Number System and Quadratic Equations

This module covers the fundamental properties of numbers and the techniques for solving quadratic equations, which are essential for algorithmic complexity analysis.

1. Number Systems

Numbers are classified into several categories:

• Natural Numbers (N): $1, 2, 3, \dots$
• Integers (Z): $\dots, -2, -1, 0, 1, 2, \dots$
• Rational Numbers (Q): Numbers that can be expressed as $p/q$.
• Irrational Numbers: Numbers like $\sqrt{2}, \pi$.
• Real Numbers (R): The set of all rational and irrational numbers.
• Complex Numbers (C): Numbers of the form $a + ib$, where $i = \sqrt{-1}$.

2. Quadratic Equations

A quadratic equation is of the form:

The Quadratic Formula

The roots of the equation are given by:

The Discriminant (D)

The term $D = b^2 - 4ac$ determines the nature of the roots:

1. If $D > 0$: Two distinct real roots.
2. If $D = 0$: Two equal real roots.
3. If $D < 0$: Two complex (imaginary) roots.

3. Sum and Product of Roots

If $\alpha$ and $\beta$ are the roots of $ax^2 + bx + c = 0$:

• Sum of roots: $\alpha + \beta = -b/a$
• Product of roots: $\alpha \cdot \beta = c/a$

Summary

These algebraic foundations are critical for competitive exams and advanced computer science topics like cryptography.