A unit for measuring angles: the angle at the centre of a circle made by an arc as long as the radius. A full turn is 2π radians, the same as 360°.
Precisely: The angle subtended at the centre of a circle by an arc whose length equals the radius; so an angle θ in radians is the arc length divided by the radius, and π radians = 180°.
How it works
Bend the radius around the edge of the circle: the angle it covers at the centre is 1 radian, a little over 57°. The whole edge is 2πr long, so it takes 2π radius-lengths to go round once, and a full turn is 2π radians.
Examples
A right angle is π/2 radians.
One radian is about 57.3°.
An arc 10 cm long on a circle of radius 5 cm makes an angle of 2 radians at the centre.
A wheel turning 3 times turns through 6π radians.
Do not confuse: π radians is 180°, not 3.14°. Writing sin 1 means the sine of 1 radian, not of 1 degree.
Used in
arc length l = rθ, the graphs of sin and cos, turning speed in physics, calculus of trigonometric functions
The distance from the centre of a circle to any point on it; also a line segment joining the centre to the circle.
Precisely: The constant distance r from the centre to every point of a circle; also any segment from the centre to a point of the circle.
How it works
All radii of one circle are equal. That one fact makes the triangle formed by a chord and the centre isosceles, and most proofs in the chapter start from it.
Examples
In a circle of radius 5 cm, every point is 5 cm from the centre, and the longest chord is 10 cm.
A chord 5 cm from the centre of a circle of radius 13 cm is 2 × √(13² − 5²) = 2 × 12 = 24 cm long.
A chord of 16 cm is 6 cm from the centre, so the radius is √(8² + 6²) = 10 cm.
CA = CB because each is a radius of the same circle, so triangle CAB is isosceles.
Do not confuse: The radius is half the diameter, not the same thing: a circle of diameter 26 cm has radius 13 cm.
Used in
chord lengths, circumference and area (Class 10), the equation of a circle (Class 11)
An action you can repeat, where you know all the results that could happen but cannot know which one will happen this time.
Precisely: A repeatable procedure whose set of possible outcomes is known in advance but whose actual outcome on any one repetition (trial) cannot be predicted.
How it works
Each repetition is a trial. List what could happen (the sample space), then either count equally likely outcomes or repeat the trial many times and record the results.
Examples
Tossing a coin is a random experiment: heads or tails, but you cannot know which.
Picking a name slip from a box for the lucky draw is a random experiment in which each student has an equal chance.
Tossing a paper cup 100 times and recording bottom, top or side is 100 trials of a random experiment.
Do not confuse: An experiment that always gives the same result (dropping a stone: it falls) is not random. Random does not mean the outcomes must be equally likely.
Used in
probability, experiments in science, games of chance
Not being able to predict exactly what will happen, even when you know everything that could happen.
Precisely: The property of a process whose individual outcomes cannot be predicted, although the possible outcomes, and often their long-run frequencies, can be known.
How it works
A single toss is unpredictable, but many tosses show a steady pattern: about half heads. Probability measures that long-run pattern. Rain is random because it depends so sensitively on temperature, humidity, wind and pressure.
Examples
The randomness of a coin toss makes it a fair way to decide which cricket team bats first.
Because of randomness, your friend can guess heads or tails but cannot know for certain.
The randomness of rain means a forecast can only give a chance of rain, not a promise.
Do not confuse: Randomness has no memory: after six heads in a row, the next toss is still 50-50 (the gambler's fallacy says otherwise and is wrong).
A way of comparing two amounts by division, written like 3 : 2 ("3 to 2").
Precisely: The comparison a : b of two quantities of the same kind, equal to the fraction a/b; it is unchanged when both parts are multiplied or divided by the same non-zero number.
How it works
Simplify by dividing both parts by their HCF. To share an amount in a ratio, add the parts, find one part's worth, and multiply. Both quantities must be in the same units before comparing.
Examples
A class with 18 boys and 12 girls has boys to girls in the ratio 18 : 12 = 3 : 2.
Two incomes in the ratio 9 : 7 can be written ₹9x and ₹7x; with x = 2000 they are ₹18,000 and ₹14,000.
A map scale of 1 : 50,000 is a ratio: 1 cm on the map is 50,000 cm (0.5 km) on the ground.
Do not confuse: Order matters: 3 : 2 is not 2 : 3. And compare in the same units: 50 cm : 2 m is 1 : 4, not 25 : 1.
Used in
sharing money and mixtures, maps and scale factors, similar triangles, trigonometric ratios
Any number you can write as one integer divided by another (not by 0), like 3/4, −2/5 or 7 (which is 7/1).
Precisely: A number of the form p/q, where p and q are integers and q ≠ 0; the set is written ℚ, for quotient.
How it works
Every integer is rational (5 = 5/1), and so is every fraction and its negative. Many fractions name the same rational number (1/2 = 2/4 = 3/6), so we usually take the one in lowest terms. Its decimal either stops or repeats.
Examples
−3/4 is a rational number: p = −3 and q = 4 are integers.
0.375 is a rational number, because it equals 3/8.
The repeating decimal 0.454545… is a rational number: it equals 5/11.
Adding two rational numbers always gives another one: 2/5 + 3/10 = 7/10.
Do not confuse: Rational does not mean "a fraction that is not whole": 5, 0 and −10 are rational too. And √2 looks simple but is not rational.
Used in
fractions and decimals, the number line, solving equations
Any number that has a place on the number line: all the rational numbers and all the irrational numbers together.
Precisely: A member of ℝ, the union of the rational and the irrational numbers; the real numbers fill the number line with no gaps.
How it works
The rationals are dense but leave gaps at points like √2 and π. Putting the irrationals into those gaps gives one unbroken line, and every point on it is a real number. Every length and every measurement is a real number.
Examples
−7, 3/4, 0.333… and √2 are all real numbers.
π is a real number: it has a point on the number line a little past 3.14.
The square root of −1 is not a real number, because no real number times itself is negative.
Do not confuse: "Real" is a name, not a judgement: imaginary numbers are just as useful, they only lie off the number line.
Used in
the number line, measurement, every later chapter of algebra and geometry
A rule that makes each term of a sequence from the term (or terms) before it, starting from a given first term.
Precisely: A description of a sequence by its starting term(s) together with a formula for tₙ in terms of earlier terms, such as t₁ = 1, tₙ = tₙ₋₁ + 3 for n ≥ 2.
How it works
Start with the given first term and apply the rule again and again. It shows how the sequence grows step by step, but to reach the 100th term you must work through the 99 before it.
Examples
The recursive formula t₁ = 1, tₙ = tₙ₋₁ + 3 gives 1, 4, 7, 10, 13, ...
The recursive formula u₁ = 1, uₙ = 2uₙ₋₁ + 3 gives 1, 5, 13, 29, 61, 125, 253, so 133 is not a term.
The recursive formula s₁ = 3, sₙ = sₙ₋₁(sₙ₋₁ − 1) gives 3, 6, 30, 870.
The recursive formula of a GP with r = 2 starting at 3 is t₁ = 3, tₙ = 2tₙ₋₁.
Do not confuse: A recursive formula needs a starting term; tₙ = tₙ₋₁ + 3 alone fits 1, 4, 7, ... and also 2, 5, 8, ...
Used in
APs and GPs, the Virahānka-Fibonacci sequence, computer programs (loops)
OpenStax: College Algebra 2e, 9.1 Sequences and Their Notations (recursive formula)
reflection
ri-FLEK-shunClass 9
A mirror image. Reflecting in the y-axis turns (x, y) into (−x, y), and reflecting in the x-axis turns (x, y) into (x, −y).
Precisely: A transformation that maps each point to its mirror image across a line, so that the line is the perpendicular bisector of the segment joining a point and its image.
How it works
The mirror line stays fixed. Each point moves straight across it, to the same distance on the other side, so only the coordinate measured across the line changes sign.
Examples
The reflection of (3, 4) in the y-axis is (−3, 4).
The reflection of (2, −5) in the x-axis is (2, 5).
A shape and its reflection have the same size and shape, but face opposite ways, like your left and right hands.
Reflecting triangle AMD in the y-axis gives a reflection A'M'D' with the same side lengths.
Do not confuse: A reflection flips a shape over; a rotation turns it round a point. Reflecting in the y-axis changes only the sign of x.
Used in
coordinate geometry, symmetry, transformations of graphs (Class 11)
OpenStax: College Algebra 2e, 3.5 Transformation of Functions (reflections)
reflection symmetry (line symmetry)
ri-FLEK-shun SIM-uh-treeFoundation
A shape has reflection symmetry when it can be folded along a line so that the two halves match exactly.
Precisely: A figure has reflection symmetry about a line when the reflection in that line maps the figure onto itself; the line is called a line (or axis) of symmetry.
How it works
Every point on one side of the line has a mirror point on the other side, at the same distance from the line. A square has 4 lines of symmetry, a regular pentagon 5, and a circle infinitely many: every diameter is one.
Examples
Folding a paper circle so that its edges meet shows a line of reflection symmetry: a diameter.
A square has reflection symmetry in 4 lines: its 2 diagonals and the 2 lines joining the midpoints of opposite sides.
The letter A has reflection symmetry in a vertical line; the letter S has none.
A regular hexagon has reflection symmetry in 6 lines.
Do not confuse: Reflection symmetry is a property of one shape; a reflection is the move itself, which can carry a shape to a new place.
Used in
circles and polygons, rangoli, art and buildings, graphs of even functions (Class 11)
A relation from A to B is a set of ordered pairs (a, b), picked from A × B, that links some elements of A to some elements of B.
Precisely: A relation R from a non-empty set A to a non-empty set B is a subset of A × B; its domain is the set of first entries and its range the set of second entries.
How it works
A rule decides which pairs belong: "is a factor of", "is the square of". Only the pairs that fit the rule are kept. A function is a special relation in which every element of A appears as a first entry exactly once.
Examples
The relation "is less than" from {1, 2} to {2, 3} is {(1, 2), (1, 3), (2, 3)}.
The relation "is the square of" from {1, 4, 9} to {1, 2, 3} is {(1, 1), (4, 2), (9, 3)}.
With 2 elements in A and 3 in B, there are 2⁶ = 64 possible relations from A to B.
Every function is a relation, but not every relation is a function.
Do not confuse: A relation may link one element to many others; a function links each input to exactly one output.
Used in
functions, Class 12 relations (reflexive, symmetric, transitive)
How often something happened, as a fraction of the total number of tries.
Precisely: The number of times an event occurs divided by the total number of trials or observations; it is the experimental probability of the event.
How it works
Count the times the event happened (its frequency) and divide by the total. The answer is between 0 and 1, like a probability, and it gets closer to the true probability as the trials grow in number.
Examples
A die rolled 50 times shows a 4 exactly 8 times: the relative frequency of a 4 is 8/50 = 0.16.
15 of 50 students like football: the frequency is 15, and the relative frequency is 15/50 = 0.3.
8 of 30 sweets in a sample are green, a relative frequency of 8/30 ≈ 0.267.
Do not confuse: The frequency is the count (15); the relative frequency is the count divided by the total (15/50 = 0.3).
Used in
experimental probability, survey results, statistics and data handling
OpenStax: Introductory Statistics 2e, 1.3 Frequency, Frequency Tables, and Levels of Measurement
repeating decimal
ri-PEE-ting DES-i-mulClass 9
A decimal that never ends, where one block of digits repeats forever, like 0.454545… or 0.1666…
Precisely: A non-terminating decimal whose digits from some point on are one block repeated without end; it is shown with a bar over the block, and every such decimal is rational.
How it works
In long division by q, only the remainders 1 to q − 1 can appear. So a remainder must come back, and from then on the digits loop. In a pure repeating decimal the loop starts right after the point (0.4545…); in a general one some digits come first (0.1666…).
Examples
5/11 = 0.454545… is a repeating decimal with the block 45.
1/6 = 0.1666… is a general repeating decimal: the 1 comes first, then 6 repeats.
To turn the repeating decimal x = 0.777… into a fraction: 10x − x = 7, so x = 7/9.
0.999… is a repeating decimal that is exactly equal to 1.
Do not confuse: A decimal that goes on forever without a repeating block (like π = 3.14159…) is not a repeating decimal; it is irrational.
Used in
converting decimals to fractions, telling rational from irrational numbers, long division
Precisely: A real number α is a root of the equation ax² + bx + c = 0 when aα² + bα + c = 0; the roots of an equation p(x) = 0 are the zeroes of the polynomial p(x).
How it works
Put the value into the equation: if both sides come out equal, it is a root. A quadratic equation has at most two roots, which may be equal, and sometimes no real ones.
Examples
1 is a root of 2x² − 3x + 1 = 0, because 2 − 3 + 1 = 0.
The roots of 6x² − x − 2 = 0 are 2/3 and −1/2, from (3x − 2)(2x + 1) = 0.
The prayer hall equation 2x² + x − 300 = 0 has roots 12 and −12.5; only 12 makes sense as a breadth.
Do not confuse: A root of an equation is not a square root: √9 = 3 is a square root, while the roots of x² = 9 are 3 and −3.
Used in
solving equations, zeroes of polynomials, checking answers
A shape has rotational symmetry when turning it about its centre, by less than a full turn, makes it look exactly the same.
Precisely: A figure has rotational symmetry of angle θ (0° < θ < 360°) about a point when a rotation by θ about that point maps the figure onto itself.
How it works
Turn the shape and watch for the moment it fits its own outline again. A square fits after 90°, 180° and 270°; a regular pentagon after every 72°. A circle fits after every angle at all, which is why a spinning wheel looks unchanged.
Examples
A square has rotational symmetry of 90° about its centre.
A regular hexagon has rotational symmetry of 60°: it fits its outline 6 times in one full turn.
A circle has complete rotational symmetry: a turn through any angle about its centre works.
Turning a chord about the centre gives another chord of the same length, at the same distance from the centre, because of the circle's rotational symmetry.
Do not confuse: Rotational symmetry is about turning; reflection symmetry is about folding. The letter S has rotational symmetry (a half turn) but no line of symmetry.
Used in
circles and regular polygons, wheels, fans and designs, transformations (Class 11)
Replacing a number with a nearby simpler one, such as the nearest ten, hundred or 2 decimal places.
Precisely: Replacing a number by the nearest value with fewer non-zero digits at a chosen place; a digit 5 or more in the next place rounds up, less than 5 rounds down.
How it works
Look at the digit just after the place you are rounding to: 5 to 9 rounds up, 0 to 4 keeps the place as it is; everything after it becomes 0 (or is dropped after the decimal point).
Examples
4,768 rounding to the nearest hundred gives 4,800 (the tens digit 6 is 5 or more).
π = 3.14159… after rounding to 2 decimal places is 3.14.
2.5 by rounding to the nearest whole number is 3 (5 rounds up).
Do not confuse: Round once, from the original number: rounding 3.447 straight to 1 decimal place gives 3.4, but rounding first to 3.45 and then again gives 3.5.
Used in
estimation, money (to the nearest rupee), measurements and answers to a given accuracy