Algebra

Meaning & Simple Explanation

Think of PMI like a row of falling dominoes. If you want to prove that every domino in an infinite line will fall, you don't need to push each one individually. You only need to prove two things:

• The Base Case: The first domino falls.
• The Inductive Step: If any one domino falls, it will inevitably knock down the next one.

Key Points

• Base Case: Prove the statement is true for n = 1.
• Assumption: Assume the statement is true for n = k.
• Inductive Step: Use that assumption to prove the statement is true for n = k + 1.

Examples

• Sum of Natural Numbers: Proving 1 + 2 + 3 + ... + n = n(n+1)/2.
• Divisibility: Proving 7ⁿ - 1 is always divisible by 6 for any natural number n.

Real-Life Use: Software Loop Verification

In computer science, engineers use mathematical induction to verify that a "loop" in a piece of code will behave correctly for any number of iterations, ensuring logic doesn't break at large scales.

Definition

A complex number is written as z = a + bi, where a is the Real part and b is the Imaginary part. We use i (the imaginary unit), where i = √-1.

Representation and the Argand Plane

Since complex numbers have two parts, we use the Argand Plane for plotting:

• Real Axis (X-axis): Represents the value of a.
• Imaginary Axis (Y-axis): Represents the value of b.

Polar Form

The Polar Form is: z = r(cos θ + i sin θ), where r is the modulus and θ is the argument.

Formulas

Examples

• Converting to Polar: If z = 1 + i, then r = √2 and θ = 45°.
• Finding Modulus: For z = 3 - 4i, |z| = √(3² + (-4)²) = 5.

Real-Life Use: Electrical Engineering (AC Circuits)

Engineers use complex numbers to represent Impedance in AC circuits. The real part represents resistance, and the imaginary part represents reactance.

Meaning & Explanation

An inequality is a statement that two expressions are not necessarily equal (e.g., x > 5 or x ≤ 10).

Algebraic and Graphical Solutions

• Algebraic: If you multiply or divide by a negative number, you must flip the inequality sign.
• Graphical: The solution is represented as a "region" or shaded area on a graph.

Examples

Algebraic: 2x + 3 < 11  =>  2x < 8  =>  x < 4
Graphical: y > x + 1 is a shaded region above the dashed line y = x + 1
        

Real-Life Use: Business Budgeting

Finding the "feasible region" for spending (e.g., 5x + 2y ≤ 100) allows business owners to see all possible combinations of resources they can afford.

The Fundamental Principle of Counting

If one task can be done in m ways and another in n ways, both can be done in m × n ways. Factorial (n!) is the product of all positive integers up to n.

Permutations vs. Combinations

Real-Life Use: Cybersecurity

Password creation relies on Permutations. Because the order of digits matters, the total possible combinations are massive, making hacking difficult.

Meaning

Allows expansion of expressions like (a + b)ⁿ without manual multiplication.

• Pascal's Triangle: Used to find Binomial Coefficients.
• General Term: T(r+1) = ⁿCᵣ · aⁿ⁻ʳ · bʳ

Example

Expand (x + 2)³ using Pascal's Triangle (1, 3, 3, 1):
= 1(x³) + 3(x² · 2¹) + 3(x¹ · 2²) + 1(2³)
= x³ + 6x² + 12x + 8
        

Definitions

• Arithmetic Progression (AP): Constant difference (d). e.g., 2, 5, 8...
• Geometric Progression (GP): Constant ratio (r). e.g., 2, 6, 18...

Comparison Table

1. What is the value of i²?
   (a) 1 (b) -1 (c) 0 (d) √-1
   Answer: (b)
2. In a permutation, does order matter?
   (a) Yes (b) No
   Answer: (a)
3. The common difference in the AP: 5, 8, 11, 14 is:
   (a) 2 (b) 3 (c) 5 (d) -3
   Answer: (b)
4. If a=2, b=3, find the Arithmetic Mean (AM).
   (a) 2.5 (b) √6 (c) 5 (d) 1
   Answer: (a)

1. Q: Prove by PMI: 1 + 3 + 5 + ... + (2n-1) = n².
   Answer: Base case n=1: 1=1². Assume n=k: S_k = k². For n=k+1: S_{k+1} = k² + (2(k+1)-1) = k² + 2k + 1 = (k+1)². Proved.
2. Q: Find the modulus of z = 5 + 12i.
   Answer: |z| = √(5² + 12²) = √(25 + 144) = √169 = 13.
3. Q: Solve the inequality: 3x - 5 ≤ x + 7.
   Answer: 2x ≤ 12 => x ≤ 6.

• P&C: Permutation = Position matters. Combination = Committee (order doesn't matter).
• Complex Numbers: i is "imaginary"—you need a special "plane" (Argand) to find it.
• Inequalities: Flip the sign when dividing by a negative (like a seesaw flipping!).

Modern Tools: WolframAlpha, MATLAB, and Python (NumPy) are industry standards for Algebra.