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Class 11 Math Complex Numbers and Quadratic Equations Notes
Chapter 3: Trigonometric Functions Notes
Chapter 5: Linear Inequalities Notes
Key Points
Imaginary number
The square root of a negative real number is called an imaginary number.
Ex – √(-2), √(-5) etc.
Iota (i)
The quantity √(-1) is an imaginary unit and it is denoted by ‘i’, called iota.
Complex number
A number of the form x + iy where
x is called real part and y is called imaginary part of the complex number.
i.e.,
Re(Z) = x
Im(Z) = y
Purely real
A complex number is purely real if its imaginary part is 0.
Purely imaginary
A complex number is purely imaginary if its real part is 0.
Modulus of a complex number
The positive square root of the sum of square of real part and square of imaginary part is called modulus (absolute value) of Z and it is denoted by |Z|.
It represents distance of Z from origin in the set of complex numbers C.
Argand plane and polar representation
Any complex number Z can be represented geometrically by a point (x, y) in a plane called argand plane, complex plane or gaussian plane.
x-axis is called real axis
y-axis is called imaginary axis
Argument or amplitude
The angle made by line joining point Z to the origin with the positive direction of X-axis in an anti-clockwise sense.
Denoted by the symbol arg(Z) or amp(Z)
arg(Z) = θ = tan⁻¹(y/x)
Principal value of argument
The unique value of θ such that −π < θ ≤ π is called the principal value of the amplitude or principal argument.
Principal value of argument:
x > 0 and y > 0 → arg(Z) = θ
x < 0 and y > 0 → arg(Z) = π − θ
x < 0 and y < 0 → arg(Z) = −(π − θ)
x > 0 and y < 0 → arg(Z) = −θ