Class 9 Maths: Number Systems, Polynomials, Coordinate Geometry, Linear Equations, Euclid's Geometry & Lines and Angles: quick revision, 20 bites

Real Numbers = Rationals + Irrationals

Every number on the line is real.

The **real number system** is the backbone of Class 9 Maths — every other chapter builds on it.

Takeaway. N ⊂ W ⊂ Z ⊂ Q ⊂ R, and irrationals sit alongside Q inside R.

Laws of Exponents for Real Numbers

Same base? Just add the powers.

Takeaway. These five laws let you simplify any expression with rational exponents.

Rationalising the Denominator

Kill the root at the bottom.

To remove a surd from a denominator, multiply numerator and denominator by its **conjugate**.

Takeaway. Conjugate multiplication turns an irrational denominator into a rational one.

Polynomials: Degree, Coefficients, Types

Degree decides the polynomial's name.

Takeaway. Degree names the polynomial; zeros are where p(x) hits zero.

Remainder & Factor Theorems

Plug in, don't divide.

Takeaway. Remainder theorem = substitute; factor theorem = substitute and check for zero.

Algebraic Identities You Must Memorise

Six identities, endless marks.

Takeaway. These identities convert factorisation and expansion problems into one-line answers.

The Cartesian Plane: Four Quadrants

Two axes, four zones, infinite points.

Two perpendicular number lines — **x-axis (horizontal)** and **y-axis (vertical)** — meet at the **origin O(0, 0)** and divide the plane into four quadrants.

Takeaway. Signs of (x, y) tell you exactly which quadrant a point lies in.

Plotting (x, y) — Order Matters

x first, y second. Always.

Takeaway. Coordinate order (x, y) is fixed — never reverse it.

Linear Equations in Two Variables

Two variables, infinite solutions.

Takeaway. A linear equation in two variables has infinitely many solutions, all lying on one straight line.

Lines Parallel to the Axes

x = a is vertical, y = b is horizontal.

Takeaway. x = constant → vertical; y = constant → horizontal.

Euclid's Axioms vs Postulates

Axioms are obvious; postulates are assumed.

Takeaway. Axioms are universal truths; postulates are geometry-specific assumptions.

Euclid's Five Postulates

Five rules that built geometry.

Takeaway. These five postulates are the foundation of Euclidean geometry.

Angle Pairs: Complementary to Linear Pair

Sum tells you the pair's name.

Takeaway. Complementary = 90°, supplementary = 180°, linear pair = adjacent supplementary.

Parallel Lines & the Transversal Theorems

One transversal, five equal-angle rules.

Takeaway. If two parallel lines are cut by a transversal, corresponding and alternate angles are equal; co-interior angles add to 180°.

Angle Sum Property of a Triangle

Three angles, one sum: 180°.

Takeaway. The three interior angles of any triangle always add up to 180°.

Exam Strategy: How to Score Full Marks

Show steps, draw figures, label everything.

Takeaway. Clear steps + labelled figures = full marks, even if the final answer is slightly off.

Remember the Quadrant Signs with 'All Silver Tea Cups'

ASTC — the quadrant cheat code.

Going **anti-clockwise** from Quadrant I, the signs of (x, y) follow the pattern **ASTC**.

Takeaway. ASTC tells you the sign pattern of coordinates in each quadrant instantly.

Don't Confuse Rational and Irrational

√4 is rational. √2 is not.

Takeaway. A square root is irrational only if the number under it is not a perfect square.

Where These Chapters Show Up in Real Life

Maths you actually use daily.

Takeaway. Every chapter in this paper powers a real technology or daily decision.

Universal 4-Step Approach to Any Maths Problem

Read, plan, solve, verify.

Takeaway. A disciplined 4-step routine prevents 90% of silly mistakes in exams.