A number that says how stretched a conic is: e = c ÷ a. A circle has 0, an ellipse between 0 and 1, a parabola exactly 1, and a hyperbola more than 1.
Precisely: For an ellipse or hyperbola, the ratio e = c/a of the distance from the centre to a focus to the distance from the centre to a vertex; equivalently, for any conic, the constant ratio of a point's distance from the focus to its distance from the directrix.
How it works
Keep a fixed and slide the foci apart. With the foci together (c = 0) the ellipse is a circle, e = 0. As they move out towards the vertices, c grows towards a and e grows towards 1, so the ellipse gets flatter and flatter.
Examples
The Earth's orbit has eccentricity about 0.017, so it is almost a circle.
Halley's comet moves on an ellipse of eccentricity about 0.97, very long and thin.
The ellipse x²/25 + y²/9 = 1 has c = 4, so its eccentricity is 4/5.
Every hyperbola has an eccentricity greater than 1.
Do not confuse: Eccentricity is a ratio, with no unit, not a length; two ellipses of different sizes can have the same eccentricity.
Used in
describing orbits, telling the conics apart, finding foci from a and e
Precisely: An event containing exactly one outcome of an experiment; the probabilities of all the elementary events of an experiment add up to 1.
How it works
List the outcomes: each one on its own is an elementary event. Events like "an even number" group several outcomes and are not elementary. Adding the probabilities of all single outcomes covers everything that can happen, so it gives 1.
Examples
Tossing a coin, "heads" and "tails" are the two elementary events: P(H) + P(T) = 1/2 + 1/2 = 1.
Rolling a die, "getting a 3" is an elementary event, but "getting a number greater than 4" is not: it has two outcomes.
For a bag with yellow, red and blue balls drawn one at a time, P(Y) + P(R) + P(B) = 1 over its elementary events.
Do not confuse: Elementary events need not be equally likely: a bag of 3 red and 1 blue ball has P(red) = 3/4 and P(blue) = 1/4, still adding to 1.
Used in
theoretical probability, checking that probabilities add to 1
Solving a pair of equations by adding or subtracting them so that one unknown disappears.
Precisely: A method for a pair of linear equations: multiply the equations by suitable non-zero numbers so that one variable has equal (or opposite) coefficients, then subtract (or add) to eliminate it, solve, and substitute back.
How it works
Make the coefficients of one variable match, then take one equation from the other. If what is left is a false statement like 0 = 4 there is no solution; if it is always true like 0 = 0 there are infinitely many.
Examples
By the elimination method, 9x − 4y = 2000 times 3 and 7x − 3y = 2000 times 4 give 27x − 12y = 6000 and 28x − 12y = 8000; subtracting, x = 2000.
So the two incomes are ₹18,000 and ₹14,000, found by the elimination method.
Adding x + 3y = 6 and 2x − 3y = 12 eliminates y at once (the elimination method): 3x = 18, so x = 6.
Do not confuse: Multiply every term of an equation, the constant too: 3 × (9x − 4y = 2000) is 27x − 12y = 6000, not = 2000.
Used in
pairs of linear equations, word problems, solving with matrices (Class 12)
OpenStax: Elementary Algebra 2e, 5.3 Solve Systems of Equations by Elimination
ellipse
i-LIPSClass 11
An oval curve: all the points whose distances from two fixed points (the foci) add up to the same number.
Precisely: The set of all points in a plane the sum of whose distances from two fixed points (the foci) is a constant 2a; with its centre at the origin and foci on the x-axis its equation is x²/a² + y²/b² = 1, where b² = a² − c².
How it works
Pin the two ends of a loose string to a board, pull it tight with a pencil and move the pencil round: the string keeps the two distances adding to the same length, so the pencil draws an ellipse. Moving the pins apart makes it flatter; putting them together makes a circle.
Examples
The Earth moves round the Sun in an ellipse, with the Sun at one focus.
A circle seen at a slant looks like an ellipse.
The ellipse x²/25 + y²/9 = 1 is 10 wide and 6 tall.
A whisper at one focus of an ellipse-shaped room can be heard at the other.
Do not confuse: An ellipse is not any oval (an egg shape is not an ellipse); it has exactly two lines of symmetry and a fixed sum of distances to the foci.
Used in
orbits of planets and comets, whispering galleries, gears and machine parts
Outcomes that each have the same chance of happening.
Precisely: Outcomes of a random experiment with equal probabilities; with n equally likely outcomes, each has probability 1/n.
How it works
You assume it when nothing makes one outcome likelier than another: a fair coin, a fair die, a well-shuffled pack. Only then can you find a probability by counting favourable outcomes and dividing.
Examples
Heads and tails are equally likely with a fair coin.
The spinner stops at 1 to 8 with equally likely outcomes, so it points to a number greater than 2 with probability 6/8 = 3/4.
Red and blue are not equally likely from a bag of 3 red and 7 blue marbles: blue has probability 7/10.
A car starting or not starting are not equally likely outcomes: most cars start.
Do not confuse: Having two possible results does not make them equally likely: "the car starts or does not" is not a 50-50 chance.
A statement that two things are equal, with an equals sign, like 2x + 3 = 11.
Precisely: A statement of equality between two expressions; its solutions are the values of the variables that make it true.
How it works
Most equations are true only for some values: 2x + 3 = 11 only when x = 4. Solving means finding those values. An equation that is true for every value is called an identity.
Examples
The equation x² − 1 = 24 is true only for x = 5 and x = −5.
For a pool of area 96 m² whose breadth is 4 m less than its length x, the equation is x(x − 4) = 96.
(x + y)² = x² + 2xy + y² is an equation that holds for all x and y, so it is also an identity.
Do not confuse: An expression (2x + 3) has no equals sign and nothing to solve; an equation (2x + 3 = 11) does.
Used in
solving problems, linear equations, quadratic equations (Class 10)
Two triangles whose three angles match one for one.
Precisely: Triangles whose corresponding angles are equal; equiangular triangles are always similar (the AAA criterion), as Thales observed.
How it works
Since the angles of a triangle add up to 180°, two equal pairs force the third pair to be equal too. Then every pair of corresponding sides is in the same ratio.
Examples
A small triangle cut off by a line parallel to one side is one of two equiangular triangles with the whole triangle.
A pole and its shadow, and a tower and its shadow at the same moment, make equiangular triangles.
Any two equilateral triangles are equiangular triangles (all angles 60°), so they are similar.
Do not confuse: Equiangular triangles need not be the same size; equilateral means equal sides within one triangle, which is a different idea.
Used in
similar triangles, heights by shadows, trigonometric ratios
Working out a rough answer quickly, with easy nearby numbers, to know roughly what the exact answer should be.
Precisely: Finding an approximate value of a calculation by replacing the numbers with rounded ones, to check or to plan.
How it works
Round each number to something easy (a ten, a hundred, one significant figure), work it out, and compare with your exact answer: if they are far apart, look for a mistake.
Examples
48 × 21 is about 50 × 20 = 1000 by estimation; the exact answer, 1008, is close.
397 + 612 is about 400 + 600 = 1000 by estimation (exactly 1009).
By estimation, 22/7 × 7 × 7 is about 3 × 49 ≈ 150, so an answer of 1540 for a circle's area of radius 7 must be wrong.
Do not confuse: An estimate is a deliberate rough answer, not a guess: it is worked out from rounded numbers.
Used in
checking answers, shopping and budgets, measurements and science
One outcome, or a group of outcomes, that you are interested in.
Precisely: A subset E of the sample space S of a random experiment; its probability with equally likely outcomes is n(E)/n(S).
How it works
Describe the event in words, then pick out the outcomes from the sample space that fit it. Count them and divide by the number of all outcomes, if they are equally likely.
Examples
Rolling a die, the event "a number greater than 4" is E = {5, 6}, with probability 2/6 = 1/3.
Tossing two coins, the event "at least one head" is {HH, HT, TH}.
Picking a yellow fruit from {Apple, Banana, Orange} is the event {Banana}.
Two dice are rolled: the event "the sum is a prime greater than 5" has 8 of the 36 outcomes, probability 2/9.
Do not confuse: An outcome is one single result; an event can hold many outcomes, one, or none at all (an impossible event, probability 0).
Used in
probability questions, tree diagrams, probability in Class 10 to 12
A probability worked out from what actually happened when you tried something many times.
Precisely: Experimental probability of an event = number of times the event occurred ÷ total number of trials; it is the relative frequency of the event.
How it works
Do the experiment (or use past records), count how often the event happened, and divide by the number of trials. With few trials it can be far from the theoretical value; with many it settles close to it.
Examples
A die rolled 12 times shows a 3 three times: the experimental probability of a 3 is 3/12 = 0.25, though the theoretical value is 1/6 ≈ 0.167.
Tossing a paper cup 100 times gives the experimental probability of each landing: bottom, top or side.
If 150 of 1,000 tyres last more than 14,000 km, the experimental probability that a tyre does is 150/1000 = 0.15.
Do not confuse: Experimental probability comes from data and can change from one experiment to the next; theoretical probability comes from counting and stays fixed.
Used in
situations with no equally likely outcomes, insurance and quality control, science experiments
A rule that gives a term of a sequence straight from its position number, without needing the terms before it.
Precisely: A formula for tₙ as an expression in n alone, such as tₙ = 4n − 3.
How it works
Substitute the position: the 20th term of tₙ = 4n − 3 is 4 × 20 − 3 = 77, with no need to list the first 19. That makes far-off terms quick, and lets you solve tₙ = some number to find a position.
Examples
The explicit formula uₙ = 2n − 1 gives the odd numbers.
For the taxi fares 240, 280, 320, ... (₹200 plus ₹40 a km), the explicit formula is 200 + 40n, so 10 km costs ₹600.
The explicit formula tₙ = n(n + 1)/2 gives the 80th triangular number, 3240, at once.
Do not confuse: An explicit formula uses only n; a recursive formula uses earlier terms. Both can describe the same sequence.
Used in
APs and GPs, far-off terms, testing whether a number is a term