$2z+3w=a+ib$ if $z=5-2i$ and $w=-1+3i$, then find $a+b$.

1. Complex Numbers:

(i) -1 is an imaginary quantity and is denoted by $i$.

(ii) $i^2=-1,i^3=-i,i^4=1$ and, $i^{\pm n}=i^{\pm k},n\in N$ where $k$ is the remainder when $n$ is divided by $4.$

(iii) For any positive real number $a,$ -a=ia.

(iv) For any two real numbers $a$ and $b,$ $\sqrt{a}\sqrt{b}=\left\{\begin{array}{l}\sqrt{ab}, \text{if at least one of} \mathrm{a} \text{and} \mathrm{b} \text{is positive}\\ -\sqrt{ab},\mathrm{ } \text{if} \mathrm{ }\mathrm{a}\mathrm{ }<\mathrm{ }0,\mathrm{ }b\mathrm{ }<\mathrm{ }0.\end{array}\right.$

(v) If $a,b$ are real numbers, then a number $z=a+ib$ is called a complex number.

(vi) Real number $a$ is known as the real part of $z$ and $b$ is known as its imaginary part. We write $a=\mathrm{Re}\left(z\right), b=\mathrm{lm}\left(z\right)$.

(vii) A complex number $z$ is purely real if $\mathrm{lm}\left(z\right)=0$ and $z$ is purely imaginary if $\mathrm{Re}\left(z\right)=0$

2. Equality of complex numbers:

Two complex numbers $z_1=a_1+i{b}_1$ and $z_2=a_2+i{b}_2$ are said to be equal if and only if  $\mathrm{Re}\left(z_1\right)=\mathrm{Re}\left(z_2\right)$ and $\mathrm{Im}\left(z_1\right)=\mathrm{Im}\left(z_2\right)$ i.e., $a_1=a_2$ and $b_1=b_2$.

3. Argand or Gaussian or Complex plane:

A complex number $z=x+iy$ can be represented by a point $P$ on the plane which is known as the Argand or Gaussian or Complex plane.

4. Algebra of complex numbers:

For any two complex numbers, z1=a1+ib1 and z2=a2+ib2.

(i) Addition: $z_1+z_2=(a_1+a_2)+i(b_1+b_2)$

(ii) Subtraction: $z_1-z_2=\left(a_1-a_2\right)+i(b_1-b_2)$

(iii) Multiplication: $z_1{z}_2=(a_1{a}_2-b_1{b}_2)+i(a_1{b}_2+a_2{b}_1)$

(iv) Reciprocal: 1z1=a1a12+b12-ib1a12+b12

(v) Division: z1z2= z11z2=a1+ib1a2a22+b22-ib2a22+b22=a1a2+b1b2a22+b22+ia2b1-a1b2a22+b22

5. Inverse of a complex number:

(i) Addition in Complex number is commutative and associative. Complex number $0=0+i0$ is the identity element for addition and every complex number $z=a+ib$ has its additive inverse $-z=-a-ib$

(ii) Every non-zero complex number $z=a+ib$ has its multiplicative inverse 1z (also known as reciprocal of $z$) such that 1z=a-iba2+b2=z¯z2.

6. Conjugate of a complex number and its properties:

(i) The conjugate of a complex number $z=a+ib$ is denoted by z¯ and is equal to $a-ib$

(ii) For any three complex numbers $z,z_1,z_2$ we have

(a) $\left(\overline{z}\right)=z$

(b) $z+\overline{z}=2\mathrm{Re}\left(z\right)$

(c) $z-\overline{z}=2i\mathrm{Im}\left(z\right)$

(d) $z=\overline{z}\iff z$ is purely real.

(e) $z+\overline{z}=0 \iff z$ is purely imaginary.

(f) $z\overline{z}=\{\mathrm{Re}\left(z\right){\}}^2+{\left\{\mathrm{Im}\left(z\right)\right\}}^2={\left|z\right|}^2$

(g) z1± z2¯=z1¯±z2¯

(h) $\overline{z_1{z}_2}=\overline{z_1} \overline{z_2}$

(i) z1z2¯=z1¯z2¯, z2 ≠ 0

7. Modulus of a complex number and its properties:

(i) The modulus of a complex number $z=a+ib$ is denoted by $\left|z\right|$ and is defined as z=a2+b2= Rez2+Imz2

(ii) If $z, z_1, z_2$ are three complex numbers, then

(a) $\left|z\right|=0\iff z=0$ i.e. $\mathrm{Re}\left(z\right)=\mathrm{Im}\left(z\right)=0$

(b) z=z¯=-z

(c) $-\left|z\right|\le \mathrm{Re}\left(z\right)\le \left|z\right|;-\left|z\right|\le \text{lm}\left(z\right)\le \left|z\right|$

(d) $z\overline{z}={\left|z\right|}^2={\left|\overline{z}\right|}^2$

(e) $\left|\text{lm}\left(z^n\right)\right|\le n\left|\text{lm}\left(z\right)\right|{\left|z\right|}^{n-1},n\in N$

(f) Rez+Imz≤2z

(g) $\left|z_1{z}_2\right|=\left|z_1\right|\left|z_2 \right|$

(h) $\left|\frac{z_1}{z_2}\right|=\frac{\left|z_1\right|}{\left|z_2 \right|}$

8. Argument or Amplitude of a complex number:

(i) The angle $\theta$ which $OP$ makes with the positive direction of $x$-axis in anti-clockwise sense is called the argument or amplitude of $z$ and is denoted by $\arg \left(z\right)$ or $\mathrm{amp}\left(z\right)$.

(ii) $\tan \theta \mathrm{ }=\frac{\mathrm{I}\mathrm{m}\left(z\right)}{\mathrm{R}\mathrm{e}\left(z\right)}\Rightarrow \theta ={\tan }^{-1}\left(\frac{\mathrm{Im}\left(z\right)}{\mathrm{Re}\left(z\right)}\right)$

(iii) Argument of a complex number is not unique, since if $\theta$ be a value of the argument, so also is $2n\pi +\theta$ where $n\in I$.

9. Principal value of $\arg \left(z\right)$:

The value$\theta$  of the argument, which satisfies the inequality $–\pi <\theta \le \pi$ is called the principal value of argument. Principal values of argument $z$ will be $\theta , \pi -\theta , -\pi +\theta$ and$-\theta$  according as the point $z=a+ib$ lies in the first, second, third and fourth quadrants respectively, where, $\theta =\left|\frac{b}{a}\right|$.

(i) Argument of the complex number $0$ is not defined.

(ii) Principal value of argument of a purely real number is $0$ if the real number is positive and is $\mathrm{\pi }$ if the real number is negative.

(iii) Principal value of argument of a purely imaginary number is $\frac{\mathrm{\pi }}{2}$ if the imaginary part is positive and is $\frac{-\mathrm{\pi }}{2}$ if the imaginary part is negative.

10. Equation of Complex Form of a Circle:

The locus of $z$ that satisfies the equation $\left|z-z_0\right|=r$ where $z_0$ is a fixed complex number and $r$ is a fixed positive real number consists of all points $z$ whose distance from $z_0$ is $r$ .

(i) $\left|z-z_0\right|<r$ represents the points interior of the circle.

(ii) $\left|z-z_0\right|>r$ represents the points exterior of the circle.

11. Representation of a complex number:

(i) $z=x+iy=r\left(\cos \theta +i\sin \theta \right)$ is called as polar form of a complex number.

(ii) The Euler's notation is $e^{i\theta }=\cos \theta +i\sin \theta$. $z=r{e}^{i\theta }$ is known as the Euler's form of $z$.

12. De’ Moivre’s Theorem:

(i) If $n$ is any rational number, then $(\cos \theta +i \sin \theta {)}^n=\cos n\theta +i \sin n\theta$.

(ii) If $z=r(\cos \theta +i\sin \theta )$ and $n$ is a positive integer, then $z^{\frac{1}{n}}=r^{\frac{1}{n}} \left[\cos \left(\frac{2k\pi +\theta }{n}\right)+i \sin \left(\frac{2k\pi +\theta }{n}\right)\right]$, where $k=0,1,2,3,\dots ,(n-1)$.

13. ${\mathit{n}}^{\mathbf{th}}$ roots of unity and its properties:

Given a positive integer $n$, a complex number $z$ is called an $n^{\mathrm{th}}$ root of unity if and only if $z^n=1$.

(i) In polar form equation can be written as $z^n =\cos \left(0+2k\mathrm{\pi }\right)+i \sin \left(0+2k\mathrm{\pi }\right)=e^{i2k\mathrm{\pi }}, k=0,1,2,....$.

(ii) In de Moivre’s theorem, $z^n =\cos \left(\frac{2k\pi }{n}\right)+i \sin \left(\frac{2k\pi }{n}\right)=e^{i\frac{2k\pi }{n}}, k=0,1,2,....n-1$.

(iii) If we denote the complex number by $\omega$, then ${\omega }^n=1$.

(iv) The $n^{\mathrm{th}}$ roots of unity are $1,\omega ,{\omega }^2......{\omega }^{n-1}$.

(v) All the $n$ roots of $n^{\mathrm{th}}$ roots unity are in Geometrical Progression.

(vi) The sum of all the $n^{\mathrm{th}}$ roots of unity is $1+\omega +{\omega }^2+.....+{\omega }^{n-1}=0$.

(vii) The product of all the $n^{\mathrm{th}}$ roots of unity is $1·\omega ·{\omega }^2.....{\omega }^{n-1}={\left(-1\right)}^{n-1}$.

14. Cube roots of unity and its properties: 

(i) Cube roots of unity are $1,\omega ,{\omega }^2$.

(ii) $1+\omega +{\omega }^2=0$.

(iii) $1·\omega ·{\omega }^2=1$.

(iv) $1+{\omega }^r+{\omega }^{2r}=\left\{\begin{array}{l}0 \text{if} r \text{is not a multiple of} 3\\ 3 \text{if} r \text{is a multiple of} 3\end{array}\right.$.