A number or expression that divides another exactly: 3 is a factor of 12, and x + 2 is a factor of x² + 4x + 4.
Precisely: A is a factor of B when B = A × C for some C of the same kind (a whole number for numbers, a polynomial for polynomials).
How it works
For numbers, check that the division leaves no remainder. For expressions, write the expression as a product: x² + 7x + 12 = (x + 3)(x + 4), so x + 3 and x + 4 are its factors.
Examples
The factors of 12 are 1, 2, 3, 4, 6 and 12.
2 is a common factor of 50p², 60pq and 18q².
x + 2 is a factor of x² + 4x + 4, because x² + 4x + 4 = (x + 2)(x + 2).
Do not confuse: A factor divides the number (3 divides 12); a multiple is divided by it (12 is a multiple of 3).
Used in
HCF and LCM, factorisation, simplifying fractions and rational expressions
OpenStax: Prealgebra 2e, 2.4 Find Multiples and Factors
factor tree
FAK-ter treeFoundation
A branching drawing that splits a number into factors, and those into more factors, until only primes are left at the ends.
Precisely: A tree diagram whose root is a number and whose branches split each number into a pair of factors, ending in leaves that are all prime; the product of the leaves is the prime factorisation.
How it works
Write the number at the top. Split it into any two factors, draw a branch to each, and keep splitting any factor that is not prime. Circle the primes. Different trees for the same number always end with the same primes.
Examples
A factor tree for 60 can start 6 × 10, then 2 × 3 and 2 × 5: the primes are 2, 2, 3, 5.
Another factor tree for 60 starts 4 × 15, yet it ends with the same primes 2, 2, 3 and 5.
A factor tree for 32760 ends in 2, 2, 2, 3, 3, 5, 7 and 13.
Do not confuse: The factor tree is a way of working; the prime factorisation is the answer it gives.
Used in
prime factorisation, HCF and LCM, simplifying square roots
OpenStax: Prealgebra 2e, 2.5 Prime Factorization and the Least Common Multiple (factor tree)
factorial
fak-TOR-ee-ulClass 11
n factorial, written n!, is the product of all the whole numbers from 1 up to n. So 4! = 4 × 3 × 2 × 1 = 24.
Precisely: For a positive whole number n, n! = 1 × 2 × 3 × … × n; by agreement 0! = 1, so that the counting formulas also work when nothing is left over.
How it works
n! counts the ways to put n different things in a row: n choices for the first place, n − 1 for the second, and so on down to 1 for the last. Each factorial is the one before times the new number: 5! = 5 × 4!.
Examples
5 factorial is 5! = 120.
Six people can stand in a queue in 6! = 720 ways, a factorial.
A factorial grows fast: 10! is 3628800.
By agreement the factorial of 0 is 1.
Dividing one factorial by another cancels a lot: 8! ÷ 6! = 8 × 7 = 56.
Do not confuse: n! is not n × 2 or n²: 4! is 24, not 8 or 16. And (2n)! is not 2 × n!.
Used in
permutations and combinations, the binomial theorem, probability
Writing an expression as a product of simpler ones: x² + 7x + 12 = (x + 3)(x + 4).
Precisely: Expressing a number or a polynomial as a product of factors; the reverse of expansion.
How it works
First take out any common factor. Then look for an identity (a perfect square, a difference of two squares, a cube) or split the middle term. Check by expanding back.
Examples
Factorisation of x² + 4x + 4 by the identity (a + b)²: it is (x + 2)².
Factorisation of 50p² + 60pq + 18q²: take out 2, then 2(25p² + 30pq + 9q²) = 2(5p + 3q)².
Factorisation of x² − 5x + 6: find two numbers with sum −5 and product 6, so (x − 2)(x − 3).
A pool with x² − 4x − 96 = 0 gives, by factorisation, (x − 12)(x + 8) = 0, so the length is 12 m.
Do not confuse: Factorisation changes a sum into a product; expansion changes a product into a sum. The value stays the same both ways.
Used in
solving quadratic equations, simplifying rational expressions, finding lengths from areas
OpenStax: Elementary Algebra 2e, 7.1 Greatest Common Factor and Factor by Grouping
fair (unbiased)
fair; un-BY-ustClass 9
A coin, die or draw is fair when no outcome is favoured: each has the same chance.
Precisely: An experiment is fair (unbiased) when its elementary outcomes are equally likely, as for a symmetrical coin tossed freely or a well-shuffled pack.
How it works
Symmetry is the reason: a coin that is the same on both sides, with nothing to make it fall one way more often, gives heads and tails equal chances. A bent coin or a weighted die is biased.
Examples
A fair coin gives heads with probability exactly 1/2.
A fair die gives each number from 1 to 6 with probability 1/6.
A coin toss is a fair way to decide which cricket team bats first, because neither captain can predict the result.
Do not confuse: Fair does not mean the results come out even: a fair coin can give 7 heads in 10 tosses. It means each outcome has the same chance every time.
Used in
theoretical probability, games and sports, sampling without bias
An outcome that belongs to the event you are asking about.
Precisely: An outcome of a random experiment that is an element of the event E under study; with equally likely outcomes, P(E) = (number of favourable outcomes) ÷ (number of possible outcomes).
How it works
Read the event, go through the sample space, and mark each outcome that fits. "Favourable" means "counts for the event", not "good".
Examples
For "an even number" on a die, the favourable outcomes are 2, 4 and 6.
For "a B" from the letters of PROBABILITY, there are 2 favourable outcomes out of 11.
For "a multiple of 3" on a spinner numbered 1 to 8, the favourable outcomes are 3 and 6, so the probability is 2/8 = 1/4.
Do not confuse: A favourable outcome need not be a happy one: for the event "the team loses", losing is the favourable outcome.
A special fixed point of a parabola, ellipse or hyperbola. The curve is defined by distances from it; the plural is foci.
Precisely: A fixed point used to define a conic: a parabola is the set of points as far from the focus as from the directrix, and an ellipse or hyperbola has two foci whose distances from any point on it have a constant sum or difference.
How it works
A parabola sends every ray that comes in parallel to its axis to its focus, which is why a dish antenna's receiver sits there. An ellipse sends a ray from one focus to the other.
Examples
The focus of y² = 12x is (3, 0).
A car headlight's bulb sits at the focus of its parabola-shaped mirror.
The Sun is at one focus of the Earth's elliptical orbit.
The ellipse x²/25 + y²/9 = 1 has a focus at (4, 0) and another at (−4, 0).
Do not confuse: The focus is inside the curve, not on it; the vertex is the point on the curve nearest the focus.
Used in
dish antennas and headlights, orbits, the equations of conics
A shape whose small parts look like the whole shape, however far you zoom in.
Precisely: A figure with self-similarity at every scale, usually built by repeating one simple rule for ever.
How it works
Start with a simple shape and apply a rule (such as "cut out the middle part") to every piece, again and again. Counting pieces or measuring areas at each stage often gives a geometric progression.
Examples
The Sierpiński triangle is a fractal made by removing middle triangles again and again.
The Sierpiński carpet is a fractal: each stage keeps 8 of 9 smaller squares, so the pieces go 1, 8, 64, ... and the area 1, 8/9, 64/81, ...
A cauliflower or a head of broccoli is close to a fractal: each floret looks like a small copy of the whole head.
Snowflakes, coastlines and tree branches have fractal patterns.
Do not confuse: A repeating wallpaper pattern repeats side by side at one size; a fractal repeats inside itself at smaller and smaller sizes.
Used in
patterns in nature, computer graphics and films, geometric progressions
A part of a whole, written as one number over another, like 3/4 (three out of four equal parts).
Precisely: A number written p/q with q ≠ 0: the whole divided into q equal parts, of which p are taken; equivalently, p ÷ q.
How it works
Multiplying or dividing top and bottom by the same number gives an equivalent fraction (2/3 = 4/6). A fraction is in lowest terms when its top and bottom are coprime. A fraction of an amount is the amount ÷ q × p.
Examples
2/5 of 30 sweets is 30 ÷ 5 × 2 = 12 sweets: a fraction of an amount.
12/18 simplifies to the fraction 2/3 by dividing top and bottom by 6.
The fraction 3/4 equals 0.75 and 75%.
Do not confuse: In a fraction the parts must be equal: one slice of a cake cut into 4 unequal pieces is not 1/4.
Used in
sharing and measuring, rational numbers, ratio and probability
How many times a value (or a class of values) occurs in the data.
Precisely: The number of observations equal to a value, or falling in a class interval, written fᵢ; the frequencies add up to n, the total number of observations.
How it works
Tally each value as you go through the data, then count the tallies. The frequency table that results is the starting point for the mean, median, mode and every graph of the data.
Examples
If 7 students scored from 40 to 55, the frequency of that class is 7.
In 4, 8, 6, 5, 3, 8 the frequency of 8 is 2.
The frequencies of the marks table, 2, 3, 7, 6, 6, 6, add to 30 students.
Do not confuse: Frequency is a count (7); relative frequency is that count divided by the total (7/30).
Used in
frequency tables and bar graphs, grouped data, experimental probability (relative frequency)
OpenStax: Introductory Statistics 2e, 1.3 Frequency, Frequency Tables, and Levels of Measurement
fundamental principle of counting
fun-duh-MEN-tul PRIN-si-pul uv KOUN-tingClass 11
If one thing can be done in m ways and after it another thing in n ways, both can be done in m × n ways.
Precisely: If an event can occur in m different ways, following which another event can occur in n different ways, then the two events in succession can occur in m × n ways; the rule extends to any number of events.
How it works
List the choices as a grid: one row for each of the m first choices, and in each row the n second choices. The grid has m rows of n, so m × n boxes.
Examples
By the fundamental principle of counting, 3 shirts and 4 trousers make 12 outfits.
A code of 3 letters from 26 has 26 × 26 × 26 = 17576 choices by the fundamental principle of counting.
With 4 roads from A to B and 3 from B to C, the fundamental principle of counting gives 12 routes.
The fundamental principle of counting is why n things can be arranged in n! ways.
Do not confuse: It multiplies when both things are done. When only one or the other is done, add instead.
Used in
permutations, combinations, probability, codes and number plates
Every whole number greater than 1 can be written as a product of primes in exactly one way, apart from the order.
Precisely: Every composite number can be expressed as a product of primes, and this factorisation is unique apart from the order in which the prime factors occur.
How it works
It has two halves: every such number can be broken into primes (existence), and there is only one way to do it (uniqueness). The uniqueness is what makes it powerful: if 5 is not in the prime factorisation, no rearrangement can bring it in.
Examples
By the Fundamental Theorem of Arithmetic, 32760 = 2³ × 3² × 5 × 7 × 13 is the only way to write 32760 as a product of primes.
The Fundamental Theorem of Arithmetic shows that 4ⁿ never ends in 0: 4ⁿ = 2²ⁿ has no prime 5.
An early form of the Fundamental Theorem of Arithmetic is Proposition 14 of Book IX of Euclid's Elements; the first correct proof was by Carl Friedrich Gauss (1777 to 1855).
Do not confuse: It says the factorisation is unique, not that it is easy to find: splitting a very large number into primes can take computers years.
Used in
HCF and LCM, proofs that √2 and √3 are irrational, which fractions give terminating decimals