Two linear equations in the same two unknowns, solved together to find values that fit both.
Precisely: Two equations a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0; a solution is a pair (x, y) that satisfies both, the point where their lines meet.
How it works
Draw both lines and read where they cross, or solve by substitution or elimination. The ratios tell you in advance: a₁/a₂ ≠ b₁/b₂ gives one solution; a₁/a₂ = b₁/b₂ = c₁/c₂ gives infinitely many; a₁/a₂ = b₁/b₂ ≠ c₁/c₂ gives none.
Examples
Akhila's rides: the pair of linear equations x − 2y = 0 and 3x + 4y = 20 gives x = 4 rides and y = 2 games.
The pair of linear equations x + 3y = 6 and 2x − 3y = 12 has the solution x = 6, y = 0.
Aftab's ages: the pair of linear equations s − 7t + 42 = 0 and s − 3t = 6 gives s = 42 and t = 12.
Do not confuse: A pair of equations can have one solution, none, or infinitely many; it is not always exactly one.
Used in
word problems with two unknowns, mixtures, ages, costs and speeds, systems of equations with matrices (Class 12)
OpenStax: Elementary Algebra 2e, 5.1 Solve Systems of Equations by Graphing
parabola
puh-RAB-uh-luhClass 10
The U-shaped curve you get when you draw the graph of a quadratic polynomial.
Precisely: The graph of y = ax² + bx + c with a ≠ 0: it opens upwards when a > 0 and downwards when a < 0, and is symmetric about a vertical line.
How it works
The places where it meets the x-axis are the zeroes of the quadratic. It can cut the x-axis twice (two zeroes), touch it once (two equal zeroes), or miss it (no real zeroes), matching a discriminant above, equal to, or below 0.
Examples
The graph of y = x² − 3x − 4 is a parabola that opens upwards and cuts the x-axis at −1 and 4.
A parabola y = −x² + 4 opens downwards, because a = −1 is negative.
A ball thrown across a field follows a path close to a parabola.
The parabola y = x² + 1 never meets the x-axis, so x² + 1 has no real zeroes.
Do not confuse: Not every U shape is a parabola: a chain hanging between two posts makes a different curve (a catenary). A parabola comes from a squared term.
Used in
graphs of quadratics, projectile motion in physics, dish antennas and torch reflectors, conic sections (Class 11)
OpenStax: College Algebra 2e, 4.2 Key Terms (parallel lines)
Pascal's triangle
pas-KAHLZ TRY-ang-gulClass 11
A triangle of numbers with 1 at the top and down each side, where every other number is the sum of the two just above it. Row n holds the coefficients of (a + b)ⁿ.
Precisely: The triangular array whose n-th row (counting from row 0) is ⁿC₀, ⁿC₁, …, ⁿCₙ; each inside entry is the sum of the two above it, by Pascal's rule ⁿCᵣ₋₁ + ⁿCᵣ = ⁿ⁺¹Cᵣ.
How it works
Start with 1, then 1 1. Each new row begins and ends with 1, and each number inside is the two above it added: 1 2 1, then 1 3 3 1, then 1 4 6 4 1.
Precisely: A ratio or fraction written with denominator 100, shown with the sign %: p% = p/100.
How it works
To find p% of an amount, multiply by p and divide by 100. To turn a fraction into a percentage, multiply it by 100. Percentages make different totals easy to compare.
Examples
18% of ₹250 is 250 × 18 ÷ 100 = ₹45, a percentage of an amount.
3/8 as a percentage is 3/8 × 100 = 37.5%.
A price rising from ₹80 to ₹100 has gone up by 20/80 × 100 = 25%, a percentage increase.
Do not confuse: A percentage increase of 25% followed by a decrease of 25% does not bring you back: 80 → 100 → 75.
Used in
marks and results, discounts, profit and loss, interest, probability as a percentage
A number or expression that is something times itself three times: 27 = 3 × 3 × 3, and p³ + 6p²q + 12pq² + 8q³ = (p + 2q)³.
Precisely: A number that is the cube of an integer, or a polynomial that is the cube of another polynomial, such as a³ + 3a²b + 3ab² + b³ = (a + b)³ and a³ − 3a²b + 3ab² − b³ = (a − b)³.
How it works
Picture a cube of edge a + b cut into pieces: two cubes (a³ and b³) and six cuboids (three of volume a²b and three of volume ab²). So check that the first and last terms are cubes (a³ and b³) and the middle two are 3a²b and 3ab². For (a − b)³ the signs take turns: +, −, +, −.
Examples
p³ + 6p²q + 12pq² + 8q³ is a perfect cube: p³ + 3(p)²(2q) + 3(p)(2q)² + (2q)³ = (p + 2q)³, so a cube with this volume has side p + 2q.
1331 is a perfect cube: 11³ = (10 + 1)³ = 1000 + 300 + 30 + 1.
x³ + 3x² + 3x + 1 is a perfect cube, (x + 1)³, but x³ + 1 is not: its two middle terms are missing.
Do not confuse: (a + b)³ is not a³ + b³: the six cuboids, 3a²b + 3ab², are missing. A perfect square of a binomial has three terms; a perfect cube of a binomial has four.
Used in
finding the side of a cube from its volume, factorisation, quick cubes of numbers, the binomial theorem (Class 11)
OpenStax: Elementary Algebra 2e, 7.4 Factor Special Products
perimeter
puh-RIM-i-terFoundation
The total length all the way round the border of a shape.
Precisely: The length of the closed boundary of a plane figure; for a polygon, the sum of its side lengths.
How it works
Imagine a tiny insect walking once round the border without turning back: the distance it walks is the perimeter. Add up the sides of a polygon; for a circle the perimeter has its own name, the circumference.
Examples
A square of side a has perimeter 4a, and a rectangle of sides a and b has perimeter 2(a + b).
An equilateral triangle of side 5 cm has perimeter 15 cm.
Doubling the side of a square doubles its perimeter: the ratio of perimeter to side stays 4 : 1 for every square.
A semicircle of radius 7 cm, with its diameter, has perimeter πr + 2r = 22 + 14 = 36 cm (taking π ≈ 22/7).
Do not confuse: Perimeter is a length (cm, m); area is the space inside (cm², m²). Two rectangles can have the same perimeter and different areas: 10 by 10 and 18 by 2 both have perimeter 40.
Used in
fencing and borders, running tracks, Heron's formula (the semi-perimeter)
OpenStax: Prealgebra 2e, 9.4 Use Properties of Rectangles, Triangles, and Trapezoids (perimeter)
permutation
pur-myoo-TAY-shunClass 11
An arrangement of things in a definite order. ABC and BAC are different permutations of the same three letters.
Precisely: An ordered arrangement of r objects chosen from n distinct objects; their number is ⁿPᵣ = n! ÷ (n − r)!.
How it works
Fill the places one at a time: n choices for the first place, n − 1 for the next, and so on. Multiplying the choices counts the permutations.
Examples
The letters A, B and C have 6 permutations: ABC, ACB, BAC, BCA, CAB, CBA.
Choosing a captain and a vice-captain from 11 players is a permutation: 11 × 10 = 110 ways.
A 4-digit PIN with no repeated digit is one of ¹⁰P₄ = 5040 permutations.
The finishing order of a race is a permutation of the runners.
Do not confuse: In a permutation order matters; in a combination it does not. A team is a combination; a team with a captain named is partly a permutation.
Used in
arrangements in a row, ranks and posts, codes, probability
The line that cuts a line segment into two equal halves and crosses it at a right angle.
Precisely: The line through the midpoint of a segment AB that is perpendicular to AB; it is exactly the set of points equidistant from A and B.
How it works
Fold the paper so that A lands on B: the crease is the perpendicular bisector. Any point P on it has PA = PB, and any point with PA = PB lies on it. That is why the perpendicular bisectors of the sides of a triangle meet at the centre of the circle through its corners.
Examples
The perpendicular bisector of a chord always passes through the centre of the circle.
Every point on the perpendicular bisector of AB is as far from A as it is from B.
The perpendicular bisectors of AB and AC meet at one point O, the circumcentre of triangle ABC.
For three points on one line, the perpendicular bisectors of AB and BC are parallel, so no circle passes through all three.
Do not confuse: A median of a triangle also goes through the midpoint of a side, but it joins it to the opposite corner and is usually not perpendicular to the side.
Used in
finding the centre of a circle, the circumcircle of a triangle, constructions, coordinate geometry
The shortest distance from a point to a line: measured along the path that meets the line at a right angle.
Precisely: The length of the perpendicular segment from a point to a line; for the point (x₁, y₁) and the line Ax + By + C = 0 it is |Ax₁ + By₁ + C| ÷ √(A² + B²).
How it works
Every other path from the point to the line is the slanting side of a right triangle whose other side is the perpendicular, and the slanting side of a right triangle is always the longest. So the perpendicular is the shortest way.
Examples
The perpendicular distance of (3, −5) from 3x − 4y − 26 = 0 is 3/5.
The height of a triangle is the perpendicular distance from a corner to the opposite side.
A line touches a circle when its perpendicular distance from the centre equals the radius.
The width of a strip between two parallel lines is the perpendicular distance between them.
Do not confuse: The distance to a line is the perpendicular distance, not the distance to any point on it you happen to pick.
Used in
heights of triangles, tangents to circles, the gap between parallel lines
Two lines that cross at a right angle. When neither is vertical, their slopes multiply to −1.
Precisely: Two lines that meet at an angle of 90°; two non-vertical lines with slopes m₁ and m₂ are perpendicular exactly when m₁m₂ = −1, and a vertical line is perpendicular to every horizontal one.
How it works
Turning a line through a right angle turns its step of 1 across and m up into a step of m across and 1 down, so the new slope is −1/m. That is why the two slopes multiply to −1: each is the other turned upside down with its sign changed.
Examples
The lines y = 2x + 1 and y = −½x + 3 are perpendicular lines: 2 × (−½) = −1.
The x-axis and the y-axis are perpendicular lines.
The altitude of a triangle and the side it falls on are perpendicular lines.
Slopes 3 and ⅓ multiply to 1, not −1, so those lines are not perpendicular lines.
Do not confuse: Parallel lines have equal slopes and never meet; perpendicular lines have slopes multiplying to −1 and meet at a right angle.
Used in
altitudes and perpendicular bisectors, tangents and radii of circles, right angles in coordinate proofs
The number you get when you divide the distance round any circle by the distance across it; it is about 3.14.
Precisely: The constant ratio C/D of the circumference of a circle to its diameter, the same for every circle; π = 3.14159265... is irrational.
How it works
Every circle, big or small, has the same C/D ratio, so it gets one name, π. Then C = πd = 2πr and the area is πr². Its digits never end and never repeat, so every fraction or decimal for π, like 22/7 or 3.14, is only an approximation.
Examples
Wrap a thread round a cotton reel 20 times, measure it, and divide by 20 × the diameter: you get about 3.14, which is pi.
Āryabhaṭa (499 CE) gave 3.1416 for pi and called it āsanna, "approaching".
Zu Chongzhi's 355/113 = 3.1415929... was the closest fraction for pi in the world for over 800 years.
In 1706 William Jones chose the Greek letter π for pi, the first letter of perimetros, "perimeter".
Pi Day is 14 March (3.14), and Pi Approximation Day is 22 July (22/7).
Do not confuse: π is not 22/7: 22/7 = 3.142857... is a little bigger. Write π ≈ 22/7, never π = 22/7.
Used in
circumference and area of circles, arcs and sectors, trigonometry and radians (Class 11), almost every branch of science
What a digit is worth because of where it stands in a number.
Precisely: The value of a digit determined by its position: in the decimal system each place is worth 10 times the place to its right (ones, tens, hundreds, …; tenths, hundredths, … after the point).
How it works
The same digit means different amounts in different places: the 7 in 70 is seventy, in 700 seven hundred. In Indian grouping, after the thousands come lakhs and crores: 1,00,000 is one lakh, the same as 100,000 (a hundred thousand).
Examples
In 4,75,382 the place value of 7 is 70,000 (seven ten-thousands).
In 3.14 the place value of 4 is 4 hundredths, 0.04.
Adding in columns works because each column holds one place value: ones under ones, tens under tens.
Do not confuse: Place value depends on the position (70,000); face value is just the digit itself (7).
Used in
reading and writing numbers, column addition and subtraction, decimals, rounding
OpenStax: Prealgebra 2e, 1.1 Introduction to Whole Numbers (place value)
point of contact
point uv KON-taktClass 10
The one point where a tangent touches a circle.
Precisely: The common point of a circle and a tangent to it; the radius to the point of contact is perpendicular to the tangent.
How it works
Join the centre to the point of contact: that radius is the shortest distance from the centre to the tangent, which is why it meets the tangent at a right angle.
Examples
Where a wheel touches the road is the point of contact.
The radius OT to the point of contact T makes a right angle with the tangent PT.
The two tangents from P touch the circle at two points of contact, Q and R, with PQ = PR.
Do not confuse: A tangent has one point of contact; a secant has two points of intersection, not points of contact.
Used in
tangents and circles, rolling wheels, constructions
The whole group you want to know about, such as every student in a school.
Precisely: The complete set of individuals or items about which a statistical question is asked; a sample is a part of it.
How it works
Asking everyone is usually too slow or costly, so you ask a sample and use its results to estimate for the whole population. The estimate is better when the sample is bigger and more like the population.
Examples
The population is all 1,500 students of the school; one class of 50 is a sample of it.
40% of a sample chose mango, so for a population of 1,500 students you would buy about 600 mangoes.
In a sample of 40 students, 9 chose the Sports Club, so in a population of 800 about 9/40 × 800 = 180 would.
Do not confuse: In statistics a population need not be people: all the sweets in a bag, or all the tyres a company makes, can be a population.
Used in
surveys and polls, estimating from samples, statistics (Class 10 to 12)
OpenStax: Introductory Statistics 2e, 1.1 Definitions of Statistics, Probability, and Key Terms
power (exponent)
POW-erFoundation
A small raised number that says how many times to multiply a number by itself: 2⁵ = 2 × 2 × 2 × 2 × 2.
Precisely: In aⁿ, a is the base and n the exponent (index, power); for a whole number n it is a multiplied by itself n times, with a⁰ = 1 and a⁻ⁿ = 1/aⁿ (a ≠ 0).
How it works
The laws follow from counting the factors: aᵐ × aⁿ = aᵐ⁺ⁿ, aᵐ ÷ aⁿ = aᵐ⁻ⁿ, (aᵐ)ⁿ = aᵐⁿ. Powers of 10 give place values and standard form.
Examples
In 2⁵ = 32, 2 is the base and 5 the power (exponent).
Multiplying adds the powers: 2³ × 2⁴ = 2⁷ = 128.
A negative power means a reciprocal: 2⁻³ = 1/2³ = 1/8.
The speed of light is about 3 × 10⁸ m/s, a power of 10 in standard form.
Do not confuse: 2⁵ is 32, not 2 × 5 = 10. And a⁰ = 1, not 0.
Used in
squares and cubes, standard form in science, prime factorisation, growth and GPs
OpenStax: Prealgebra 2e, 10.2 Use Multiplication Properties of Exponents
prime factorisation
pryme fak-ter-eye-ZAY-shunFoundation
Writing a number as a product of prime numbers only, like 60 = 2 × 2 × 3 × 5.
Precisely: The expression of a natural number greater than 1 as a product of primes, usually written with powers in increasing order of the primes: 32760 = 2³ × 3² × 5 × 7 × 13.
How it works
Divide by the smallest prime that goes in, write it down, and repeat with what is left (or draw a factor tree) until only primes remain. Then collect equal primes into powers. By the Fundamental Theorem of Arithmetic the result is the same whichever way you start.
Examples
The prime factorisation of 140 is 2² × 5 × 7.
The prime factorisation of 3825 is 3² × 5² × 17.
From the prime factorisation 96 = 2⁵ × 3 and 404 = 2² × 101, the HCF is 2² = 4.
4ⁿ = 2²ⁿ never ends in 0: its prime factorisation has no 5, and a number ending in 0 must have 2 and 5.
Do not confuse: A factorisation like 60 = 6 × 10 is not a prime factorisation: 6 and 10 are not prime.
Used in
HCF and LCM, simplifying fractions and roots, deciding if a decimal terminates, proofs about divisibility
OpenStax: Prealgebra 2e, 2.5 Prime Factorization and the Least Common Multiple
prime number
pryme NUM-berFoundation
A whole number greater than 1 whose only factors are 1 and itself, like 2, 3, 5, 7 and 11.
Precisely: A natural number p > 1 that has exactly two factors, 1 and p.
How it works
To test a number, try dividing it by the primes 2, 3, 5, 7, … up to its square root; if none divides it exactly, it is prime. Every whole number greater than 1 is either a prime or a product of primes, so primes are the building blocks of all whole numbers.
Examples
13 is a prime number: its only factors are 1 and 13.
2 is the only even prime number; every other even number has 2 as a factor.
3803 and 3607 are prime numbers, and 12,34,56,789 = 3² × 3803 × 3607.
There are infinitely many prime numbers: the list never ends.
Do not confuse: 1 is not a prime number: it has only one factor. A number like 91 = 7 × 13 looks prime but is composite.
Used in
prime factorisation, HCF and LCM, proving numbers irrational, codes and computer security
OpenStax: Prealgebra 2e, 2.5 Prime Factorization and the Least Common Multiple (prime number)
probability
prob-uh-BIL-i-teeClass 9
A number from 0 to 1 that measures how likely something is to happen: 0 means it cannot happen, 1 means it is sure to happen.
Precisely: A measure P(E) of the likelihood of an event E, with 0 ≤ P(E) ≤ 1; with equally likely outcomes, P(E) = number of favourable outcomes ÷ number of possible outcomes.
How it works
There are two objective ways to find it. Count: if all outcomes are equally likely, divide the favourable ones by all of them (theoretical probability). Or experiment: repeat the action many times and divide the number of times the event happened by the number of trials (experimental probability).
Examples
The probability of heads when a fair coin is tossed is 1/2.
The probability of rolling a 4 on a fair die is 1/6 ≈ 0.167, or about 16.7%.
A letter picked at random from PROBABILITY is a B with probability 2/11 ≈ 0.182: 2 Bs among 11 letters.
Two coins are tossed: the probability of at least one head is 3/4, since 3 of HH, HT, TH, TT have a head.
Do not confuse: A probability says what happens in the long run, not what will happen next: a 1/6 chance does not mean a 4 will turn up exactly once in every 6 rolls.
Used in
weather forecasts, games and sports, insurance and business, science and research, statistics (Class 10 to 12)
A line from 0 to 1 on which you place events by how likely they are: impossible, less likely, even chance, more likely, certain.
Precisely: The interval from 0 (impossible) to 1 (certain) on which every probability lies, with 0.5 marking an even chance.
How it works
Put the event where its probability is, like a point on a number line. Below 0.5 it is less likely than not; above 0.5 it is more likely than not; exactly 0.5 is an even chance. A probability can also be written as a percentage: 0.75 is 75%.
Examples
Rolling a number greater than 6 on a die sits at 0 on the probability scale: impossible.
Rolling a 3 is at 1/6 on the probability scale: less likely, but not impossible.
Heads on a fair coin is at 0.5 on the probability scale: an even chance.
Drawing a card numbered 2 to 10 from a pack of 52 is at 36/52 ≈ 0.69 on the probability scale: more likely.
Do not confuse: No probability is below 0 or above 1. A "150% chance" or a probability of −0.2 is always a mistake.
Used in
comparing chances, describing risk in words and numbers
A way to prove something by first pretending it is false, then showing that this leads to something impossible.
Precisely: A proof that assumes the negation of the statement, derives by valid steps a contradiction (a statement and its opposite both true), and so concludes the statement is true.
How it works
Assume the opposite. Reason carefully, step by step. When you reach a clash with something known or already assumed, the steps were fine, so the assumption must be wrong.
Examples
Hippasus used proof by contradiction to show √2 is irrational: assuming √2 = p/q in lowest terms leads to p and q both being even.
By proof by contradiction: if there were a largest natural number N, then N + 1 would be larger, which is impossible.
The same proof by contradiction shows √3 is irrational, using "divisible by 3" in place of "even".
Do not confuse: It is not a proof by example: showing a claim fails for one case disproves it, but showing it holds for many cases proves nothing.
Used in
irrational numbers, number theory, geometry proofs (Class 10 onward)
When two ratios are equal (2 : 3 = 4 : 6); and when two quantities change together in a fixed way.
Precisely: Four numbers are in proportion, a : b :: c : d, when a/b = c/d, so that ad = bc. Two quantities are in direct proportion when their ratio stays fixed (y = kx), and in inverse proportion when their product stays fixed (xy = k).
How it works
In direct proportion, doubling one doubles the other, so use the unitary method: find the value for 1, then multiply. In inverse proportion, doubling one halves the other: more workers, fewer days.
Examples
2 : 3 :: 4 : 6 is a proportion, because 2 × 6 = 3 × 4.
If 3 pens cost ₹45, then 7 pens cost ₹45 ÷ 3 × 7 = ₹105: cost is in direct proportion to the number of pens.
If 6 workers take 10 days, 12 workers take 5 days: days are in inverse proportion to workers.
Do not confuse: Not every pair that grows together is proportional: a taxi fare of ₹200 + ₹40 per km grows with distance, but 10 km does not cost twice 5 km.
Used in
unitary method and shopping, speed, time and work, similar triangles, science laws
OpenStax: Prealgebra 2e, 8.6 Solve Proportion and Similar Figure Applications
Pythagoras theorem
py-THAG-or-us THEER-umClass 9
In a right triangle, the square on the longest side equals the two other squares added together.
Precisely: In a right-angled triangle with legs a and b and hypotenuse c, a² + b² = c².
How it works
Square the two shorter sides and add them; the total is the square of the hypotenuse. It only works when the angle between the two shorter sides is exactly 90°.
Examples
By Pythagoras theorem, legs of 6 cm and 8 cm give a hypotenuse of 10 cm.
Carpenters use Pythagoras theorem with a 3-4-5 triangle to check that a corner is square.
The distance formula is Pythagoras theorem written with coordinates.
Do not confuse: It does not work for triangles without a right angle; there the cosine rule is needed.
Used in
the distance formula, trigonometry, heights and distances