Math Formulas for Class 11 Chapter 5 Complex Numbers and Quadratic Equations

Math Formulas for Class 11 Chapter 5 Complex Numbers and Quadratic Equations

Imaginary Numbers

The square root of a negative real number is called imaginary number, e.g. etc. The quantity is an imaginary unit and it is denoted by ‘ i ‘ called iota.

Integral Power of IOTA(i)

$$
i=\sqrt{-1}, i^2=-1, i^3=-i, i^4=1
$$
So, $$i^{4 n+1}=i, i^{4 n+2}=-1, i^{4 n+3}=-i, i^{4 n}=1$$
– For any two real numbers a and b, the result $$\sqrt{a} \times \sqrt{b}=\sqrt{a b}$$ is true only, when atleast one of the given numbers is either zero or positive. $$\sqrt{-a} \times \sqrt{-b} \neq \sqrt{a b}$$ So, $$i^2=\sqrt{-1} \times \sqrt{-1} \neq 1$$
– ‘ i ‘ is neither positive, zero nor negative.
– Important Result $$i^n+i^{n+1}+i^{n+2}+i^{n+3}=0$$

Algebra of Complex Numbers

Let and be any two complex numbers.
(i) Addition of Complex Numbers
$$
\begin{aligned}
z_1+z_2 & =\left(x_1+i y_1\right)+\left(x_2+i y_2\right) \\
& =\left(x_1+x_2\right)+i\left(y_1+y_2\right)
\end{aligned}
$$
Properties of addition
– Closure is also a complex number.
– Commutative $$z_1+z_2=z_2+z_1$$
– Associative $$z_1+\left(z_2+z_3\right)=\left(z_1+z_2\right)+z_3$$
– Existence of additive identity
$$
z+0=z=0+z
$$
Note: Here, 0 is additive identity.
– Existence of Additive inverse
$$
z+(-z)=0=(-z)+z
$$
Note: Here, ‘-z’ is additive inverse.
(ii) Subtraction of Complex Numbers
$$
\begin{aligned}
z_1-z_2 & =\left(x_1+i y_1\right)-\left(x_2+i y_2\right) \\
& =\left(x_1-x_2\right)+i\left(y_1-y_2\right)
\end{aligned}
$$

(iii) Multiplication of Complex Numbers
$$
\begin{aligned}
z_1 z_2 & =\left(x_1+i y_1\right)\left(x_2+i y_2\right) \\
& =\left(x_1 x_2-y_1 y_2\right)+i\left(x_1 y_2+x_2 y_1\right)
\end{aligned}
$$
Properties of multiplication
– Closure is also a complex number.
– Commutative $$z_1 z_2=z_2 z_1$$
– Associative $$z_1\left(z_2 z_3\right)=\left(z_1 z_2\right) z_3$$
– Existence of multiplicative identity
$$
z \cdot 1=z=1 \cdot z
$$
Here, 1 is multiplicative identity.
– Existence of multiplicative inverse For every non-zero complex number ‘z’, there exists a complex number such that
– Distributive law $$z_1\left(z_2+z_3\right)=z_1 z_2+z_1 z_3$$
(iv) Division of Complex Numbers $$\frac{z_1}{z_2}=\frac{x_1+i y_1}{x_2+i y_2}=\frac{\left(x_1 x_2+y_1 y_2\right)+i\left(x_2 y_1-x_1 y_2\right)}{x_2^2+y_2^2}$$ where,

Conjugate of a Complex Number

Let , if ‘ i ‘ is replaced by (-i), then it is said to be conjugate of the complex number z and denoted by , i.e. .
Properties of Conjugate
(i) $$\overline{(\bar{z})}=z$$
(ii) $$z+\bar{z}=2 \operatorname{Re}(z), z-\bar{z}=2 i \operatorname{Im}(z)$$
(iii) if z is purely real $$z=\bar{z}$$
(iv) If z is purely imaginary $$z+\bar{z}=0$$
(v) $$\overline{z_1+z_2}=\bar{z}_1+\bar{z}_2$$
(vi) $$\overline{z_1-z_2}=\bar{z}_1-\bar{z}_2$$
(vii) $$\overline{z_1 z_2}=\bar{z}_1 \cdot \bar{z}_2$$
(viii) $$\overline{\left(\frac{z_1}{z_2}\right)}=\frac{\bar{z}_1}{\bar{z}_2}, \bar{z}_2 \neq 0$$
(ix) $$z \cdot \bar{z}=\{\operatorname{Re}(z)\}^2+\{\operatorname{Im}(z)\}^2$$
(x) $$z_1 \bar{z}_2+\bar{z}_1 z_2=2 \operatorname{Re}\left(\bar{z}_1 z_2\right)=2 \operatorname{Re}\left(z_1 \bar{z}_2\right)$$
(xi) If $$z=f\left(z_1\right)$$, then $$\bar{z}=f\left(\bar{z}_1\right)$$
(xii) $$(\bar{z})^n=\left(\overline{z^n}\right)$$

Modulus (Absolute Value) of a Complex Number

Let be a complex number. Then, the positive square root of the sum of square of real part and square of imaginary part is called modulus (absolute values) of z and it is denoted by |z| i.e. $$|z|=\sqrt{x^2+y^2}$$

Properties of Modulus
(i) $$|z| \geq 0$$
(ii) If , then z=0 i.e. $$\operatorname{Re}(z)=0=\operatorname{Im}(z)$$
(iii) $$-|z| \leq \operatorname{Re}(z) \leq|z|$$ and $$-|z| \leq \mid \operatorname{Im} z) \leq|z|$$
(iv) $$|z|=|\bar{z}|=|-z|=|-\bar{z}|$$
(v) $$z \cdot \bar{z}=|z|^2$$
(vi) $$\left|z_1 z_2\right|=\left|z_1\right|\left|z_2\right|$$
(vii) $$\left|\frac{z_1}{z_2}\right|=\frac{\left|z_1\right|}{\left|z_2\right|}, z_2 \neq 0$$
(viii) $$\left|z_1+z_2\right|^2=\left|z_1\right|^2+\left|z_2\right|^2+2 \operatorname{Re}\left(z_1 \bar{z}_2\right)$$
(ix) $$\left|z_1-z_2\right|^2=\left|z_1\right|^2+\left|z_2\right|^2-2 \operatorname{Re}\left(z_1 \bar{z}_2\right)$$
(x) $$\left|z_1+z_2\right| \leq\left|z_1\right|+\left|z_2\right|$$
(xi) $$ \left|z_1-z_2\right| \geq\left|z_1\right|-\left|z_2\right|$$