiplication: - Apply the distributive property (a + ib)(c + id), and remember that (i^2 = -1).
Example: - (2 + 3i)(1 + 4i) = 2(1) + 2(4i) + 3i(1) + 3i(4i) = 2 + 8i + 3i - 12 = -10 + 11i
(iii) Division: - Multiply numerator and denominator by the conjugate of the denominator.
Example: - (2 + 3i)/(1 + 4i) = (2 + 3i)(1 - 4i)/(1 + 4i)(1 - 4i)} = (2 + 3i)(1 - 4i)/{1^2 - (4i)^2} = (-10 - 5i)/17 = -10/17 – 5/17i
3. Modulus and Argument: -
(i) Modulus: - Formula: - |z| = sqrt(a^2 + b^2),where z = a + ib.
Example: - z = 3 + 4i, |z| = sqrt(3^2 + 4^2) = sqrt{9 + 16} = 5.
(ii) Argument: - Formula: - theta = tan^-1(b/a).
Example: - z = 1 + i, theta = tan^-1 (1/1) = pi/4
Always check the quadrant of the complex number to assign the correct argument value.
4. Using Conjugates: - If z = a + ib, then overline{z} = a - ib.
The product of a complex number and its conjugate gives a real number.\
Z*overline{z} = (a + ib)(a - ib) = a^2 + b^2
Use the conjugate when dividing complex numbers.
5. Polar and Euler’s Form: -
(i) Converting to Polar Form: - Use the modulus r = |z| and argument theta = arg(z).
Example: - z = 1 + i, |z| = sqrt{1^2 + 1^2} = sqrt{2}, theta = pi/4 So, z = sqrt{2} [cos(pi/4) + i sin(pi/4)].
(ii) Multiplication/Division in Polar Form: -
Multiplication: - Multiply moduli and add arguments.
z1z2 = r1r2[cos(theta1 + theta2) + i sin(theta1 + theta2)]
Division: - Divide moduli and subtract arguments.
z1z2 = r1r2[cos(theta1 - theta2) + i sin(theta\1 - theta2)]
6. De Moivre’s Theorem: - For problems involving powers or roots, use De Moivre’s theorem.
z^n = r^n [cos (ntheta) + i sin (ntheta)]
7. Powers of i: - Memorize the cycle of i.
i^1 = i, i^2 = -1, i^3 = -i, i^4 = 1
• Use this to simplify powers of i in any expression.
General Tips: -
• Step-by-Step: - Break problems down into smaller steps and work on each part.
• Check Units: - For arguments, ensure you are in the correct quadrant.
• Visualize: - Plotting complex numbers on the Argand plane helps in understanding the modulus and argument.
By mastering these steps and concepts, you can easily solve complex number problems.
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