The flat surface with the x-axis and y-axis drawn on it, where every point has a pair of coordinates. It is also called the coordinate plane or the xy-plane.
Precisely: A plane with two perpendicular number lines meeting at the origin, in which each point corresponds to exactly one ordered pair of real numbers.
How it works
Each point matches one ordered pair, and each ordered pair matches one point. That two-way match lets pictures (points, lines, shapes) be turned into numbers and equations, and back again.
Examples
The Cartesian plane is named after René Descartes, who with Pierre de Fermat joined algebra and geometry in 1637.
On the Cartesian plane, the equation y = 2x + 1 is a straight line.
Shalini drew Reiaan's room on a Cartesian plane, so every piece of furniture had an address.
Do not confuse: The plane is the whole flat surface, not just the two axes drawn on it.
Used in
coordinate geometry, graphs of equations, maps and computer screens
The fixed point in the middle of a circle; every point of the circle is the same distance from it.
Precisely: The point from which all points of a circle are equidistant; it lies on every diameter and on the perpendicular bisector of every chord.
How it works
To find the centre of a paper circle, fold it so that the edges meet, open it, and fold it again another way: each crease is a diameter, and the two creases cross at the centre. On a drawing, draw two chords and their perpendicular bisectors; they meet at the centre.
Examples
Fold a paper circle in half twice, in two different ways: the two creases cross at the centre.
The perpendicular bisector of any chord passes through the centre of the circle.
All circles through two points A and B have their centre somewhere on the perpendicular bisector of AB.
The centre of the circle through the corners of a right-angled triangle is the midpoint of its hypotenuse.
Do not confuse: The centre is inside the circle, not on it. "Centre" is the spelling NCERT uses; American books write "center".
Used in
finding the middle of round objects, circle theorems, constructions
The point where the three medians of a triangle meet: its balance point. Its coordinates are the averages of the corners' coordinates.
Precisely: The common point of the three medians of a triangle, dividing each median in the ratio 2 : 1 from the vertex; its coordinates are ((x₁ + x₂ + x₃)/3, (y₁ + y₂ + y₃)/3), with a z term in space.
How it works
Each median joins a corner to the middle of the opposite side. The three medians always cross at one point, two thirds of the way along each from its corner, and a cardboard triangle balances on a pin at exactly that point.
Examples
The centroid of the triangle with corners (0, 0), (6, 0) and (0, 9) is (2, 3).
A triangle cut from card balances on a pencil point at its centroid.
The centroid is two thirds of the way from each corner to the middle of the opposite side.
In space, the centroid of (3, −5, 7), (−1, 7, −6) and (1, 1, 2) is (1, 1, 1).
Do not confuse: The centroid is where the medians meet; the centre of a triangle's circumcircle (where the perpendicular bisectors meet) is usually a different point.
Used in
coordinate geometry, centre of mass in physics, the section formula
A straight line segment joining two points on a circle.
Precisely: A line segment whose two end points both lie on a circle.
How it works
The perpendicular from the centre cuts a chord into two equal halves, so half the chord, its distance from the centre and a radius make a right triangle: (half chord)² + (distance)² = radius². The longer the chord, the closer it is to the centre; the longest chord goes through the centre and is the diameter.
Examples
A thread pulled tight between two points on a wheel is a chord of the wheel.
In a circle of radius 15 cm, a chord 9 cm from the centre is 2 × √(225 − 81) = 2 × 12 = 24 cm long.
Equal chords subtend equal angles at the centre, and they are equally far from it.
A chord that passes through the centre is a diameter, the longest chord of all.
Do not confuse: A chord is the straight segment across the inside; an arc is the curved piece of the circle between the same two points.
Used in
circle theorems, finding lengths in circles, trigonometry (Class 11)
A perfectly round curve: every point on it is the same distance from one fixed point in the middle, the centre.
Precisely: The set of all points in a plane that are at a given distance (the radius) from a given point (the centre) in that plane.
How it works
Tie a string to a peg, pull it tight and walk round: the free end draws a circle, because its distance from the peg never changes. The circle is only the curve; the flat part inside it, together with the curve, is the circular region.
Examples
A raindrop falling on a pond makes ripples, each one a circle around the point where the drop landed.
The circle with centre A and radius 3 cm is every point of the paper that is exactly 3 cm from A.
Through any three points that are not on one line there is exactly one circle (Theorem 1 of the chapter).
A circle looks the same after any turn about its centre, which is why a rolling wheel never seems to change.
Do not confuse: A circle is the round line only; the region inside it is the circular region. An oval (ellipse) is round too, but its points are not all at one distance from a single centre.
Used in
wheels, clocks and gears, chords and angles in circles, area and circumference (Class 10), the equation of a circle (Class 11)
The centre of the circle that passes through all three corners of a triangle.
Precisely: The point where the perpendicular bisectors of the sides of a triangle meet; it is equidistant from the three vertices.
How it works
Draw the perpendicular bisectors of two sides; where they cross is the circumcentre (the third one passes through it too). Where it lands tells you the kind of triangle: inside an acute-angled triangle, outside an obtuse-angled one, and at the midpoint of the hypotenuse of a right-angled one.
Examples
A triangle with angles 70°, 60° and 50° is acute-angled, so its circumcentre lies inside it.
In a triangle with an angle of 100°, the circumcentre lies outside the triangle.
In a right triangle with hypotenuse 10 cm, the circumcentre is the midpoint of the hypotenuse, 5 cm from every corner.
Do not confuse: The circumcentre (where the perpendicular bisectors meet) is not the centroid (where the medians meet); the two are the same point only in an equilateral triangle.
Used in
the circumcircle, constructions, coordinate geometry (Class 10 and 11)
The circle that passes through all the corners of a triangle (or of any polygon that has such a circle).
Precisely: The unique circle through the three vertices of a triangle; it circumscribes the triangle, and the triangle is inscribed in it.
How it works
Its centre is the circumcentre, and its radius is the distance from there to any corner. Every triangle has exactly one, because three non-collinear points have exactly one circle through them. A quadrilateral has one only when it is cyclic.
Examples
Every triangle has exactly one circumcircle.
The circumcircle of a right triangle has the hypotenuse as a diameter.
A regular hexagon inscribed in a circumcircle of radius r has sides that are also r long.
A rectangle has a circumcircle centred where its diagonals cross; a parallelogram that is not a rectangle has none.
Do not confuse: The circumcircle passes through the corners, outside the triangle; the incircle sits inside and touches the sides.
Used in
triangles and circles, cyclic quadrilaterals, constructions
The perimeter of a circle: the distance all the way round it.
Precisely: The length of a circle, C = πd = 2πr, where d is the diameter and r the radius.
How it works
The ratio of circumference to diameter is the same for every circle; that ratio is π. So multiply the diameter by π (or the radius by 2π). Going the other way, divide the circumference by 2π to get the radius.
Examples
A circle of radius 7 cm has circumference 2 × 22/7 × 7 = 44 cm.
A car tyre of diameter 56 cm has circumference 176 cm, so over 10 km it turns about 5,682 times.
A circle whose circumference is 44 cm has radius 44 ÷ (2 × 22/7) = 7 cm.
If the circumferences of two circles are in the ratio 5 : 4, their radii are also in the ratio 5 : 4.
Do not confuse: The circumference is the length round the circle (cm); the area is the space inside it (cm²). Doubling the radius doubles the circumference but makes the area four times as big.
Used in
wheels and distance travelled, running tracks, arc length, the area of a circle
A range of values, like 10 to 25, used to group data; its width is the class size.
Precisely: An interval of values into which data are grouped, with a lower and an upper class limit; the class size (width) h is upper limit − lower limit.
How it works
Split the whole range of the data into equal, non-overlapping intervals and count how many values fall in each. By convention a value on a boundary goes into the class that starts with it (40 goes in 40 to 55, not 25 to 40).
Examples
Marks grouped as 10 to 25, 25 to 40, 40 to 55, … use class intervals of size 15.
The class interval 55 to 70 has lower limit 55, upper limit 70 and size 15.
Ages 0 to 10, 10 to 20, 20 to 30 are class intervals of size 10.
Do not confuse: The class size is the width (15), not the number of values in the class, which is its frequency.
Used in
grouped frequency tables, histograms, the mean, median and mode of grouped data
The number multiplying the variable in a term. In 4x + 5y + 3, the coefficient of x is 4.
Precisely: In a term such as a·xⁿ, the constant factor a that multiplies the power of the variable.
How it works
Look at one term and read the number written in front of the letter, sign included. A letter on its own (x) has coefficient 1, and −x² has coefficient −1.
Examples
In 4x + 5y + 3, the coefficient of y is 5.
The coefficient of x³ in 5y³ + 2x³ − 8 is 2.
In −x², the coefficient is −1, even though no 1 is written.
The taxi fare 25 + 12d rupees has coefficient 12: every extra kilometre adds ₹12.
Do not confuse: The coefficient multiplies the variable; the constant term stands alone. In 4x + 3, 4 is the coefficient and 3 is the constant.
Used in
polynomials, identities and factorisation, the slope in y = ax + b
The standard deviation as a percentage of the mean, CV = σ ÷ x̄ × 100. The set with the smaller CV is the more consistent one.
Precisely: CV = (σ ÷ x̄) × 100 for a set with non-zero mean; a unitless measure of relative spread used to compare the variability of different sets.
How it works
Dividing the spread by the size of the values makes sets of different sizes comparable: a spread of 10 around a mean of 1000 (CV 1) is far steadier than a spread of 10 around a mean of 20 (CV 50).
Examples
A set with σ = 5 and mean 50 has coefficient of variation 10.
The batsman with the lower coefficient of variation is the more consistent scorer.
The coefficient of variation has no units, so weights and heights can be compared.
Do not confuse: A larger standard deviation does not always mean more variable: compare the CVs when the means differ.
Used in
comparing consistency, quality control in factories, finance (risk against return)
Two lines that lie exactly on top of each other: they are the same line.
Precisely: Lines whose equations are multiples of each other, a₁/a₂ = b₁/b₂ = c₁/c₂, so every point of one is on the other.
How it works
Multiply one equation by a number and you get the other. Every solution of one is a solution of the other, so the pair has infinitely many solutions.
Examples
x + y = 5 and 2x + 2y = 10 are coincident lines: the second is the first times 2.
5x − 3y = 11 and −10x + 6y = −22 are coincident lines (multiply the first by −2).
2x + 3y − 9 = 0 and 4x + 6y − 18 = 0 are coincident lines, with ratios 1/2 = 1/2 = 1/2.
Do not confuse: Coincident lines meet everywhere; parallel lines never meet. Both have equal slopes, but only coincident lines have the same intercept.
Precisely: Three or more points are collinear when a single straight line passes through all of them; otherwise they are non-collinear.
How it works
Any two points always lie on one line, so the word matters from three points on. Three non-collinear points make a triangle and have exactly one circle through them. Three collinear points have no circle through them at all, because a line cuts a circle in at most two points.
Examples
(1, 1), (2, 2) and (3, 3) are collinear: all three lie on the line y = x.
No circle passes through three collinear points A, B and C.
A, B and C are not collinear, so the perpendicular bisectors of AB and AC meet, and exactly one circle passes through all three.
The corners of a triangle are never collinear; if they were, the "triangle" would be flat, with no area.
Do not confuse: Collinear (on one line) is not concyclic (on one circle). Three collinear points can never be concyclic.
Used in
circles through points, triangles, coordinate geometry (Class 10)
OpenStax: College Algebra 2e, 9.2 Arithmetic Sequences (common difference)
common ratio
KOM-un RAY-shee-ohClass 9
The fixed number you multiply by each time in a geometric progression.
Precisely: The constant r = tₙ ÷ tₙ₋₁ between consecutive terms of a GP (with no term equal to 0).
How it works
Divide any term by the one before it. If every such ratio is the same, the sequence is a GP and that value is r. If r is bigger than 1 the terms grow; between 0 and 1 they shrink; if r is negative the signs take turns.
Examples
In 3, 6, 12, 24, ... the common ratio is 2.
In 5, 15/4, 45/16, 135/64, ... the common ratio is 3/4.
In 1, −1, 1, −1, ... the common ratio is −1.
A ball bouncing back to 3/4 of its height each time makes heights 18, 13.5, 10.125, ... ft with common ratio 0.75.
Do not confuse: The common ratio is found by dividing, the common difference by subtracting. 2, 4, 8 has common ratio 2, not common difference 2.
Used in
geometric progressions, growth and decay, compound interest (later)
OpenStax: College Algebra 2e, 9.3 Geometric Sequences (common ratio)
complementary events
kom-pli-MEN-tuh-ree i-VENTSClass 10
An event and its opposite, "E" and "not E": exactly one of them must happen.
Precisely: The event Ē (not E), made of all outcomes not in E, is the complement of E; since exactly one of them happens, P(E) + P(not E) = 1.
How it works
When an event is awkward to count, count its opposite and take it from 1. "At least one" questions are often easiest this way.
Examples
If the probability that Reshma wins a match is 0.62, the probability that Sangeeta wins is 1 − 0.62 = 0.38, because the two are complementary events.
Drawing an ace from 52 cards has probability 4/52 = 1/13, so its complementary event, not an ace, has probability 12/13.
Rolling a 6 and rolling "not 6" are complementary events: 1/6 + 5/6 = 1.
Do not confuse: Complementary is not the same as "different": rolling a 1 and rolling a 2 cannot both happen, but they are not complements, since neither may happen.
Used in
probability shortcuts, "at least one" problems, the complement rule
A number of the form a + ib, where a and b are real numbers and i is the number whose square is −1. a is its real part and b its imaginary part.
Precisely: An ordered pair of real numbers (a, b), written a + ib, with addition and multiplication defined so that i² = −1; the real numbers are the complex numbers with b = 0.
How it works
Work with a + ib as with any bracket of algebra, and whenever i² turns up, replace it by −1. Every quadratic equation has two roots among the complex numbers, even when it has none among the real numbers.
Examples
3 + 4i is a complex number with real part 3 and imaginary part 4.
Every real number, such as 7, is also a complex number: 7 + 0i.
The roots of x² + 1 = 0 are the complex numbers i and −i.
A complex number a + ib can be drawn as the point (a, b).
Do not confuse: The imaginary part of 3 + 4i is 4, not 4i; the number beside i is the imaginary part.
Used in
solving quadratics with negative discriminants, electric circuits, waves and signals
A whole number greater than 1 that has more factors than just 1 and itself, like 4, 6, 9 and 15.
Precisely: A natural number greater than 1 that is not prime, so it can be written as a product of two smaller natural numbers.
How it works
If you can find any factor other than 1 and the number, it is composite. Every composite number splits into primes in exactly one way (the Fundamental Theorem of Arithmetic).
Examples
91 is a composite number: 91 = 7 × 13.
7 × 11 × 13 + 13 is a composite number, because it equals 13 × (7 × 11 + 1) = 13 × 78 = 1014.
7 × 6 × 5 × 4 × 3 × 2 × 1 + 5 = 5045 is a composite number: it is 5 × 1009.
Do not confuse: 1 is neither prime nor composite. And an odd number can be composite: 9 = 3 × 3.
OpenStax: Prealgebra 2e, 2.5 Prime Factorization and the Least Common Multiple (composite number)
compound angle
KOM-pound ANG-gulClass 11
An angle written as a sum or difference of two angles, like A + B or A − B. Special formulas give its sine, cosine and tangent from those of A and B.
Precisely: An angle formed as the algebraic sum of two or more angles; its trigonometric ratios are found with the addition formulas, such as sin(A + B) = sin A cos B + cos A sin B.
How it works
You cannot just add the sines: sin(30° + 60°) = sin 90° = 1, but sin 30° + sin 60° is about 1.37. The addition formulas mix the sines and cosines of both angles to get the right answer.
Examples
75° is the compound angle 45° + 30°, so sin 75° = sin 45° cos 30° + cos 45° sin 30°.
15° is the compound angle 45° − 30°.
cos(A + B) is the cosine of a compound angle.
In tan(A − B), the angle A − B is a compound angle.
Do not confuse: sin(A + B) is not sin A + sin B. The ratio of a compound angle needs its own formula.
Used in
exact values of angles like 15° and 75°, proving identities, double angle formulas (A + A)
OpenStax: Precalculus 2e, 7.2 Sum and Difference Identities
concyclic (points)
kon-SY-klikClass 9
Points that all lie on one circle.
Precisely: A set of points is concyclic when a single circle passes through all of them.
How it works
Any three non-collinear points are concyclic, so the question really starts at four. Four points A, B, C and D are concyclic when AB subtends equal angles at C and D on the same side of it, or when the quadrilateral they form has opposite angles adding up to 180°.
Examples
The four corners of a square are concyclic; the centre of their circle is where the diagonals cross.
If ∠ACB = ∠ADB with C and D on the same side of AB, then A, B, C and D are concyclic.
A quadrilateral with angles 80°, 110°, 100° and 70°, in order, has concyclic corners, because 80° + 100° = 180° and 110° + 70° = 180°.
Three concyclic points can never be collinear.
Do not confuse: Concyclic (on one circle) is not collinear (on one line). And any three points not on a line are concyclic, so the idea only tells you something new for four or more points.
A solid with a circular base that narrows to a point at the top, like an ice-cream cone or a birthday cap.
Precisely: A right circular cone of radius r, height h and slant height l = √(r² + h²); curved surface area πrl, total surface area πr(l + r), volume ⅓πr²h.
How it works
Three cones of water fill a cylinder with the same base and height exactly, which is where the ⅓ comes from. Its curved surface unrolls into a sector of a circle of radius l.
Examples
A cone of radius 3 cm and height 4 cm has slant height 5 cm and volume ⅓ × 3.14 × 9 × 4 = 37.68 cm³.
A toy rocket is a cone mounted on a cylinder.
A cone and a cylinder with the same base and height: the cylinder holds exactly three times as much.
Do not confuse: The curved area uses the slant height l, the volume uses the height h: mixing them is the usual mistake.
Used in
funnels, tents and caps, volume and surface area, combined solids
Exactly the same shape and exactly the same size, so one would fit perfectly on top of the other.
Precisely: Two figures are congruent when one can be moved (slid, turned or flipped) onto the other exactly; congruent triangles have all corresponding sides and angles equal, written △ABC ≅ △DEF.
How it works
For triangles you need not check all six parts: SSS, SAS, ASA and RHS each prove congruence. Congruent triangles then give equal sides and angles for free, which is how many circle theorems are proved.
Examples
All circles with the same radius are congruent.
Triangles CAB and CDE with CA = CD, CB = CE and AB = DE are congruent by SSS.
Your left and right hands are congruent in size and shape, though one is a flipped copy of the other.
Do not confuse: Congruent means same shape and same size; similar means same shape, perhaps a different size. Every congruent pair is similar, but not the other way.
Used in
geometry proofs, circle theorems, constructions and designs
A curve made where a flat plane cuts a double cone: a circle, an ellipse, a parabola or a hyperbola, depending on how the plane is tilted.
Precisely: The intersection of a plane with a double right circular cone; apart from the special cases through the vertex, it is a circle, an ellipse, a parabola or a hyperbola, and each has an equation of the second degree in x and y.
How it works
Cut straight across the cone and you get a circle. Tilt the cut a little and it stretches into an ellipse. Tilt it until it runs parallel to the side of the cone and it never closes: a parabola. Tilt it further so it cuts both halves of the double cone and you get the two branches of a hyperbola.
Examples
The edge of the light from a torch on a wall is a conic section: a circle straight on, an ellipse at a slant.
The path of a thrown ball is a conic section, a parabola.
The planets move along a conic section, an ellipse, with the Sun at a focus.
Every conic section has an eccentricity: 0 for a circle, under 1 for an ellipse, 1 for a parabola, over 1 for a hyperbola.
Do not confuse: A parabola is one kind of conic section, not another name for all of them.
Used in
orbits of planets and satellites, dish antennas and headlights, bridges and cooling towers
The conjugate of a + ib is a − ib: the same number with the sign of its imaginary part changed. It is written with a bar, z̄.
Precisely: For z = a + ib, the complex number z̄ = a − ib; it is the reflection of z in the real axis, and z z̄ = a² + b² = |z|².
How it works
Multiplying a number by its conjugate removes the i: (a + ib)(a − ib) = a² + b². That is why dividing by a complex number starts by multiplying the top and bottom by the conjugate of the bottom.
Examples
The conjugate of 3 + 4i is 3 − 4i.
The conjugate of −2i is 2i, and the conjugate of 5 is 5.
A number times its conjugate is real: (1 + i)(1 − i) = 2.
On the Argand plane a number and its conjugate are mirror images in the real axis.
Do not confuse: The conjugate changes only the sign of the imaginary part, not of the real part: the conjugate of 3 + 4i is 3 − 4i, not −3 − 4i.
Used in
dividing complex numbers, the modulus, complex roots of quadratics (they come in conjugate pairs)
Coming one straight after another, with nothing missed out.
Precisely: Following each other in order without a gap: consecutive integers differ by 1, and consecutive terms of a sequence are tₙ₋₁ and tₙ.
How it works
Write the first one, then each next one in turn. In algebra, consecutive whole numbers are n, n + 1, n + 2; consecutive odd or even numbers are n, n + 2, n + 4.
Examples
7, 8 and 9 are consecutive numbers.
In an AP, consecutive terms always differ by the same amount, d.
For any three consecutive square numbers, the smallest plus the largest minus twice the middle is always 2: 25 + 49 − 72 = 2.
25 + 26 + ... + 58 is a sum of consecutive numbers: S₅₈ − S₂₄ = 1711 − 300 = 1411.
Do not confuse: Consecutive odd numbers (5, 7, 9) differ by 2, not 1. Consecutive means next in the list being used.
A pair of equations that has at least one solution: their lines meet.
Precisely: A pair of linear equations with at least one common solution: either exactly one (intersecting lines, a₁/a₂ ≠ b₁/b₂) or infinitely many (coincident lines).
How it works
Compare the ratios of the coefficients. If a₁/a₂ ≠ b₁/b₂ the lines cross once; if all three ratios are equal the lines are the same. Either way the pair is consistent.
Examples
x − 2y = 0 and 3x + 4y − 20 = 0 are a consistent pair: 1/3 ≠ −2/4, so the lines cross once, at (4, 2).
x + y = 5 and 2x + 2y = 10 are a consistent pair with infinitely many solutions (the same line twice).
y = 2x − 2 and y = 4x − 4 are a consistent pair: they meet at (1, 0).
Do not confuse: Consistent does not mean exactly one solution: a dependent pair is consistent too, with infinitely many.
Used in
deciding if a problem can be solved, graphs of lines
The statement you get by swapping the "if" part and the "then" part of another statement.
Precisely: The converse of "if P, then Q" is "if Q, then P"; it is a different statement and must be proved (or disproved) on its own.
How it works
Write the result in the form "if ..., then ...", then swap the two halves. Sometimes the converse is true as well, and then both can be said together with "if and only if"; sometimes it is false.
Examples
Theorem 2 says equal chords subtend equal angles at the centre; its converse, Theorem 3, says chords that subtend equal angles at the centre are equal.
"Opposite angles of a cyclic quadrilateral add up to 180°" has a true converse: a quadrilateral whose opposite angles add up to 180° is cyclic.
The converse of "if a number ends in 0, it is even" is "if a number is even, it ends in 0", which is false: 12 is even.
The converse of Pythagoras' theorem is true: if a² + b² = c², the triangle has a right angle opposite c.
Do not confuse: A true statement can have a false converse, so a converse always needs its own proof. The converse is not the opposite (the negation) of the statement.
Used in
geometry theorems, logic and proof, mathematical reasoning (Class 11)
The pair of numbers (x, y) that tells you exactly where a point is: first how far across, then how far up or down.
Precisely: The ordered pair (x, y) giving a point's signed distances from the y-axis and from the x-axis.
How it works
Start at the origin. Move x steps along the x-axis (right if positive, left if negative), then y steps up or down (up if positive, down if negative). Where you stop is the point.
Examples
The coordinates of B are (4.5, 0): 4.5 steps right and no steps up.
A cinema ticket "Row F, Seat 12" works like coordinates: two pieces of information pin down one seat.
To plot (−3, 2), follow the coordinates in order: 3 steps left, then 2 steps up.
Swapping the coordinates of (2, 5) gives (5, 2), which is a different point.
Do not confuse: The order matters: (2, 5) and (5, 2) are different points unless the two numbers are equal.
Used in
plotting points, maps and GPS, the distance and midpoint formulas
OpenStax: College Algebra 2e, 2.1 Key Terms (x-coordinate, y-coordinate)
coprime (numbers)
koh-PRYMEClass 10
Two whole numbers are coprime when the only factor they share is 1.
Precisely: Integers a and b are coprime (relatively prime) when HCF(a, b) = 1.
How it works
Compare their prime factorisations: if no prime appears in both, they are coprime. Neither number has to be prime itself. Proofs that √2 is irrational start by writing it as p/q with p and q coprime, a fraction in lowest terms.
Examples
8 and 15 are coprime: 8 = 2³ and 15 = 3 × 5 share no prime.
35 and 64 are coprime, although neither is a prime number.
A fraction is in its lowest terms when its top and bottom are coprime, like 3/8.
Do not confuse: Coprime is about a pair of numbers, prime is about one number. 9 and 10 are coprime though neither is prime.
Used in
lowest terms of fractions, proofs of irrationality, number theory
A fact that follows straight away from a result that has just been proved.
Precisely: A statement whose proof follows immediately, in one or two steps, from a theorem already proved.
How it works
Once a theorem is proved, put a special case into it. "The angle at the centre is twice the angle at the circle", used for a semicircle (180° at the centre), gives at once the corollary "the angle in a semicircle is 90°".
Examples
A corollary of Theorem 9: the angle a diameter subtends at any point of the circle is 90°.
Theorem 9 has a second corollary: the angles that one arc subtends at points on the rest of the circle are all equal.
From "the angles of a triangle add up to 180°" comes the corollary that each angle of an equilateral triangle is 60°.
Do not confuse: A corollary needs almost no new proof and comes after a theorem; a lemma is a small result proved before a theorem, to help prove it.
cosecant, secant and cotangent (the reciprocal ratios)
koh-SEE-kunt, SEE-kunt, koh-TAN-juntClass 10
The three ratios you get by turning sine, cosine and tangent upside down.
Precisely: cosec A = 1/sin A = hypotenuse/opposite, sec A = 1/cos A = hypotenuse/adjacent, and cot A = 1/tan A = adjacent/opposite = cos A/sin A.
How it works
Pair each with its partner: cosec with sin, sec with cos, cot with tan. A memory aid: the "co" of cosecant goes with the sine that has no "co", and the plain secant goes with cosine.
Examples
sin 30° = 1/2, so its reciprocal ratio cosecant is cosec 30° = 2.
cos 60° = 1/2, so the secant is sec 60° = 2.
tan 60° = √3, so the cotangent is cot 60° = 1/√3.
Do not confuse: cosec pairs with sin, not with cos (and sec with cos). They are reciprocals, 1 over the ratio, not the ratio of the complementary angle.
Used in
trigonometric identities (1 + tan² A = sec² A, cot² A + 1 = cosec² A), simplifying expressions
OpenStax: Algebra and Trigonometry 2e, 7.2 Right Triangle Trigonometry (reciprocal functions)
cosine (cos)
KOH-syneClass 10
For an angle in a right triangle, the adjacent side divided by the hypotenuse.
Precisely: cos A = (side adjacent to A) ÷ hypotenuse, for an acute angle A of a right triangle; cos A = sin (90° − A).
How it works
It falls from 1 at 0° to 0 at 90°. It is the sine of the complementary angle, which is where the name comes from: Aryabhata called it kotijya, Edmund Gunter named it cosinus, and Jonas Moore first wrote "cos" in 1674.
Examples
cos 60° = 1/2, the same as sin 30°, because 30° and 60° add to 90° (the cosine is the sine of the complementary angle).
If tan A = 4/3, the hypotenuse is 5 and the cosine is cos A = 3/5.
A 10 m ladder at 60° to the ground stands 10 × cos 60° = 5 m from the wall, using the cosine.
Do not confuse: cos A uses the adjacent side, sin A the opposite: swapping them is the most common slip. sec A = 1/cos A.
Used in
heights and distances, the cosine rule (Class 11), work done by a force in physics
The number that, multiplied by itself three times, gives the number you started with: ³√27 = 3.
Precisely: The real number b with b³ = a, written ³√a; every real number, negative or positive, has exactly one real cube root.
How it works
For perfect cubes, prime-factorise and take a third of each power, or use the last-digit trick (each last digit of a cube comes from one last digit of the root). Negative numbers have negative cube roots.
Examples
³√27 = 3 is a cube root, because 3 × 3 × 3 = 27.
The cube root of −8 is −2, since (−2)³ = −8.
³√1,75,616 = 56: the cube root ends in 6, and 5³ ≤ 175 < 6³.
Do not confuse: Unlike square roots, cube roots of negative numbers are real. And ³√8 is 2, not 8 ÷ 3.
Used in
volumes (the edge of a cube from its volume), solving x³ = a, surds
A polynomial whose highest power of x is 3, like x³ − 4x or 2x³ − 5x² − 14x + 8.
Precisely: A polynomial of degree 3, of the form ax³ + bx² + cx + d with a ≠ 0.
How it works
Its graph is an S-shaped curve that crosses the x-axis once, twice or three times, so it has 1, 2 or 3 zeroes (at most 3). For zeroes α, β, γ of ax³ + bx² + cx + d: α + β + γ = −b/a, αβ + βγ + γα = c/a, and αβγ = −d/a.
Examples
x³ − 4x is a cubic polynomial with zeroes −2, 0 and 2.
The cubic polynomial 2x³ − 5x² − 14x + 8 has zeroes 4, −2 and 1/2; their sum is 5/2 = −(−5)/2.
x³ is a cubic polynomial with only one zero, 0.
Do not confuse: Cubic means degree 3, not three terms: x³ is cubic with one term, while x² + x + 1 has three terms but is quadratic.
Used in
volumes (a cube of side x has volume x³), graphs, zeroes and coefficients
A box shape with six rectangular faces, like a brick or a matchbox.
Precisely: A solid bounded by six rectangles, with length l, breadth b and height h; volume lbh, total surface area 2(lb + bh + hl), diagonal √(l² + b² + h²). A cube is a cuboid with l = b = h.
How it works
Opposite faces are equal pairs, which is why the surface area has three products, each doubled. The diagonal uses Pythagoras twice.
Examples
A cuboid 3 cm by 4 cm by 12 cm has volume 144 cm³, surface area 192 cm² and a diagonal of 13 cm.
A room is a cuboid: its four walls have area 2h(l + b).
Shanta's shed is a cuboid with half a cylinder on top.
Do not confuse: A cuboid has rectangular faces; a cube has square faces (all edges equal). Every cube is a cuboid, but not every cuboid is a cube.
Used in
rooms, tanks and boxes, volume and surface area, packing problems
A running total of the frequencies, adding each class to all the classes before it.
Precisely: The cumulative frequency of a class is the sum of the frequencies of that class and all the classes before it (the "less than" type); the last one equals n.
How it works
Keep adding as you go down the table. It shows how many values lie below each upper limit, which finds the median class (where it first passes n/2) and draws the cumulative frequency curve (ogive).
A four-sided shape whose four corners all lie on one circle.
Precisely: A quadrilateral whose vertices are concyclic; its opposite angles add up to 180°, and any quadrilateral whose opposite angles add up to 180° is cyclic.
How it works
Each of two opposite angles is half the angle that its arc makes at the centre, and the two arcs together go once round the centre, 360°. Half of 360° is 180°, so the two opposite angles add up to 180°.
Examples
In a cyclic quadrilateral ABCD with ∠A = 75°, the opposite angle ∠C is 180° − 75° = 105°.
In the cyclic quadrilateral PQRS, ∠P = (2x + 10)° and ∠R = (3x − 20)° add up to 180°, so 5x − 10 = 180, x = 38, ∠P = 86° and ∠R = 94°.
A cyclic quadrilateral with sides 5, 5, 12 and 12 is a kite made of two 5-12-13 right triangles, so its area is 2 × ½ × 5 × 12 = 60 square units.
Every rectangle is a cyclic quadrilateral; a parallelogram that is not a rectangle is not one.
Do not confuse: The test is opposite angles adding up to 180°, not opposite angles being equal. A parallelogram with angles 70° and 110° has equal opposite angles, but 70° + 70° is not 180°, so it is not cyclic.
Used in
angle problems in circles, geometry proofs, Brahmagupta's area formula (for cyclic quadrilaterals)
A solid with two equal circles at the ends joined by a curved side, like a tin or a pipe.
Precisely: A right circular cylinder of radius r and height h; curved surface area 2πrh, total surface area 2πr(r + h), volume πr²h.
How it works
Cut the curved side straight down and unroll it: it becomes a rectangle whose length is the circle's circumference, 2πr, and whose width is h. The volume is the base circle's area times the height.
Examples
A cylinder of radius 7 cm and height 10 cm holds 1540 cm³, taking π ≈ 22/7.
The label on a tin is the curved surface of a cylinder: 2πrh.
A juice glass is a cylinder, but a raised hemisphere at its bottom makes it hold less than it seems.
Do not confuse: The curved surface area 2πrh leaves out the two circles; the total adds 2πr². A hollow pipe has an inner and an outer radius.
Used in
tins, pipes, tanks and wells, volume and surface area, combined solids