Definitions: Mathematics 11 NBF

Model Textbook of Mathematics for Class XI is published by National Book Foundation (NBF), Islamabad, Pakistan. NBF can be considered as Federal Textbook Board Islamabad. The book has total of nine (9) chapters.

Definition of the book provide the quick overview of the book.

Complex Number: A complex number is an expression of the form $x+iy$, where $x,y\in\mathbb{R}$ and $i^2=1$. Set of all complex numbers is usually denoted by $\mathbb{C}$. Every complex number $x+i y$ has two parts $x$ and $y$. $x$ is called the real part and $y$ is called the imaginary part i.e., $Re(z)=x$ and $Im(z)=y$.

Conjugate of a Complex Number: Conjugate of a complex number $z=x+i y$ is denoted by $\bar{z}$ and is defined as $\bar{z}=x-i y$.

Magnitude of Complex Number: If $z=x+i y$ is a complex number, then its magnitude, denoted by $|z|$, is defined as $|z|=\sqrt{x^{2}+y^{2}}$.

Real & Imaginary Parts: If $z=x+iy$, then

• $Re(z)=x$ and $Im(z)=y$.
• $Re(z^{-1})= \dfrac{Re(z)}{|z|^2}$ and $Im(z^{-1}) = -\dfrac{Im(z)}{|z|^2}$.
• $\text{Re}(z^{-2}) = \frac{(\text{Re}(z))^2 - (\text{Im}(z))^2}{|z|^4}$ and $\text{Im}(z^{-2}) = \frac{-2 \text{Re}(z) \text{Im}(z)}{|z|^4}$.

If $z_1=x_1+iy_1$ and $z_2=x_2+iy_2$, then

• $\text{Re}\left(\frac{x_{1}+i y_{1}}{x_{2}+i y_{2}}\right)=\frac{x_{1} x_{2}+y_{1} y_{2}}{x_{2}^{2}+y_{2}^{2}}$ and $\text{Im}\left(\frac{x_{1}+i y_{1}}{x_{2}+i y_{2}}\right)=\frac{x_{2} y_{1}-x_{1} y_{2}}{x_{2}^{2}+y_{2}^{2}}$.
• $Re\left(\left(\frac{x_1 + i y_1}{x_2 + i y_2}\right)^{-1}\right) = \frac{x_1 x_2 + y_1 y_2}{x_1^2 + y_1^2}$ and $Im\left(\left(\frac{x_1 + i y_1}{x_2 + i y_2}\right)^{-1}\right) = \frac{x_1 y_2 - x_2 y_1}{x_1^2 + y_1^2}$.
• $$\begin{align} &Re\left(\left(\frac{x_{1}+i y_{1}}{x_{2}+i y_{2}}\right)^{-2}\right) \\ &= \frac{\left(x_{2}^{2}-y_{2}^{2}\right)\left(x_{1}^{2}-y_{1}^{2}\right) + 4 x_{2} x_{1} y_{2} y_{1}}{\left(x_{1}^{2}+y_{1}^{2}\right)^{2}}\end{align}$$

$$\begin{align}&Im\left(\left(\frac{x_{1}+i y_{1}}{x_{2}+i y_{2}}\right)^{-2}\right)\\& = \frac{-2 \left[x_{1} y_{1} \left(x_{2}^{2}-y_{2}^{2}\right) - x_{2} y_{2} \left(x_{1}^{2}-y_{1}^{2}\right)\right]}{\left(x_{1}^{2}+y_{1}^{2}\right)^{2}}.\end{align}$$

• $\text{Re}\left(\left(\frac{x_1 + i y_1}{x_2 + i y_2}\right)^2\right) = \frac{\left(x_1^2 - y_1^2\right)\left(x_2^2 - y_2^2\right) + 4 x_1 x_2 y_1 y_2}{\left(x_2^2 + y_2^2\right)^2}$ and $\text{Im}\left(\left(\frac{x_1 + i y_1}{x_2 + i y_2}\right)^2\right) = \frac{2 \left[x_1 y_1 \left(x_2^2 - y_2^2\right) - x_2 y_2 \left(x_1^2 - y_1^2\right)\right]}{\left(x_2^2 + y_2^2\right)^2}.$

Polar Form of Complex Numbers: The form of complex number: $z=r(\cos \theta+i \sin \theta)$ is called polar form of a complex number. Here $r$ is modulus of complex number, i.e., $r=|z|$ and $\theta$ is called argument of $z$ denoted by $\arg(z)$.

Principal Argument: The value of the $\arg(z)$ between $-\pi$ and $\pi$ or equal to $\pi$ is called pricipal argument, denoted by $Arg(z)$.

Euler Identity: The identity $e^{i \theta}=\cos \theta+i \sin \theta$, where $\theta \in \mathbb{R}$, is called Euler identity.