Mathematics — Class 9

System using two perpendicular axes (x and y) to describe any point in a plane by exactly two numbers. Formalised by René Descartes in 1637. Named after him.

Pair of numbers locating a point. x = horizontal distance from y-axis (abscissa), y = vertical distance from x-axis (ordinate). ORDER MATTERS: (3,4) ≠ (4,3).

Q1: (+,+) — Q2: (−,+) — Q3: (−,−) — Q4: (+,−). Origin (0,0) is where axes meet. Points on the axes are NOT in any quadrant.

Arab scholar who studied Indian Siddhāntas and used coordinate ideas to calculate positions of cities across Asia. An early user of coordinate geometry for geography.

The point (0, 0) where the x-axis and y-axis intersect. Starting reference point of the coordinate system. Any point is described relative to the origin.

Polynomial of degree 1. Form: ax + b where a ≠ 0. Graph is always a straight line. E.g., 2x + 3, 5 − 4y, x/2 − 7.

Highest power of the variable. x² + 5x + 3 → degree 2 (quadratic). 3x³ − 4 → degree 3 (cubic). ax + b → degree 1 (linear). Constant (5) → degree 0.

Quantity increasing by fixed amount over equal time intervals. Graph = upward straight line. E.g., Saving ₹5 every day — savings after n days = 5n (a linear polynomial).

The value of x that makes p(x) = 0. For ax + b = 0, zero = −b/a. Also called the ROOT. Every linear polynomial has exactly ONE zero. Graph crosses x-axis at this point.

Any number expressible as p/q where p and q are integers and q ≠ 0. Always gives terminating or repeating decimal. E.g., 3/4 = 0.75 (terminating), 5/11 = 0.454545... (repeating). Brahmagupta formalised rules for rational numbers.

Cannot be written as p/q. Gives non-terminating, non-repeating decimal. E.g., √2, π, √3. First proof: Hippasus (~400 BCE) proved √2 is irrational. π is irrational — proved by Lambert in 1761. 22/7 ≈ π but 22/7 ≠ π.

Union of all rational and irrational numbers. Form a complete, continuous, unbroken number line. Every real physical measurement has a home on the real number line. Includes: Natural numbers ⊂ Integers ⊂ Rationals ⊂ Reals.

Distance of a number from 0 on the number line. Always non-negative. |5/3| = 5/3. |−5/3| = 5/3. |0| = 0. Represents magnitude without direction.

Between ANY two rational numbers, there are infinitely many rational numbers. To find a rational between a/b and c/d: compute their average = (a/b + c/d)/2. This can be repeated infinitely — rationals are DENSE on the number line.

(a + b)² = a² + 2ab + b². Proved geometrically — square of side (a+b) = 2 squares (a², b²) + 2 rectangles (ab each). E.g., 102² = (100+2)² = 10000+400+4 = 10404.

(a − b)² = a² − 2ab + b². E.g., 98² = (100−2)² = 10000−400+4 = 9604. Quick mental calculation!

(a + b)(a − b) = a² − b². "Difference of two squares." E.g., 102 × 98 = (100+2)(100−2) = 100²−2² = 10000−4 = 9996.

a² + b² + c² + 2ab + 2bc + 2ca. Extension of (a+b)². Useful when three terms are present. Can also be derived geometrically using a square partitioned into 9 regions.

Set of all points on a plane equidistant from a given point (centre). That distance = radius. A chord passing through centre = diameter = 2 × radius. All points on circle are called the circumference.

A line segment joining any two points on the circle. Diameter is the longest chord (passes through centre). The angle subtended by a chord at the centre is called the central angle.

Arc length = (θ/360) × 2πr, where θ = central angle in degrees, r = radius. For a sector: Area = (θ/360) × πr². These formulas relate the fraction of the full circle to the full circumference/area.

Set of all points satisfying a given condition. A circle = locus of all points equidistant from the centre. The locus concept helps define geometric shapes precisely as a set of points.

C = 2πr = πd. π ≈ 22/7 ≈ 3.14159. π is irrational (22/7 is only an approximation). Better approximation: 355/113. Pi Day = March 14 (3/14) or July 22 (22/7).

A = πr². Derived by cutting circle into thin sectors and rearranging into a near-rectangle with length πr and height r. Area of sector = (θ/360) × πr². Area of segment = Area of sector − Area of triangle.

s = (a+b+c)/2 (semi-perimeter). Area = √[s(s-a)(s-b)(s-c)]. Useful when height is unknown. Named after Heron of Alexandria. For a right-angled triangle: Area = ½ × base × height.

Area of sector = (θ/360°) × πr². E.g., A clock's minute hand (r=7cm) sweeps 60° in 5 min. Area swept = (60/360) × π × 7² = (1/6) × 154 = 25.67 cm².

P(event) = Favourable outcomes ÷ Total equally likely outcomes. Range: 0 to 1. P=0 → impossible, P=1 → certain, P=0.5 → even chance (50-50).

P(event) + P(not event) = 1. P(A) + P(A') = 1. E.g., P(heads) = 1/2, so P(not heads) = 1 − 1/2 = 1/2. Useful shortcut when P(not A) is easier to calculate.

The set of ALL possible outcomes of an experiment. E.g., Die: {1,2,3,4,5,6}. Coin: {H,T}. Two coins: {HH,HT,TH,TT}. Probability is calculated relative to sample space.

Each outcome has the same chance of occurring. E.g., Unbiased die: each face (1-6) has probability 1/6. Fair coin: heads and tails both have probability 1/2.

Sequence with constant difference (d) between consecutive terms. E.g., 2,5,8,11 (d=3). 10,7,4,1 (d=−3). Graph of an AP is always a straight line.

tₙ = a + (n−1)d. a = first term, d = common difference, n = position. E.g., AP: 3,7,11… → t₁₀ = 3 + 9×4 = 39. Also written as tₙ = (a−d) + nd = linear in n.

Fixed amount added to each term in an AP. d = t₂ − t₁ = t₃ − t₂ = any term − previous term. d can be positive (increasing AP), negative (decreasing AP), or zero (constant sequence).

Sₙ = n/2 × [2a + (n−1)d] = n/2 × (first term + last term). E.g., Sum of first 10 natural numbers = 10/2 × (1+10) = 55. Famous formula attributed to Gauss.