Class IX Maths SA-1: Number Systems, Polynomials, Coordinate Geometry, Linear Equations & Lines and Angles: quick revision, 20 bites

Rational vs Irrational: The Decimal Test

Decimals reveal a number's true identity.

Every real number is either **rational** or **irrational** — and the decimal expansion tells you which.

Takeaway. If the decimal neither terminates nor repeats, the number is irrational — no exceptions.

What Is a Rational Number?

Ratio of two integers — that's the whole story.

Takeaway. The denominator q must never be zero — that single condition defines the set.

Infinitely Many Rationals Between Any Two

Squeeze a rational — there's always room.

Between any two rational numbers, there are **infinitely many** rational numbers — not just one.

Takeaway. Density of rationals: no 'next' rational exists after any given one.

Laws of Exponents for Real Numbers

Same base? Add, subtract, multiply the powers.

Takeaway. Fractional exponents are just radicals in disguise — rewrite before simplifying.

Rationalising the Denominator

Kill the surd at the bottom.

To simplify $\frac{1}{7+3\sqrt{2}}$, multiply numerator and denominator by the **conjugate** of the denominator.

Takeaway. Conjugate = same terms, opposite middle sign. Always multiply top and bottom.

Degree Decides the Polynomial Type

One number classifies every polynomial.

Takeaway. Degree = highest power of the variable. That single number names the polynomial.

Coefficient vs Constant Term

Don't confuse the number in front with the one alone.

Takeaway. Coefficient of x = the number attached to x, not the leading or constant number.

Factor Theorem: The Zero Test

Plug in the value — zero means factor.

Takeaway. Factor theorem turns 'is it a factor?' into a simple substitution.

The Three Cubic Identities You Must Know

Three identities unlock every cubic problem.

Takeaway. When x + y + z = 0, the identity collapses to $x^3+y^3+z^3 = 3xyz$ — a favourite exam shortcut.

Factorising $x^3 - y^3$ Without Tears

SOAP it: Same, Opposite, Always Positive.

For $x^3 - y^3 = (x-y)(x^2 + xy + y^2)$, remember **SOAP**:

Takeaway. SOAP keeps the signs straight for both sum and difference of cubes.

The Four Quadrants of the Cartesian Plane

Signs of (x, y) tell you the quadrant.

Takeaway. Signs of coordinates instantly place a point — no plotting needed.

Standard Form of a Linear Equation

Always rearrange to ax + by + c = 0.

Takeaway. Standard form always has zero on the right — signs of a, b, c matter.

Angle Pairs on a Straight Line

Straight line = 180°. Always.

Takeaway. Any two angles on a straight line add to 180° — the most used fact in geometry.

Parallel Lines: The Angle Toolkit

One transversal, many equal angles.

Takeaway. Equal for corresponding/alternate; supplementary for co-interior.

Bisectors and Perpendiculars

Two geometry words, two clear jobs.

Takeaway. Bisector splits angles; perpendicular splits at 90°; collinear means same line.

Decimal Expansion: Terminating or Not?

Denominator's factors decide the decimal.

Takeaway. Factorise the denominator — 2s and 5s only means terminating.

Not Every Square Root Is Irrational

√225 is not irrational — check first.

Takeaway. Perfect square under the root → rational. Otherwise → irrational.

Proving AB = CD from AC = BD

Subtract the shared segment.

This uses **Euclid's axiom**: if equals are subtracted from equals, the remainders are equal.

Takeaway. Segment proofs work by breaking each side into parts and cancelling the common piece.

Why Coordinate Geometry Matters

Your GPS speaks in quadrants.

Takeaway. Every plotted point is a real-world location in disguise.

Quadrant Signs: The 'CAST' Shortcut

Start top-right, go anticlockwise.

Quadrants are numbered **anticlockwise** starting from the top-right:

Takeaway. Anticlockwise from top-right: I → II → III → IV.