Complex Numbers - Revision Notes

 CBSE Class 11 Mathematics

Revision Notes
Chapter-5
COMPLEX NUMBERS AND QUADRATIC EQUATIONS

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1. Algebra, Modulus and Conjugate of Complex Numbers
2. Argand Plane and Polar Representation
3. Quadratic Equations

• ⇒$\Rightarrow$     i2=−1$i^2=-1$
• Imaginary Number: Square root of a negative number is called an Imaginary number. For example, −5−−−√,−16−−−−√,$\sqrt{-5},\sqrt{-16},$ etc. are imaginary numbers.
• Integral power of Iota (i$i$) : ip(p>4)=i4q+r$i^p\left(p>4\right)=i^{4q+r}$ = (i4)q.ir=ir,${\left(i^4\right)}^q.i^r=i^r,$ where −1−−−√=i$\sqrt{-1}=i$ and i4=1$i^4=1$
• Complex Number: A number of the form a + ib,$\text{a }+\text{ib},$ 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 z.$z.$
• Real part of z=a+ib$z=a+ib$ is a$a$ and is denoted by Re(z)=a$Re(z)=a$.
• Imaginary part of z=a+ib$z=a+ib$ is b$b$ and is written as Im(z)=b$Im(z)=b$.
• Equality of complex numbers: Two complex numbers z1=a+ib$z_1=a+ib$ and z2=c+id$z_2=c+id$ are said to be equal, if a=c$a=c$ and b=d$b=d$.
• 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 z=a+ib$z=a+ib$ is z¯¯¯=a−ib$\overline{z}=a-ib$ 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 z=x+iy$z=x+iy$ is denoted by |z|=x2+y2−−−−−−√$\left|z\right|=\sqrt{x^2+y^2}$.
• Argument of a complex number x+iy$x+iy$ : Arg(x+iy)=$(x+iy)=$ tan−1yx${\tan }^{-1}\frac{y}{x}$.
• Representation of complex number as ordered pair: Any complex number a+ib$a+ib$ can be written in ordered pair as (a,b)$\left(a,b\right)$, where a is the real past and b is the imaginary part of a complex number.
• Let z1 = a + ib and z2 = c + id. Then$\text{Let }{\text{z}}_1=\text{ a }+\text{ ib and }{\text{z}}_2=\text{ c }+\text{ id}.\text{ Then}$

          (i) z1+ z2= (a + c) + i (b + d)${\text{z}}_1\text{+ }{\text{z}}_2\text{= (a + c) + i (b + d)}$

          (ii) z1z2= (ac -bd) + i (ad +bc)${\text{z}}_1{\text{z}}_2\text{= (ac -bd) + i (ad +bc)}$

• Division of a complex number: If z1=a+ib$z_1=a+ib$ and z2=c+id$z_2=c+id$, then,

          z1z2=a+ibc+id=(a+ib)(c−id)(c+id)(c−id)$\frac{z_1}{z_2}=\frac{a+ib}{c+id}=\frac{\left(a+ib\right)\left(c-id\right)}{\left(c+id\right)\left(c-id\right)}$= ac+bdc2+d2+ibc−adc2+d2$\frac{ac+bd}{c^2+d^2}+i\frac{bc-ad}{c^2+d^2}$

•   For any non-zero complex number z = a + ib (a ≠ 0, b ≠ 0),$\text{z }=\text{ a }+\text{ ib }\left(\text{a }\ne 0,\text{ b }\ne 0\right),$ there exists the complex number  aa2+b2+i−ba2+b2$\frac{a}{a^2+b^2}+i\frac{-b}{a^2+b^2}$denoted by 1z$\frac{1}{z}$or  z−1$z^{-1}$, called the multiplicative inverse of z such that (a + ib) (a2a2+b2+i−ba2+b2)=1+io=1$\text{(a + ib) }\left(\frac{a^2}{a^2+b^2}+i\frac{-b}{a^2+b^2}\right)\text{=1+io=1}$

• Polar form of a complex number: The polar form of the complex number z = x + iy is r (cosθ + i sinθ)$\text{z }=\text{ x }+\text{ iy is r }\left(\text{cos}\theta +\text{ i sin}\theta \right)$, where r=x2+y2−−−−−−√$r=\sqrt{x^2+y^2}$ (the modulus of z) and  cosθ =xr,$\text{cos}\theta =\frac{x}{r},$  sinθ =yr,$\sin \theta =\frac{y}{r},$. (θ is known as the argument of z. The value of θ, such that is called the principal argument of  z.
• Important properties:      (i) |z1|+|z2|≥|z1+z2|$\left|z_1\right|+\left|z_2\right|\ge \left|z_1+z_2\right|$,      (ii) |z1|−|z2|≤|z1+z2|$\left|z_1\right|-\left|z_2\right|\le \left|z_1+z_2\right|$
• Fundamental Theorem of algebra: A polynomial equation of n degree has n roots.

Quadratic Equation:

• Quadratic Equation: Any equation containing a varibale of highest degree 2 is known as quadratic equation. e.g., ax2+bx+c=0$a{x}^2+bx+c=0$.
• Roots of an equation: The values of variable satisfying a given equation are called its roots. Thus, x=α$x=\alpha$ is a root of the equation p(x)=0$p\left(x\right)=0$ if p(α)=0.$p\left(\alpha \right)=0.$
• Solution of quadratic equation: The solutions of the quadratic equation ax2+bx+c=0$a{x}^2+bx+c=0$, where a,b,c∈R,$a,b,c\in \mathrm{R},$ a≠0,$a\ne 0,$  b2−4ac<0,$b^2-4ac<0,$  are given by x=−b±i4ac−b2√2a.