∞

n..... ∞ n..... ∞ The Counting Numbers

So N even = N odd

So Number of even numbers = Number of odd numbers = Number of odd+ even numbers !!!!!!!!

So !!!!!!! N even = N odd = N even + N odd Odd ain’t it

There are more Real numbers R than Natural numbers N The real numbers include rational numbers such as the counting numbers (integers) and fractions which involve integers such as 23 and −12/79 and irrational numbers such as π and √2 which are infinite decimals with a non-recurring sequence

All the points on a line

א o natural numbers

א 1 real numbers

א o natural numbers א 1 real numbers א 2 squiggly lines

Georg Cantor

From “One, Two, Three, Infinity” Gamow

Why do we have a 10 numeral “decimal system? …………etc…………………….

Why do we have a 10 numeral “decimal system? …………etc……………………. Why not 12?

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davids-pics.blogspot.com three-leaf clovers for every four leaf clover

An illustration of Cantor's diagonal argument for the existence of uncountable sets.[31] The sequence at the bottom cannot occur anywhere in the infinite list of sequences above.Cantor's diagonal argumentuncountable sets[31]

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In mathematics, there are two conventions for the set of natural numbers: it is either the set of positive integers {1, 2, 3,...} according to the traditional definition or the set of non-negative integers {0, 1, 2,...} according to a definition first appearing in the nineteenth century.mathematicssetpositiveintegers123non-negative0 Natural numbers have two main purposes: counting ("there are 6 coins on the table") and ordering ("this is the 3rd largest city in the country"). These purposes are related to the linguistic notions of cardinal and ordinal numbers, respectively. (See English numerals.) A more recent notion is that of a nominal number, which is used only for naming.countingorderingEnglish numeralsnominal number Properties of the natural numbers related to divisibility, such as the distribution of prime numbers, are studied in number theory. Problems concerning counting and ordering, such as partition enumeration, are studied in combinatorics.divisibilityprime numbers number theorypartitionenumeration combinatorics

History of natural numbers and the status of zero The natural numbers had their origins in the words used to count things, beginning with the number 1.[1][1] The first major advance in abstraction was the use of numerals to represent numbers. This allowed systems to be developed for recording large numbers. The ancient Egyptians developed a powerful system of numerals with distinct hieroglyphs for 1, 10, and all the powers of 10 up to one million. The Babylonians had a place-value system based essentially on the numerals for 1 and 10. A stone carving from Karnak, dating from around 1500 BC and now at the Louvre in Paris, depicts 276 as 2 hundreds, 7 tens, and 6 ones; and similarly for the number 4,622.numeralsEgyptianshieroglyphsBabyloniansplace-valueKarnakLouvre A much later advance in abstraction was the development of the idea of zero as a number with its own numeral. A zero digit had been used in place-value notation as early as 700 BC by the Babylonians but they omitted it when it would have been the last symbol in the number.[2] The Olmec and Maya civilization used zero developed independently as a separate number as early as 1st century BC, but this usage did not spread beyond Mesoamerica.zerodigit[2]OlmecMaya civilizationMesoamerica

The concept as used in modern times originated with the Indian mathematician Brahmagupta in 628. Nevertheless, medieval computers (e.g. people who calculated the date of Easter), beginning with Dionysius Exiguus in 525, used zero as a number without using a Roman numeral to write it. Instead nullus, the Latin word for "nothing", was employed.IndianBrahmagupta computersEasterDionysius ExiguusRoman numeral The first systematic study of numbers as abstractions (that is, as abstract entities) is usually credited to the Greek philosophers Pythagoras and Archimedes. Note that many Greek mathematicians did not consider 1 to be "a number", so to them 2 was the smallest number.[3]abstractionsentitiesGreekPythagorasArchimedes[3]

Independent studies also occurred at around the same time in India, China, and Mesoamerica. IndiaChinaMesoamerica Several set-theoretical definitions of natural numbers were developed in the 19th century. With these definitions it was convenient to include 0 (corresponding to the empty set) as a natural number. Including 0 is now the common convention among set theorists, logicians, and computer scientists. Many other mathematicians also include 0, although some have kept the older tradition and take 1 to be the first natural number[4]. Sometimes the set of natural numbers with 0 included is called the set of whole numbers or counting numbers.set-theoretical definitions of natural numbersempty setset theoristslogicianscomputer scientists[4]whole numbers

There are more Real numbers R than Natural numbers N In mathematics, the real numbers include both rational numbers, such as 42 and −23/129, and irrational numbers, such as pi and the square root of two; or, a real number can be given by an infinite decimal representation, such as , where the digits continue in some way; or, the real numbers may be thought of as points on an infinitely long number line.mathematics rational numbers irrational numberspisquare root of two decimal representationnumber line

n.. ∞ n.. ∞

So Number of even numbers = Number of odd numbers

So Number of even numbers

There are more Real numbers R than Natural numbers N These descriptions of the real numbers, while intuitively accessible, are not sufficiently rigorous for the purposes of pure mathematics. The discovery of a suitably rigorous definition of the real numbers— indeed, the realization that a better definition was needed—was one of the most important developments of 19th century mathematics. Popular definitions in use today include equivalence classes of Cauchy sequences of rational numbers; Dedekind cuts; a more sophisticated version of "decimal representation"; and an axiomatic definition of the real numbers as the unique complete Archimedean ordered field. These definitions are all described in detail below.equivalence classesCauchy sequencesDedekind cutscompleteArchimedean orderedfield

A real number may be either rational or irrational; either algebraic or transcendental; and either positive, negative, or zero. Real numbers are used to measure continuous quantities. They may in theory be expressed by decimal representations that have an infinite sequence of digits to the right of the decimal point; these are often represented in the same form as … The ellipsis (three dots) indicate that there would still be more digits to come.rational irrationalalgebraictranscendentalpositivenegativezerocontinuousdecimal representations ellipsis

Rational and Irrational Numbers Rational Numbers A rational number is a number that can be written as a ratio. That means it can be written as a fraction, in which both the numerator (the number on top) and the denominator (the number on the bottom) are whole numbers. The number 8 is a rational number because it can be written as the fraction 8/1. Likewise, 3/4 is a rational number because it can be written as a fraction. Even a big, clunky fraction like 7,324,908/56,003,492 is rational, simply because it can be written as a fraction.

Every whole number is a rational number, because any whole number can be written as a fraction. For example, 4 can be written as 4/1, 65 can be written as 65/1, and 3,867 can be written as 3,867/1. Irrational Numbers All numbers that are not rational are considered irrational. An irrational number can be written as a decimal, but not as a fraction. An irrational number has endless non-repeating digits to the right of the decimal point. Here are some irrational numbers: π = … = … Although irrational numbers are not often used in daily life, they do exist on the number line. In fact, between 0 and 1 on the number line, there are an infinite number of irrational numbers!

In mathematics a rational number is any number that can be expressed as the quotient a/b of two integers, with the denominator b not equal to zero. Since b may be equal to 1, every integer is a rational number. The set of all rational numbers is usually denoted Q (for quotient).mathematicsintegers denominatorsetquotient The decimal expansion of a rational number always either terminates after finitely many digits or begins to repeat the same sequence of digits over and over. Moreover, any repeating or terminating decimal represents a rational number. These statements hold true not just for base 10, but also for binary, hexadecimal, or any other integer base.decimal expansiondigitsrepeatsequencebase 10binaryhexadecimalbase A real number that is not rational is called irrational.real numberirrational Irrational numbers include √2, π, and e. The decimal expansion of an irrational number continues forever without repeating. Since the set of rational numbers is countable, and the set of real numbers is uncountable, almost every real number is irrational.√2πecountableuncountablealmost every

The rational numbers can be formally defined as the equivalence classes of the quotient set Z × (Z − {0}) / ~, where the cartesian product Z × (Z − {0}) is the set of all ordered pairs (m,n) where m and n are integers with n ≠ 0, and "~" is the equivalence relation defined by (m1,n1) ~ (m2,n2) if, and only if, m1n2 − m2n1 = 0.formally equivalence classesquotient setcartesian product ordered pairsintegersequivalence relationif, and only if In abstract algebra, the rational numbers form a field. This is the archetypical field of characteristic zero, and is the field of fractions for the ring of integers. Finite extensions of Q are called algebraic number fields, and the algebraic closure of Q is the field of algebraic numbers.abstract algebrafieldcharacteristicfield of fractionsringextensionsalgebraic number fieldsalgebraic closurealgebraic numbers In mathematical analysis, the rational numbers form a dense subset of the real numbers. The real numbers can be constructed from the rational numbers by completion, using either Cauchy sequences or Dedekind cuts.mathematical analysisdense subsetcompletionCauchy sequencesDedekind cuts

Cardinality of the continuum Main article: Cardinality of the continuumCardinality of the continuum One of Cantor's most important results was that the cardinality of the continuum ( ) is greater than that of the natural numbers ( ); that is, there are more real numbers R than natural numbers N. Namely, Cantor showed that (see Cantor's diagonal argument).cardinality of the continuumCantor's diagonal argument The continuum hypothesis states that there is no cardinal number between the cardinality of the reals and the cardinality of the natural numbers, that is, (see Beth one). However, this hypothesis can neither be proved nor disproved within the widely accepted Zermelo- Fraenkel set theory, even assuming the Axiom of Choice.continuum hypothesiscardinal numberBeth oneZermelo- Fraenkel set theoryAxiom of Choice Cardinal arithmeticCardinal arithmetic can be used to show not only that the number of points in a real number line is equal to the number of points in any segment of that line, but that this is equal to the number of points on a plane and, indeed, in any finite-dimensional space.real number line segment

The first three steps of a fractal construction whose limit is a space- filling curve, showing that there are as many points in a one- dimensional line segment as in a two-dimensional square.space- filling curve The first of these results is apparent by considering, for instance, the tangent function, which provides a one-to-one correspondence between the interval (−π/2, π/2) and R (see also Hilbert's paradox of the Grand Hotel). The second result was proved by Cantor in 1878, but only became intuitively apparent in 1890, when Giuseppe Peano introduced the space-filling curves, curved lines that twist and turn enough to fill the whole of any square, or cube, or hypercube, or finite- dimensional space. These curves can be used to define a one-to-one correspondence between the points in the side of a square and those in the square. tangentone-to-one correspondenceintervalHilbert's paradox of the Grand HotelGiuseppe Peanospace-filling curveshypercubeone-to-one correspondence Cantor also showed that sets with cardinality strictly greater than exist (see his generalized diagonal argument and theorem). They include, for instance:generalized diagonal argumenttheorem the set of all subsets of R, i.e., the power set of R, written P(R) or 2Rpower set the set RR of all functions from R to R Both have cardinality (see Beth two).Beth two The cardinal equalities and can be demonstrated using cardinal arithmetic:cardinal equalitiescardinal arithmetic

Harry Kroto 2004

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