Kerala State Board New Syllabus Plus One Maths Notes Chapter 5 Complex Numbers and Quadratic Equations.

## Kerala Plus One Maths Notes Chapter 5 Complex Numbers and Quadratic Equations

we have studied linear equations in one and two variables and quadratic equations in one variable. We have seen that the equation x^{2} + 1 = 0 has no real solution since the root of a negative number does not exist in a real number. So, we need to extend the real number system to a larger number system to accommodate such numbers.

I. Complex Numbers

A number of the form a + ib, where a and b are real numbers and i = √-1.

Usually, a complex number is denoted by z, a is the real part of z denoted by Re(z) and b is the imaginary part of z denoted by Im(z).

II. Algebra of Complex Numbers

Addition: Let z_{1} = a + ib and z_{2} = c + id be two complex numbers. Then the sum z_{1} + z_{2} is obtained by adding the real and imaginary parts.

1. z_{1} + z_{2} = z_{2} + z_{1}, commutative.

2. z_{1} + (z_{2} + z_{3}) = (z_{1} + z_{2}) + z_{3}, associative.

3. 0 + i0 is the identity element.

4. -z is the inverse of z.

Multiplication: Let z_{1} = a + ib and z_{2} = c + id be two complex numbers.

Then the product z_{1}z_{2} is defined as follows:

z_{1}z_{2} = (ac – bd) + i(ad + bc).

1. z_{1}z_{2} = z_{2}z_{1}, commutative.

2. z_{1}(z_{2}z_{3}) = (z_{1}z_{2})z_{3}, associative.

3. 1 + i0 is the identity element.

4. \(\frac{1}{z}\) is the inverse of z.

5. z_{1}(z_{2} + z_{3}) = z_{1}z_{2} + z_{1}z_{3}, distributive law.

Power of ‘i’: i^{3} = -i, i^{4} = 1

In general i^{4k} = 1, i^{4k+1} = i, i^{4k+2} = -1, i^{4k+3} = -i

Identities:

The Modulus and Conjugate of a complex number:

Consider a complex number z = a + ib . Then, the conjugate of z is denoted by \(\bar{z}\), defined as \(\bar{z}\) = a – ib and the modulus of z is denoted by |z|, defined as \(\sqrt{a^{2}+b^{2}}\).

Properties:

III. Representation of Complex Number

Argand Plane:

A complex number z = a + ib which corresponds to the ordered pair (a, b) can be represented geometrically as the unique point P(a, b) in the XY-plane, where the real part is taken along the x-axis and the imaginary part along the y-axis. Such a plane is called the Argand Plane or Complex plane.

Polar Form:

Let the point P represent the non-zero complex number z = x + iy. Let the directed line segment OP be of length ‘r’ and be the angle which OP makes with the positive direction of the x-axis. Then, P is determined by the unique ordered pair of a real number (r, θ) called polar coordinate of the point P, where x = r cos θ, y = r sin θ and therefore the polar form of z can be represented as z = r(cos θ + i sin θ).

The principle argument of z is value ‘θ’ such that -x ≤ θ ≤ π, denoted by arg z.

To find the principle argument, we find tan α = |\(\frac{y}{x}\)|, 0 ≤ α ≤ \(\frac{\pi}{2}\)

The quadrant on which ‘P’ lies | arg z = |

I | α |

II | π – α |

III | α – π |

IV | -α |

Positive real axis | 0 |

Negative real axis | π |

Positive imaginary axis | \(\frac{\pi}{2}\) |

Negative imaginary axis | \(-\frac{\pi}{2}\) |