Numbers

Lucas numbers

2, 1, 3, 4, 7, 11, 18, 29, 47, 123, ... similar to Fibonacci numbers, but sequence defined by 2, 1, ... rather than [0], 1, 1, ...

Parallel Postulate

Through a given point, only one line can be drawn parallel to another, given line

complex numbers

numbers of the form a+bi, where i is sqrt -1. comprise 'real' part (a) and 'imaginary' part (b), corresponding to location on 'complex number plane' set C 'C' from 'complex'

rational numbers

numbers that can be expressed as a fraction, with integer numerator, positive integer denominator set Q Q from 'quotient'

Perfect number

positive integer which is equal to its aliquot sum eg 6 = 1x6=2x3 = 1+2+3 also equal to half the sum of all its divisors (including itself) NB - all are Mersenne primes

natural numbers

positive integers (may exclude zero) whole numbers, counting numbers set N (set N(1) or N(0) to signify start value)

Mersenne prime

prime number that is one less than a power of 2 (prime power of 2 in practice) prime q of form q=((2^p) - 1), where p is prime 3, 7, 31, 127, ... NB - all are perfect numbers [Marin Mersenne, 1536]

Golden Ratio

ratio of 2 numbers a, b where a>b and a:b = a+b:a aka Golden Mean, Golden Section 1.6180327868852

integers

set Z - numbers with no fractional part includes 'counting numbers' (positive), zero and 'negative counting numbers' Z from German 'Zahl' - 'number'

Mandelbrot set

set of complex numbers c for which the function f(c)(z) = (z^2) + c does not diverge when iterated from z = 0. Images derived from sampling points on Complex Plane: - black if above function does not diverge - other colours according to how rapidly the function diverges

aliquot sum

sum of an integer's proper, positive divisors ie positive divisors excluding itself (but including 1)

Aleph(0)

the number of Natural Numbers - a transfinite number [aka Aleph null] Coined by Cantor

real numbers

all numbers, including non-rational numbers set R 'R' from 'real'

Cantor set

an uncountable subset of the (uncountable) set of Real numbers

Mersenne number

any integer that is one less than power of 2 integer of form q=((2^p)-1) Marin Mersenne, 1536 3, 7, 15, 31, 63, 127, ...

Gaussian integers

complex numbers, whose real and imaginary parts are both integers

Lucas sequences

constant-recursive integer sequences that satisfy the recurrence relation: x(n) = P.x(n-1) - Q.x(n-2), P & Q integers in the sequence Fibonacci & Lucas numbers are examples

Goldbach's conjecture

every even number is the sum of 2 primes not yet answered

Fundamental theorem of arithmetic

every integer > 1 is the product of primes in only 1 way (except for rearrangement of order) Proof in Euclid's Elements

largest known prime number

(2^74,207,281) − 1 also a Mersenne prime

Euclid-Euler theorem

"every even perfect number can be represented in the form: p = 2^(n-1) * ((2^n)-1) where n is prime, and ((2^n)-1) is Mersenne prime" [Euclid proved that numbers of this form are always perfect numbers Euler proved that this form accounts for all perfect numbers]

Carl Friedrich Gauss

1777 - 1855 worked in a wide variety of fields in both mathematics and physics incuding number theory, analysis, differential geometry, geodesy, magnetism, astronomy and optics. His work has had an immense influence in many areas.

Georg Cantor

1845 - 1918 Russian mathematician who can be considered as the founder of set theory and introduced the concept of infinite numbers with his discovery of cardinal numbers. He also advanced the study of trigonometric series.

Algebraic Closure

A numerical field where every polynomial equation over that field has a solution that lies in that field. P(x) = a(n)x^n + a(n-1)^(n-1) + ... + a(2)^2 + a(1)x + a(0) = 0 Complex numbers are, because all P(x) with complex co-efficients (ie a(i)) have a complex solution. Real numbers are not, because some real P(x) have complex solutions Rational numbers are not, because some rational P(x) have real solutions Likewise, integer & natural number P(x).

Complex Plane

Cartesian co-ordinate representation of complex numbers, with their real parts plotted on x axis, and imaginary parts plotted on y axis

Cardinal Number

Number that denotes the magnitiude of a set

Fundamental Theorem of Algebra

The field of Complex Numbers is Algebraically Closed

Pythagoras' Theorem

The square on the hypotenuse = the sum of the squares on the other two sides aka Bride's chair, Franciscan's cowl, peacock's tail,

Fibonacci numbers

[0], 1, 1, 2, 3. 5. 8. 13, 21, 34, 55, 89, 144, ... integer sequence where each element after first 2 is the sum of preceding 2 elements F(n) = F(n-1) + F(n-2), n > 2 ratio of successive terms converges to Golden Mean [Leonardo of Pisa, aka Fibonacci, 1202 (Liber Abaci)]

bijective function

a function which is both injective and surjective (ie every element of x and y are uniquely mapped). one-to-one, with no unmapped elements

Countable set

a set with the same cardinality (number of elements) as some subset of N (natural numbers). Either 'finite' - shares cardinality with only a subset of N or 'countably infinite' - shares cardinality with N itself (ie its elements have a one-to-one correspondence with elements of N) (aka denumerable) Integers & Rational numbers are; Real numbers are not.

Pythagoras

about 569 BC - about 475 BC A Greek philosopher who made important developments in mathematics, astronomy, and the theory of music. The theorem now known as Pythagoras's theorem was known to the Babylonians 1000 years earlier but he may have been the first to prove it.

surjective function

function F(x, y) where every element element of y is mapped from an element of x (but not necessarily a unique element of x) 'many to one' with all elements of y mapped

injective function

function that preserves distictness: of the form F(x, y) where every element of x maps to exactly 1 element of y. (ie x is mapped one-to-one, but some elements of y may not be mapped)

fractal dimension

if a fractal's one-dimensional lengths are doubled, the spatial content of the fractal scales by a power that is not necessarily an integer Usually greater than topological dimension.

prime number

integer > 1 with no smaller integer factors (ie, divisible only by 1 and itself) 2, 3, 5, 7, 11

Definable sequence

integer sequence for which there is a statement P(x) that is true for that sequence x, and false for all other integer sequences. Not always computable

Computable sequence

integer sequence for which there is an algorithm which, given n, calculates a(n) for all n>0 always definable

Complete sequence

integer sequence where every positive integer can be expressed as a sum of values in the sequence, using each value only once eg powers of 2 - every positive integer is sum of powers of 2 (as expressed as binary number)

Fractal

mathematical set that exhibits a repeating pattern that displays at every scale aka expanding symmetry or evolving symmetry eg Mandelbrot set from 'fractional dimension' (coined by Mandelbrot 1975)

Minkowski dimension

method to determine the fractal dimension of a set S in Euclidean space R^n. dim(box)(S) := lim((logN(e)/log(1/e)) as e --> 0 where N(e) is the number of boxes of side length e required to cover the set. [also Minkowski-Bouligand dimension, or Box-counting dimension]

Hausdorff dimension

non-integer dimension of an object (eg fractal) note: 'classical' geometric objects such as point, line, polygon, polyhedron etc have integer dimension (0, 1, 2, 3 etc)

even number

number divisible by 2 of form n=2k, where k is an integer

odd number

number not divisible by 2, of form n=2k+1, where k is an integer

radix (base)

number of unique digits, including zero, in a numeral system

Transfinite Number

number which is greater than all finite numbers, but not infinite Coined by Cantor