# Chapter 01: Number System

Notes (Solutions) of Chapter 01: Number System, Text Book of Algebra and Trigonometry Class XI (Mathematics FSc Part 1 or HSSC-I), Punjab Text Book Board, Lahore.

### Contents & summary

- Rational numbers and irrational numbers
- Properties of real numbers
- Complex numbers
- Operation on complex numbers
- Complex numbers as ordered pairs of real numbers
- Properties of the foundamental operations on complex numbers
- A special subset of $\mathbb{C}$

- The real line
- Real plane or coordinate plane
- Geometrical representation of complex numbers, the complex plane
- To find real and imaginary parts of (i) $(x+iy)^n$ (ii) $\left(\frac{x_1+iy_1}{x_2+iy_2}\right)^n, x_2+iy_2\neq 0$

Did you know $\sqrt{-1}=i$ is misleading?

The square root of complex number has many values with two distinct values. $\sqrt{-1}$ has two distinct values $i$ and $-i$. This fact can be verified by taking square of $i$ and $-i$, which give the same answer $-1$. It is different to say $i^2=-1$ and $\sqrt{-1}=i$ unless we define $\sqrt{-1}$ as principal square root of complex number. (see detail answer here)

The square root of complex number has many values with two distinct values. $\sqrt{-1}$ has two distinct values $i$ and $-i$. This fact can be verified by taking square of $i$ and $-i$, which give the same answer $-1$. It is different to say $i^2=-1$ and $\sqrt{-1}=i$ unless we define $\sqrt{-1}$ as principal square root of complex number. (see detail answer here)

### Solutions

### Short Questions

The following short questions of this chapter was send by Mr. Akhtar Abbas.

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