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## Complex Numbers And Quadratic Equations class 11 Notes Mathematics

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**CBSE Class 11 Mathematics Revision Notes Chapter-5 COMPLEX NUMBERS AND QUADRATIC EQUATIONS**

**Algebra, Modulus and Conjugate of Complex Numbers****Argand Plane and Polar Representation****Quadratic Equations**

**Imaginary Number:**Square root of a negative number is called an Imaginary number. For example, etc. are imaginary numbers.**Integral power of Iota (**: ) = where and**Complex Number**: A number of the form where a and b are real numbers, is called a complex number, a is called the real part and b is called the imaginary part of the complex number. It is denoted by**Real part of**is and is denoted by .**Imaginary part of**is and is written as .**Equality of complex numbers**: Two complex numbers and are said to be equal, if and .**Conjugate of a complex number**: Two complex numbers are said to be conjugate of each other, if their sum is real and their product is also real. Conjugate of a complex number is i.e., conjugate of a complex number is obtained by changing the sign of imaginary part of z.**Modulus of a complex number**: Modulus of a complex number is denoted by .**Argument of a complex number**: Arg .**Representation of complex number as ordered pair**: Any complex number can be written in ordered pair as , where a is the real past and b is the imaginary part of a complex number.

(i)

(ii)

- Division of a complex number: If and , then,

=

- For any non-zero complex number there exists the complex number denoted by or , called the multiplicative inverse of z such that

**Polar form of a complex number**: The polar form of the complex number , where (the modulus of z) and . (θ is known as the argument of z. The value of θ, such that is called the principal argument of z.**Important properties**: (i) , (ii)**Fundamental Theorem of algebra**: A polynomial equation of n degree has n roots.

**Quadratic Equation**:

**Quadratic Equation**: Any equation containing a variable of highest degree 2 is known as quadratic equation. e.g., .**Roots of an equation**: The values of variable satisfying a given equation are called its roots. Thus, is a root of the equation if**Solution of quadratic equation**: The solutions of the quadratic equation , where are given by

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