A number built from i, the square root of −1, which has no place on the real number line.
Precisely: A number of the form bi, where b is a real number and i² = −1; together with the real numbers these give the complex numbers.
How it works
No real number times itself is negative, so √−1 is not on the number line. Mathematicians made a new direction, off the line, for i. Class 9 only meets the idea; it is studied in Class 11.
Examples
i is an imaginary number with i × i = −1.
The square root of −9 is the imaginary number 3i, since (3i)² = 9 × (−1) = −9.
Engineers use imaginary numbers to describe alternating current in circuits.
Do not confuse: Imaginary does not mean useless or fake; it is only a name. They are just not real numbers.
Used in
complex numbers (Class 11), quadratic equations with no real roots, electrical engineering and physics
OpenStax: Intermediate Algebra 2e, 8.8 Use the Complex Number System
impossible event
im-POS-i-bul i-VENTClass 10
An event that can never happen; its probability is 0.
Precisely: An event containing no outcome of the experiment, so the number of favourable outcomes is 0 and P(E) = 0.
How it works
Check the sample space: if no outcome fits the description, the event is impossible. Every probability then lies between this 0 and the 1 of a sure event.
Examples
Getting an 8 on one throw of a die is an impossible event: P = 0.
Drawing a green ball from a bag of red and blue balls is an impossible event.
Two dice showing a total of 13 is an impossible event; the largest total is 12.
Do not confuse: Very unlikely is not impossible: winning a lottery has a tiny probability, but not 0.
Used in
the probability scale, checking answers (no probability is below 0)
The biggest circle that fits inside a triangle, touching all three sides.
Precisely: The unique circle inside a triangle that touches each of its three sides; its radius r gives the area of the triangle as r × s, where s is the semi-perimeter.
How it works
Join the centre of the incircle to the three corners: this cuts the triangle into three triangles, each with a side as its base and r as its height. Their areas add to ½ r (a + b + c) = r s.
Examples
The 3-4-5 triangle has area 6 and semi-perimeter 6, so its incircle has radius 6 ÷ 6 = 1.
Every triangle has exactly one incircle and exactly one circumcircle.
For a regular polygon, the area is ½ × perimeter × radius of its incircle; Archimedes used this to prove that a circle has area πr².
Do not confuse: The incircle is inside and touches the sides; the circumcircle is outside and passes through the corners. For the 3-4-5 triangle they have radii 1 and 2.5.
Used in
areas of triangles and regular polygons, constructions, the area of a circle
The angle a line makes with the positive x-axis, turning anticlockwise from the axis to the line. Its tangent is the line's slope.
Precisely: The angle θ, with 0° ≤ θ < 180°, measured anticlockwise from the positive direction of the x-axis to the line; for a non-vertical line, the slope is m = tan θ.
How it works
Stand where the line crosses the x-axis, face along the positive x-axis, and turn anticlockwise until you face along the line: the turn is the inclination. A line rising to the right has an inclination below 90°; one falling to the right has an inclination between 90° and 180°, and a negative slope.
Examples
A line with inclination 45° has slope tan 45° = 1.
The x-axis and every line parallel to it have inclination 0°.
A vertical line has inclination 90°, and so it has no slope.
A line falling to the right, with inclination 135°, has slope −1.
Do not confuse: The inclination is an angle; the slope is its tangent, a number. A line at 60° has slope √3, not 60.
Used in
the slope of a line, the angle between two lines, ramps, roads and roofs
A pair of equations with no solution at all: their lines are parallel and never meet.
Precisely: A pair of linear equations with no common solution, which happens when a₁/a₂ = b₁/b₂ ≠ c₁/c₂.
How it works
The two lines have the same slope but different intercepts, so they run side by side for ever. Solving by elimination ends in a false statement, like 0 = 4.
Examples
x + 2y − 4 = 0 and 2x + 4y − 12 = 0 are an inconsistent pair: 1/2 = 2/4 but −4/−12 is 1/3, not 1/2.
x − y = 8 and 3x − 3y = 16 are an inconsistent pair: the second would need x − y = 16/3.
Two rails of a track, drawn as lines, are an inconsistent pair: they never meet.
Do not confuse: Inconsistent means no solution, not a wrong working: the equations themselves contradict each other.
Used in
deciding when a problem has no answer, graphs of lines
OpenStax: Elementary Algebra 2e, 5.1 Solve Systems of Equations by Graphing (inconsistent system)
independent events
in-di-PEN-dunt i-VENTSClass 9
Events where one happening makes no difference to the chance of the other.
Precisely: Events A and B are independent when the occurrence of one does not change the probability of the other; then P(A and B) = P(A) × P(B).
How it works
Separate tosses of a coin, or rolls of a die, are independent, because the coin or die has no memory. Drawing balls without putting them back is not independent: the first draw changes what is left.
Examples
Two tosses of a coin are independent events, so P(HH) = 1/2 × 1/2 = 1/4.
Each roll in Snakes and Ladders is one of a string of independent events: a 6 is 1/6 every time.
Drawing a red ball and then, without putting it back, a blue ball are not independent events: the second draw is from 8 balls, not 9.
Do not confuse: Independent is not the same as "cannot happen together". Heads on the first toss and heads on the second are independent and can both happen.
Used in
tree diagrams, multi-step experiments, probability in Class 12
OpenStax: Introductory Statistics 2e, 3.2 Independent and Mutually Exclusive Events
infinite series
IN-fi-nit SEER-eezClass 9
A sum that never stops: you keep adding more and more terms, and the total gets closer and closer to one number.
Precisely: A sum of infinitely many terms a₁ + a₂ + a₃ + ...; its value, when it has one, is the number that the running totals approach.
How it works
Add the terms one at a time and watch the running total. For Mādhava's series 1 − 1/3 + 1/5 − 1/7 + ..., the totals swing above and below π/4 and close in on it, so 4 times the series is exactly π.
Examples
Mādhava's infinite series π/4 = 1 − 1/3 + 1/5 − 1/7 + ... was the first exact formula for π.
Four terms of that infinite series give 4 × (1 − 1/3 + 1/5 − 1/7) ≈ 2.895, still far from π: this series closes in slowly.
The infinite series 1/2 + 1/4 + 1/8 + ... adds up to exactly 1: each term fills half of the gap that is left.
The repeating decimal 0.333... is the infinite series 3/10 + 3/100 + 3/1000 + ..., whose value is 1/3.
Do not confuse: An infinite series is a sum; a sequence is just the list of terms. And not every infinite series has a value: 1 + 1 + 1 + ... grows without end.
Used in
exact formulas for π, repeating decimals, calculus (Class 11 and 12)
A whole number that can be positive, negative or zero: …, −3, −2, −1, 0, 1, 2, 3, …
Precisely: A member of the set ℤ = {…, −2, −1, 0, 1, 2, …}: the natural numbers, zero, and the negatives of the natural numbers.
How it works
Integers extend the counting numbers to the left of 1 on the number line: zero, then the negatives. Brahmagupta called positives fortunes and negatives debts, and gave the rules: the product of two debts is a fortune.
Examples
A debt of ₹5 can be written as the integer −5.
−3, 0 and 12 are all integers, but 2.5 is not.
The symbol ℤ for the integers comes from the German word Zahlen, meaning numbers.
Do not confuse: An integer has no fractional part: −4 is an integer, −4.5 is not. And "whole number" usually means 0, 1, 2, … only.
Used in
negative numbers and temperature, coordinates, algebra
A number that cannot be written as a fraction of two integers, like √2 or π. Its decimal goes on forever without repeating.
Precisely: A real number that is not rational: it cannot be written as p/q with integers p and q, q ≠ 0.
How it works
Its decimal never ends and never falls into a repeating block. That √2 is irrational was first proved, by contradiction, around 400 BCE; that π is, by Lambert in 1761.
Examples
√2 = 1.41421356… is an irrational number: no fraction equals it exactly.
π is an irrational number, so 22/7 and 3.14 are only close to it.
The diagonal of a square of side 1 has length √2, an irrational number.
Do not confuse: A long decimal is not automatically irrational: 0.142857142857… repeats, so it is rational (1/7).