A smaller group chosen from a whole group, studied to learn about the whole group.
Precisely: A subset of a population from which data are collected in order to estimate facts about the population; choosing it is called sampling.
How it works
Collect data from the sample, work out relative frequencies, and scale them up to the population. A bigger sample that is more like the population (different classes, not just one) gives a more trustworthy estimate.
Examples
Asking 50 students of one class is a sample of the 1,500 students in the school.
In a sample of 30 sweets, 7 were yellow, so a bag of 600 probably holds about 7/30 × 600 = 140 yellow sweets.
A sample of 100 students from different classes represents the school better than 50 from one class.
Do not confuse: A sample is part of the population, not the whole of it. A sample chosen unfairly (only your friends) is biased and can mislead.
Used in
surveys and opinion polls, quality checks in factories, statistics (Class 10 to 12)
OpenStax: Introductory Statistics 2e, 1.2 Data, Sampling, and Variation in Data and Sampling
sample space
SAM-pul spayssClass 9
The list of every possible outcome of a random experiment, written inside curly brackets.
Precisely: The set S of all possible outcomes of a random experiment, each listed exactly once; the number of its elements is written n(S).
How it works
Write every outcome once, with none missing and none repeated, separated by commas inside { }. For experiments in steps, a tree diagram or a table makes sure nothing is left out. Make it as detailed as the question needs.
Examples
Tossing a coin: the sample space is S = {H, T}, with n(S) = 2.
Tossing two coins: the sample space is {HH, HT, TH, TT}, with n(S) = 4.
Counting heads when three coins are tossed: the sample space is {0, 1, 2, 3}.
Drawing two of 4 numbered balls with replacement gives a sample space of 16 outcomes; without replacement, 12.
Do not confuse: The sample space lists outcomes, not events; an event is a part (subset) of it. And n(S) counts outcomes; it is not the size of a statistical sample.
Used in
theoretical probability, tree diagrams, probability in Class 10 to 12
The number every length is multiplied by when a figure is enlarged or shrunk.
Precisely: The common ratio of corresponding sides of similar figures; on maps and plans it is called the representative fraction, such as 1 : 50,000.
How it works
Divide a length in the new figure by the matching length in the old one. Above 1 it is an enlargement, below 1 a reduction. Angles do not change; areas change by the square of the scale factor.
Examples
A map with scale factor 1 : 50,000 shows 1 km as 2 cm, because 2 cm × 50,000 = 1,00,000 cm = 1 km.
A shadow of a figure cast by a lamp is an enlargement with a scale factor greater than 1.
Triangles with sides 3, 4, 5 and 6, 8, 10 are similar with scale factor 2.
Do not confuse: Doubling the scale factor doubles lengths but makes areas four times as large.
Used in
maps and building plans, model making, enlarging photos, similar triangles
Precisely: A line that meets a circle in two distinct points; the part of it inside the circle is a chord.
How it works
A line and a circle can meet in no point (non-intersecting), one point (a tangent) or two points (a secant). Moving a secant until its two points meet turns it into a tangent.
Examples
A line crossing a circular pond from one side to the other is a secant.
Every chord lies on a secant: extend the chord in both directions.
As a secant is moved outwards, its chord gets shorter until it touches the circle as a tangent.
Do not confuse: A secant is a whole line; a chord is only the segment inside the circle. The secant ratio, sec A = 1/cos A, is a different idea with the same name.
Used in
circles and tangents, the secant-tangent theorem (later study)
A formula for the point that divides a line segment in a given ratio, such as 3 : 1.
Precisely: The point P dividing the segment from A(x₁, y₁) to B(x₂, y₂) internally in the ratio m₁ : m₂ is ((m₁x₂ + m₂x₁)/(m₁ + m₂), (m₁y₂ + m₂y₁)/(m₁ + m₂)).
How it works
Each coordinate is a weighted average: the point is m₁/(m₁ + m₂) of the way from A to B, so B's coordinate gets the weight m₁ and A's gets m₂. With the ratio 1 : 1 it becomes the midpoint formula. To find an unknown ratio, call it k : 1 and solve.
Examples
By the section formula, the point dividing (4, −3) to (8, 5) in the ratio 3 : 1 is ((24 + 4)/4, (15 − 3)/4) = (7, 3).
The section formula shows that (−4, 6) divides A(−6, 10) to B(3, −8) in the ratio 2 : 7.
Taking the ratio k : 1 in the section formula, the y-axis divides (5, −6) to (−1, −4) in the ratio 5 : 1, at (0, −13/3).
Do not confuse: The weights cross over: m₁ multiplies B's coordinates, not A's. Putting m₁ with A gives the point for the ratio m₂ : m₁ instead.
Used in
points of trisection, the centroid of a triangle, vectors and 3D geometry (Class 12)
A slice of a circle, like a slice of pizza: the region between two radii and the arc joining them.
Precisely: The region bounded by an arc of a circle and the two radii to its end points; if the arc subtends θ° at the centre, its area is πr² × θ/360.
How it works
Equal angles at the centre give equal slices, so a sector of θ° is θ/360 of the whole disc. A semicircular disc is ½ of πr²; a quarter circle (quadrant) is ¼.
Examples
A sector of radius 7 cm and angle 60° has area 22/7 × 49 × 60/360 = 77/3 ≈ 25.67 cm².
In 10 minutes the 7 cm minute hand of a clock sweeps a sector of 60°, about 25.67 cm².
For a chord subtending 90° in a circle of radius 10 cm, the minor sector is 3.14 × 100 ÷ 4 = 78.5 cm² and the major sector is 235.5 cm².
Two car wipers of length 28 cm, each sweeping 120°, clean two sectors of about 1642.67 cm² in all.
Do not confuse: A sector is bounded by two radii and an arc; a segment is bounded by a chord and an arc. The sector's perimeter includes the two radii: arc + 2r.
Used in
clocks, fans and wipers, pie charts, area of a segment
The part of a circle cut off by a chord: the region between the chord and its arc.
Precisely: The region bounded by an arc of a circle and the chord joining the end points of the arc; the smaller one is the minor segment and the larger the major segment.
How it works
Area of the minor segment = area of the sector − area of the triangle made by the two radii and the chord. The major segment is the rest of the circle.
Examples
For a chord subtending 60° in a circle of radius 15 cm, the minor segment is 117.75 − 97.31 ≈ 20.44 cm² (π ≈ 3.14, √3 ≈ 1.73).
The major segment of that circle is 706.5 − 20.44 ≈ 686.06 cm².
Angles in the same segment of a circle are equal (Class 9, Chapter 5).
Do not confuse: A segment of a circle (chord and arc) is not a line segment (a straight piece of a line), and not a sector (two radii and an arc).
Used in
areas of circle parts, designs with arcs, angles in the same segment
Precisely: For a polygon with sides a, b, c, ..., s = (a + b + c + ...)/2; the letter s in Heron's and Brahmagupta's formulas.
How it works
Add all the sides and halve the total. In Heron's formula each s − a is the amount by which the other two sides together are longer than a, halved, so it is always positive for a real triangle.
Examples
A triangle with sides 3, 4 and 5 has semi-perimeter 6.
A triangle has sides 8 cm and 11 cm and perimeter 32 cm, so the third side is 13 cm and the semi-perimeter is 16 cm.
A cyclic 4-gon with sides 5, 5, 12 and 12 has semi-perimeter 17.
Do not confuse: The semi-perimeter is half the perimeter, not half a side; mixing them up is the most common slip with Heron's formula.
Used in
Heron's formula, Brahmagupta's formula, the area r × s of a triangle with incircle radius r
Precisely: Either of the two arcs into which a diameter divides a circle; also the half-disc bounded by that arc and the diameter.
How it works
Reflecting the circle in the diameter swaps the two halves, so they are equal: each has length half of 2πr, which is πr, and each half-disc has area ½πr². The angle it makes at any point on it is 90°.
Examples
A semicircle of radius 7 cm has curved length 22/7 × 7 = 22 cm.
The bends of a 400 m running track are two semicircles, which together make one full circle.
The angle in a semicircle is 90°: join any point of it to the two ends of the diameter.
Three small semicircles along a diameter have the same total length as the big semicircle on the whole diameter.
Do not confuse: The curved length of a semicircle is πr; its perimeter, with the diameter, is πr + 2r. A quarter circle is half of a semicircle.
Used in
running tracks and arches, perimeters and areas of composite shapes, the angle in a semicircle
A list of numbers (or objects) in a fixed order, often following a rule.
Precisely: An ordered list t₁, t₂, t₃, ... whose members are called terms; it is finite if it stops and infinite if it goes on for ever.
How it works
Order matters: 1, 2, 3 is not the same sequence as 3, 2, 1. A rule tells you the terms, either straight from the position n (an explicit formula) or from the terms before (a recursive formula). Three dots, ..., mean that it carries on.
Examples
The odd numbers 1, 3, 5, 7, ... form an infinite sequence.
6, 12, 24, 48, 96 is a finite sequence of five terms.
The triangular numbers 1, 3, 6, 10, 15, ... are a sequence in which the gaps grow by 1 each time.
The primes 2, 3, 5, 7, 11, 13, ... are a sequence with no simple formula for the next term.
Do not confuse: A sequence is a list; a series is the sum of the terms of a list. A set has no order, a sequence does.
Used in
patterns, arithmetic and geometric progressions, series and limits (Class 11 and 12), computer programs
OpenStax: College Algebra 2e, 9.1 Sequences and Their Notations
series
SEER-eezClass 10
The sum you get by adding up the terms of a sequence.
Precisely: The sum of the terms of a sequence, a₁ + a₂ + a₃ + …; for the first n terms of an AP it is Sₙ = n/2 × [2a + (n − 1)d] = n/2 × (first term + last term).
How it works
For an AP, write the sum forwards and backwards and add the two lines: every column adds to the same total (first + last), and there are n columns, so twice the sum is n × (first + last). This is how Gauss added 1 to 100 as a boy.
Examples
The series 1 + 2 + 3 + … + 100 adds to 100 × 101 ÷ 2 = 5050, as Gauss found.
Shakila puts ₹100, ₹150, ₹200, … into a money box on 21 birthdays: the series totals 21/2 × (100 + 1100) = ₹12,600.
The series of the first 22 terms of the AP 8, 3, −2, … is 22/2 × [16 + 21 × (−5)] = 11 × (−89) = −979.
Do not confuse: A sequence is the list (8, 3, −2, …); a series is the sum of it (8 + 3 + (−2) + …).
Used in
totals of savings and instalments, sums of APs and GPs, infinite series (Class 11 and 12)
OpenStax: College Algebra 2e, 9.4 Series and Their Notations
set
setClass 11
A well-defined collection of things, called its elements. "Well-defined" means you can always tell whether something belongs to it.
Precisely: A well-defined collection of distinct objects; written by listing the elements in braces, {1, 2, 3}, or by a rule, {x : x is a prime less than 10}.
How it works
Order and repeats do not matter: {1, 2, 3} and {3, 1, 2, 2} are the same set. "The tall students" is not a set, since "tall" is not well-defined; "the students taller than 150 cm" is.
Examples
The vowels of English form the set {a, e, i, o, u}.
The set of prime numbers less than 10 is {2, 3, 5, 7}.
The set with no elements is the empty set, written ∅.
n(A) is the number of elements in the set A.
Do not confuse: A set is not a list: order and repeats do not count.
Used in
all of mathematics, probability (events are sets of outcomes), functions and relations
A triangle pattern made by cutting out the middle quarter of a triangle, then of each piece left, again and again for ever.
Precisely: The fractal obtained from an equilateral triangle by joining the midpoints of its sides and removing the central triangle, then repeating this on every remaining triangle; at stage n it has 3ⁿ triangles and 3/4 raised to n of the original area.
How it works
Each step replaces every black triangle by 3 smaller ones, so the count is multiplied by 3 (a GP with r = 3), while each step keeps 3/4 of the black area (a GP with r = 3/4). The count grows quickly as the area shrinks towards 0. Named after Wacław Sierpiński (1882 to 1969).
Examples
The Sierpiński triangle has 1, 3, 9, 27 black triangles at stages 0 to 3, and 243 at stage 5.
The black area of the Sierpiński triangle at stage 5 is (3/4)⁵ ≈ 0.237 of the start.
Zoom into one corner of the Sierpiński triangle and you see the whole pattern again.
Do not confuse: The number of triangles goes up (×3) while the area goes down (×3/4): two different GPs from the same picture.
A short way to write a long sum with the Greek letter Σ (sigma): Σ k from k = 1 to n means 1 + 2 + … + n.
Precisely: The notation Σ (from k = m to n) f(k), meaning f(m) + f(m + 1) + … + f(n); k is the index, m and n are the limits, and f(k) is the general term.
How it works
Read it as a small loop: start k at the bottom number, work out the expression, add it on, raise k by 1, and stop after the top number.
Examples
In sigma notation, 1² + 2² + … + 10² is Σ k² from k = 1 to 10.
Σ (2k − 1) from k = 1 to 5 in sigma notation means 1 + 3 + 5 + 7 + 9 = 25.
The sum of the first n natural numbers is written n(n + 1)/2 or, in sigma notation, Σ k.
The mean is Σ xᵢ ÷ n in sigma notation.
Do not confuse: The letter k inside is only a counter; Σ k² from 1 to 3 is 14 whatever the counter is called.
Used in
sums of squares and cubes, the mean and variance in statistics, series in calculus
OpenStax: Precalculus 2e, 11.4 Series and Their Notations
similar figures
SIM-i-ler FIG-yerzClass 10
Figures with the same shape but not necessarily the same size, like a photo and its enlargement.
Precisely: Two polygons with the same number of sides are similar when their corresponding angles are equal and their corresponding sides are in the same ratio; for triangles, △ABC ∼ △DEF.
How it works
Enlarging or shrinking a figure by a scale factor keeps every angle and multiplies every length by the same number. For polygons in general you must check both angles and ratios; for triangles, equal angles alone are enough.
Examples
All circles are similar figures, and so are all squares and all equilateral triangles.
A square and a rhombus with equal sides are not similar figures: their angles differ.
A 6 m pole casting a 4 m shadow and a tower casting a 28 m shadow at the same time form similar figures, so the tower is 6 × 28 ÷ 4 = 42 m tall.
Do not confuse: Similar is not the same as congruent: similar figures can differ in size. Two photos of the same size but different shapes are not similar.
Used in
maps and scale drawings, heights by shadows, enlarging photos, trigonometry (Class 10)
Short tests that prove two triangles similar without checking every angle and every side.
Precisely: Two triangles are similar if two angles of one equal two angles of the other (AA, or AAA); if all three pairs of sides are in the same ratio (SSS); or if one angle is equal and the sides that include it are in the same ratio (SAS).
How it works
Look for what you know: equal angles (from parallel lines or a shared angle) suggest AA; three side lengths suggest SSS; an equal angle between two known sides suggests SAS. Once similar, all the other ratios and angles follow.
Examples
By the similarity criteria, a triangle with AB = 2 cm, ∠A = 50°, AC = 4 cm is similar to one with DE = 3 cm, ∠D = 50°, DF = 6 cm (SAS: 2/3 = 4/6).
Two triangles with angles 40° and 60° each are similar by the AA similarity criteria: the third angles are both 80°.
Sides 3, 4, 5 and 6, 8, 10 pass the SSS test of the similarity criteria: every ratio is 1/2.
Do not confuse: SSS similarity needs sides in the same ratio; SSS congruence needs sides exactly equal. And there is no SSA criterion.
Used in
proofs in geometry, heights and distances, the Pythagoras theorem proof by similar triangles
For an angle in a right triangle, the opposite side divided by the hypotenuse.
Precisely: sin A = (side opposite A) ÷ hypotenuse, for an acute angle A of a right triangle; it lies between 0 and 1.
How it works
It grows from 0 at 0° to 1 at 90°. The idea goes back to Aryabhata's ardha-jya (half-chord), shortened to jya or jiva; the Latin translation sinus gave "sine", and Edmund Gunter (1581 to 1626) first wrote "sin".
Examples
sin 30° = 1/2: in a 30° right triangle the opposite side is half the hypotenuse, the sine of 30°.
If sin A = 3/4, the sine tells you the opposite side is 3 and the hypotenuse 4; the adjacent side is √7, so cos A = √7/4.
A 10 m ladder leaning at 60° to the ground reaches 10 × sin 60° = 5√3 ≈ 8.66 m up the wall, using the sine.
Do not confuse: sin A can never be more than 1 (the opposite side is never longer than the hypotenuse); cosec A = 1/sin A is never less than 1.
Used in
heights and distances, waves and sound (Class 11 physics), the sine rule for any triangle
The length along the sloping side of a cone, from its tip to the edge of its base.
Precisely: For a right circular cone of radius r and height h, the slant height is l = √(r² + h²), the hypotenuse of the right triangle formed by the height and a radius.
How it works
Slice the cone down the middle: the height, a radius and the sloping edge make a right triangle, so Pythagoras gives l. The curved surface area is then πrl.
Examples
A cone with radius 3 cm and height 4 cm has slant height √(9 + 16) = 5 cm.
The cone of Rasheed's top has radius 1.75 cm and height 3.25 cm, so its slant height is about 3.7 cm.
A tent in the shape of a cone uses canvas of area πrl, where l is its slant height.
Do not confuse: The slant height l is longer than the height h: the height goes straight down the middle, the slant height along the outside.
Used in
curved surface area of cones, tents, funnels and ice-cream cones
OpenStax: College Algebra 2e, 4.1 Key Terms (slope)
special case
SPESH-ul kayssClass 9
A narrower result that you get from a general one by adding an extra condition.
Precisely: An instance of a general result obtained by imposing an additional condition, such as setting two quantities equal or one quantity to zero.
How it works
Take the general result and put in the extra condition: b = a, c = 0, d = 0. What comes out is true because the general result is. Checking special cases is also a quick way to test a new formula.
Examples
A square is a special case of a rectangle, so A = a² is a special case of A = ab.
For an isosceles right triangle, a = b, and the special case of a² + b² = c² is c = a√2.
Heron's formula is the special case d = 0 of Brahmagupta's formula.
Testing Heron's formula on the special case of the 3-4-5 right triangle gives 6, the same as ½ × 3 × 4.
Do not confuse: A special case follows from the general result; the general result does not follow from one special case. Knowing that squares work tells you nothing certain about all rectangles.
Used in
checking formulas, proofs, formula variants in the Library
A perfectly round ball: every point on its surface is the same distance from its centre.
Precisely: The set of all points in space at a fixed distance r (the radius) from a fixed point (the centre); its surface area is 4πr² and its volume ⁴⁄₃πr³.
How it works
It is the three-dimensional version of a circle. Its surface area is exactly the curved surface of the cylinder that fits round it (4πr²), and its volume is two thirds of that cylinder's volume.
Examples
A sphere of radius 3 cm has volume ⁴⁄₃π × 27 = 36π ≈ 113.1 cm³.
The same sphere has surface area 4π × 9 = 36π ≈ 113.1 cm²: the same number, in different units.
A football, a marble and the Earth (roughly) are spheres.
Do not confuse: A sphere is the surface only; a solid ball is the sphere with its inside. A circle is flat; a sphere is not.
Used in
balls and globes, planets, melting and recasting problems
A way to factorise x² + bx + c: split bx into two parts whose numbers add to b and multiply to c.
Precisely: To factorise px² + qx + r, write q = m + n with mn = pr, split qx as mx + nx, and group the terms in pairs to take out a common factor.
How it works
It is what the algebra tiles do when the x-tiles go on two sides of the square. For x² + 7x + 12 you need numbers adding to 7 and multiplying to 12: 3 and 4.
Examples
Splitting the middle term of x² + 11x + 30 as 5x + 6x gives (x + 5)(x + 6).
For 6x² + 7x + 2, splitting the middle term as 3x + 4x (3 × 4 = 12 = 6 × 2) gives 3x(2x + 1) + 2(2x + 1) = (3x + 2)(2x + 1).
For x² − 4x − 96, splitting the middle term as −12x + 8x gives (x − 12)(x + 8).
Do not confuse: Both conditions must hold: 2 and 15 multiply to 30 but add to 17, so they cannot split 11x in x² + 11x + 30.
OpenStax: Elementary Algebra 2e, 7.3 Factor Trinomials of the Form ax^2+bx+c
square (of a number)
skwairFoundation
A number multiplied by itself: the square of 7 is 7 × 7 = 49, written 7².
Precisely: For any number a, its square is a² = a × a; it is never negative, and it is the area of a square of side a.
How it works
Squaring a negative number gives a positive one, because negative × negative is positive. For numbers like 25² or 43², tricks and identities such as (a + b)² help.
Examples
The square of 7 is 7² = 49.
The square of 25 is 625 (2 × 3 = 6, then write 25).
The square of −4 is (−4)² = 16, a positive number.
Do not confuse: The square of 7 is 49; a square root of 49 is 7. And −4² usually means −(4²) = −16, while (−4)² = 16.
OpenStax: Prealgebra 2e, 5.7 Simplify and Use Square Roots
square root
skwair rootFoundation
The number that, multiplied by itself, gives the number you started with: √49 = 7, because 7 × 7 = 49.
Precisely: A square root of a ≥ 0 is a number b with b² = a; the sign √a means the non-negative one. √a is rational only when a is a perfect square (of a rational).
How it works
For perfect squares, use prime factorisation (halve each power) or the last-digit trick. For other numbers the root is irrational and is given as a surd (√2) or a decimal approximation (1.414…).
Examples
√144 = 12 is a square root, because 12 × 12 = 144.
The square root √2 ≈ 1.414 is irrational: 1.414² = 1.999396, close to 2 but not equal.
The square root of 5476 is 74: it ends in 4 or 6, and 7² ≤ 54 < 8².
Do not confuse: x² = 49 has two solutions, 7 and −7, but √49 means only 7. And the square root of a negative number is not a real number.
Used in
Pythagoras and distances, quadratic equations, surds and irrational numbers
OpenStax: Prealgebra 2e, 5.7 Simplify and Use Square Roots
squaring a shape (quadrature)
SKWAIR-ing uh shaypClass 9
Drawing a square with exactly the same area as a given shape, using only a straight edge and a compass (or a rope and pegs).
Precisely: Constructing a square equal in area to a given figure; for a rectangle a by b the square has side √(ab).
How it works
Baudhāyana's Śhulbasūtra (about 800 BCE) squares a rectangle using ((a + b)/2)² − ((a − b)/2)² = ab: draw a right triangle whose hypotenuse is (a + b)/2 and one leg (a − b)/2; its other leg is the side of the square.
Examples
Squaring a shape such as a 9 by 4 rectangle gives a square of side 6, because ((9 + 4)/2)² − ((9 − 4)/2)² = 42.25 − 6.25 = 36.
Baudhāyana's construction for squaring a shape uses the difference of two squares in geometric form.
The Egyptians' and Baudhāyana's rule A ≈ (8d/9)² comes from approximately squaring a shape that is a circle.
Do not confuse: Squaring a rectangle or a triangle can be done exactly; squaring a circle exactly with straight edge and compass was proved impossible in 1882, because π is transcendental.
Used in
ancient altar building (the Śhulbasūtras), constructions, the history of π
The head start given to runners in the outer lanes, so that everyone runs the same distance round the bends.
Precisely: The distance between the starting points of adjacent lanes on a curved track; it equals the extra length of the outer lane's curved part.
How it works
On the straights all lanes are equally long, but on a bend an outer lane has a bigger radius, so a longer arc. For one semicircular bend, a lane w metres further out is π × w metres longer, whatever the radius of the bend.
Examples
With lanes 1.22 m wide, the stagger for one bend is π × 1.22 ≈ 3.83 m.
The stagger is the same between every pair of neighbouring lanes, because each lane is one lane width further out (if every runner keeps the same distance from the inner edge of their lane).
Without a stagger, the runner in lane 8 would run the farthest in a 400 m race.
Do not confuse: The stagger is not an advantage: the outer runner starts ahead only by the extra distance their lane adds.
A measure of how spread out data are: the square root of the average of the squared distances from the mean. It is written σ.
Precisely: The positive square root of the variance, σ = √(Σ(xᵢ − x̄)² ÷ n); it is in the same units as the data.
How it works
Find the distance of each value from the mean, square the distances, average them (that is the variance) and take the square root. Values bunched near the mean give a small σ; widely scattered values give a large one.
Examples
The data 6, 8, 10, …, 24 have standard deviation √33, about 5.74.
Heights all equal to 150 cm have standard deviation 0.
A batsman with a smaller standard deviation of scores is more consistent.
The standard deviation of 4, 6, 8, 10, 12 is 2√2.
Do not confuse: The standard deviation is the root of the variance; the variance is in squared units.
Used in
comparing spread, the coefficient of variation, probability and statistics in Class 12 and beyond
A guess at how likely something is, based on a person's own judgement rather than on counting or experiments.
Precisely: A degree of belief that a person assigns to an event from their own reading of the evidence; different people may assign different values.
How it works
People look at clues and judge. It is useful for one-off events that cannot be repeated or counted, but it is not objective: the same clues can lead two people to different answers.
Examples
"The sun is bright, so it is unlikely to rain" is a subjective probability.
A friend who says "It is very hot, so it may rain later" gives a different subjective probability from the same weather.
A fan's feeling that their team will surely win tomorrow is a subjective probability.
Do not confuse: Subjective probability comes from judgement; experimental probability comes from data and theoretical probability from counting equally likely outcomes.
Used in
everyday decisions, forecasts by experts, Bayesian statistics (later study)
Solving a pair of equations by writing one unknown in terms of the other, then putting that into the other equation.
Precisely: A method for a pair of linear equations: express one variable in terms of the other from one equation, substitute into the second to get an equation in one variable, solve it, then substitute back.
How it works
Pick the equation where a variable is easiest to get alone (a coefficient of 1 helps). Put that expression in place of the variable in the other equation, solve, and then find the other variable. Check in both equations.
Examples
By the substitution method, from x + 2y = 3 write x = 3 − 2y; then 7(3 − 2y) − 15y = 2 gives y = 19/29 and x = 49/29.
Aftab's ages by the substitution method: s = 3t + 6 into s − 7t + 42 = 0 gives 4t = 48, so t = 12 and s = 42.
The substitution method on y = 2x − 2 and y = 4x − 4 gives 2x − 2 = 4x − 4, so x = 1 and y = 0.
Do not confuse: Substitute into the other equation: putting x = 3 − 2y back into x + 2y = 3 only gives 3 = 3.
Used in
pairs of linear equations, word problems, a line meeting a curve (Class 11)
OpenStax: Elementary Algebra 2e, 5.2 Solving Systems of Equations by Substitution
subtend (an angle)
sub-TENDClass 9
A chord or an arc subtends an angle at a point when the lines from its two ends to that point make that angle.
Precisely: A segment or arc AB subtends the angle APB at a point P: the angle between the lines joining P to the end points A and B.
How it works
Stand at the point and look at the two ends of the chord: the angle between your two lines of sight is the angle the chord subtends there. The same chord subtends different angles at different points, but at all points of the circle on the same side of it, the angles are equal.
Examples
The chord BC subtends the angle BAC at the centre A.
Equal chords subtend equal angles at the centre.
A diameter subtends a right angle at every other point of the circle.
In a circle of radius 12 cm, a chord that subtends 60° at the centre is 12 cm long, because the triangle it makes with the centre is equilateral.
Do not confuse: The angle is at the point, not at the chord: "AB subtends ∠APB at P". And the angle at the centre is twice the angle at the circle, not equal to it.
Used in
circle theorems, heights and distances (Class 10), astronomy (the angle the Moon subtends at your eye)
One cube plus or minus another, which always splits into a short bracket times a long one: x³ − y³ = (x − y)(x² + xy + y²) and x³ + y³ = (x + y)(x² − xy + y²).
Precisely: Expressions of the form x³ + y³ or x³ − y³, factorised by the identities x³ + y³ = (x + y)(x² − xy + y²) and x³ − y³ = (x − y)(x² + xy + y²).
How it works
Multiply out (x − y)(x² + xy + y²) with the distributive property: each middle term appears once with + and once with −, so they all cancel and only x³ − y³ is left. A memory aid for the signs: the short bracket copies the sign between the cubes, the long bracket starts its middle term with the opposite sign, and its last sign is always +.
Examples
x³ − 8 is a difference of cubes: x³ − 2³ = (x − 2)(x² + 2x + 4).
27a³ + 1 is a sum of cubes: (3a)³ + 1³ = (3a + 1)(9a² − 3a + 1).
By the difference of cubes, 10³ − 9³ = (10 − 9)(100 + 90 + 81) = 271.
Its three-letter cousin is x³ + y³ + z³ − 3xyz = (x + y + z)(x² + y² + z² − xy − yz − zx). With x + y + z = 10, xyz = 25 and x² + y² + z² = 38, it gives the sum of cubes x³ + y³ + z³ = 145.
Do not confuse: x³ + y³ is not (x + y)³: (x + y)³ has the extra terms 3x²y + 3xy². And the long bracket x² + xy + y² is not the perfect square x² + 2xy + y², so it does not factorise further.
Used in
factorisation, simplifying rational expressions, quick arithmetic with cubes
OpenStax: Elementary Algebra 2e, 7.4 Factor Special Products (sums and differences of cubes)
sum to infinity
sum too in-FIN-i-teeClass 11
The number that the sums of more and more terms of a GP settle on, when the ratio is between −1 and 1. It is a ÷ (1 − r).
Precisely: For a geometric series with first term a and common ratio r, |r| < 1, the limit of the partial sums Sₙ as n grows without bound, equal to a/(1 − r); when |r| ≥ 1 the sums do not settle and there is no sum to infinity.
How it works
The sum of n terms is a(1 − rⁿ)/(1 − r). When r is between −1 and 1, rⁿ shrinks towards 0 as n grows, so the sum gets as close as you like to a/(1 − r).
Examples
The sum to infinity of 1 + ½ + ¼ + … is 1 ÷ (1 − ½) = 2.
0.333… is the sum to infinity of 3/10 + 3/100 + …, which is ⅓.
A ball that bounces to ¾ of its height each time travels a finite distance: a sum to infinity.
1 + 2 + 4 + … has no sum to infinity, since its ratio is 2.
Do not confuse: A sum to infinity is not adding forever by hand; it is the value the sums approach. Only GPs with |r| < 1 have one.
Used in
recurring decimals as fractions, bouncing balls and repeated sharing, infinite series in calculus
The total area of all the outside surfaces of a solid: how much paper would wrap it, or paint would cover it.
Precisely: The total surface area (TSA) is the area of the whole boundary of a solid; the curved (lateral) surface area (CSA) leaves out the flat top and base.
How it works
Unfold the solid into flat pieces (its net) and add their areas. A cylinder unrolls into a rectangle 2πr by h plus two circles; a cone opens into a sector. For combined solids, add only the surfaces you can actually see.
Examples
A cylinder of radius 7 cm and height 10 cm has curved surface area 2 × 22/7 × 7 × 10 = 440 cm² and total 748 cm².
Rasheed's top (a cone on a hemisphere, 5 cm tall and 3.5 cm wide) needs about 39.6 cm² of colour: the surface area of the hemisphere plus the cone's curved part.
Mayank's bird-bath, a cylinder 1.45 m tall of radius 30 cm with a hemispherical hollow, has surface area 2πr(h + r) = 3.3 m².
Do not confuse: When solids are joined, the faces that touch are no longer outside, so the surface area of the join is not the sum of the two total surface areas.
Used in
painting and wrapping, making tins and tents, heat loss in science