__ Evolution of number system - Journey from counting to real numbers. __

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evolution to number system - journey from counting to real number

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# Natural number

Natural numbers can be used for counting (one apple, two apples, three apples, ...) from top to bottom.

In mathematics, the **natural numbers** are the ordinary whole numbers used for 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 later notion is that of a nominal number, which is used only for naming.

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.

There is no universal agreement about whether to include zero in the set of natural numbers: some define the natural numbers to be the **positive integers** {1, 2, 3, ...}, while for others the term designates the **non-negative integers** {0, 1, 2, 3, ...}. The former definition is the traditional one, with the latter definition first appearing in the 19th century. Some authors use the term "natural number" to exclude zero and "whole number" to include it; others use "whole number" in a way that excludes zero, or in a way that includes both zero and the negative integers.

## 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.

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 over one million. 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. The Babylonians had a place-value system based essentially on the numerals for 1 and 10.

A much later advance was the development of the idea that zero can be considered as a number, with its own numeral. The use of a zero digit in place-value notation (within other numbers) dates back as early as 700 BC by the Babylonians, but they omitted such a digit when it would have been the last symbol in the number.^{[1]} The Olmec and Maya civilizations used zero as a separate number as early as the 1st century BC, but this usage did not spread beyond Mesoamerica. The use of a numeral zero in modern times originated with the Indian mathematician Brahmagupta in 628. However, zero had been used as a number in the medieval computus (the calculation of the date of Easter), beginning with Dionysius Exiguus in 525, without being denoted by a numeral (standard Roman numerals do not have a symbol for zero); instead *nulla* or *nullae*, genitive of *nullus*, the Latin word for "none", was employed to denote a zero value.^{[2]}

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]}

Independent studies also occurred at around the same time in India, China, and Mesoamerica.^{[citation needed]}

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**. On the other hand, *integer* being Latin for *whole*, the integers usually stand for the negative and positive whole numbers (and zero) altogether.

## Notation

Mathematicians use **N** or (an N in blackboard bold, displayed as â„• in Unicode) to refer to the set of all natural numbers. This set is countably infinite: it is infinite but countable by definition. This is also expressed by saying that the cardinal number of the set is aleph-null .

Typically, if a mathematician uses for the set and he needs in the same scientific context this set including , then he mostly writes for the latter.

On the other hand, if he uses for the set and he needs in the same scientific context this set excluding , then he mostly writes or for the latter.

To be unambiguous about whether zero is included or not, sometimes an index (or superscript) "0" is added in the former case, and a superscript "" or subscript "" is added in the latter case:

Some authors who exclude zero from the naturals use the terms *natural numbers with zero*, *whole numbers*, or *counting numbers*, denoted **W**, for the set of nonnegative integers. Others use the notation **P** for the positive integers if there is no danger of confusing this with the prime numbers. In that case, a popular notation is to use a script ** P** for positive integers (which extends to using script

**for negative integers, and script**

*N***for zero).**

*Z*Set theorists often denote the set of all natural numbers including zero by a lower-case Greek letter omega: ω. This stems from the identification of an ordinal number with the set of ordinals that are smaller. One may observe that adopting the von Neumann definition of ordinals and defining cardinal numbers as minimal ordinals among those with same cardinality, one gets . Lowercase omega **ω** is also similar to **W**.

## Algebraic properties

The addition (+) and multiplication (×) operations on natural numbers have several algebraic properties:

- Closure under addition and multiplication: for all natural numbers
*a*and*b*, both*a*+*b*and*a*×*b*are natural numbers. - Associativity: for all natural numbers
*a*,*b*, and*c*,*a*+ (*b*+*c*) = (*a*+*b*) +*c*and*a*× (*b*×*c*) = (*a*×*b*) ×*c*. - Commutativity: for all natural numbers
*a*and*b*,*a*+*b*=*b*+*a*and*a*×*b*=*b*×*a*. - Existence of identity elements: for every natural number
*a*,*a*+ 0 =*a*and*a*× 1 =*a*. - Distributivity of multiplication over addition for all natural numbers
*a*,*b*, and*c*,*a*× (*b*+*c*) = (*a*×*b*) + (*a*×*c*) - No zero divisors: if
*a*and*b*are natural numbers such that*a*×*b*= 0 then*a*= 0 or*b*= 0

## Properties

One can recursively define an addition on the natural numbers by setting *a* + 0 = *a* and *a* + *S*(*b*) = *S*(*a* + *b*) for all *a*, *b*. Here *S* should be read as "successor". This turns the natural numbers (**N**, +) into a commutative monoid with identity element 0, the so-called free monoid with one generator. This monoid satisfies the cancellation property and can be embedded in a group. The smallest group containing the natural numbers is the integers.

If we define 1 := *S*(0), then *b* + 1 = *b* + *S*(0) = *S*(*b* + 0) = *S*(*b*). That is, *b* + 1 is simply the successor of *b*.

Analogously, given that addition has been defined, a multiplication × can be defined via *a* × 0 = 0 and *a* × S(*b*) = (*a* × *b*) + *a*. This turns (**N**^{*}, ×) into a free commutative monoid with identity element 1; a generator set for this monoid is the set of prime numbers. Addition and multiplication are compatible, which is expressed in the distribution law: *a* × (*b* + *c*) = (*a* × *b*) + (*a* × *c*). These properties of addition and multiplication make the natural numbers an instance of a commutative semiring. Semirings are an algebraic generalization of the natural numbers where multiplication is not necessarily commutative. The lack of additive inverses, which is equivalent to the fact that **N** is not closed under subtraction, means that **N** is *not* a ring; instead it is a semiring (also known as a *rig*).

If we interpret the natural numbers as "excluding 0", and "starting at 1", the definitions of + and × are as above, except that we start with *a* + 1 = *S*(*a*) and *a* × 1 = *a*.

For the remainder of the article, we write *ab* to indicate the product *a* × *b*, and we also assume the standard order of operations.

Furthermore, one defines a total order on the natural numbers by writing *a* ≤ *b* if and only if there exists another natural number *c* with *a* + *c* = *b*. This order is compatible with the arithmetical operations in the following sense: if *a*, *b* and *c* are natural numbers and *a* ≤ *b*, then *a* + *c* ≤ *b* + *c* and *ac* ≤ *bc*. An important property of the natural numbers is that they are well-ordered: every non-empty set of natural numbers has a least element. The rank among well-ordered sets is expressed by an ordinal number; for the natural numbers this is expressed as "ω".

While it is in general not possible to divide one natural number by another and get a natural number as result, the procedure of *division with remainder* is available as a substitute: for any two natural numbers *a* and *b* with *b* ≠ 0 we can find natural numbers *q* and *r* such that

*a* = *bq* + *r* and *r* < *b*.

The number *q* is called the *quotient* and *r* is called the *remainder* of division of *a* by *b*. The numbers *q* and *r* are uniquely determined by *a* and *b*. This, the Division algorithm, is key to several other properties (divisibility), algorithms (such as the Euclidean algorithm), and ideas in number theory.

## Generalizations

Two generalizations of natural numbers arise from the two uses:

- A natural number can be used to express the size of a finite set; more generally a cardinal number is a measure for the size of a set also suitable for infinite sets; this refers to a concept of "size" such that if there is a bijection between two sets they have the same size. The set of natural numbers itself and any other countably infinite set has cardinality aleph-null ().
- Linguistic ordinal numbers "first", "second", "third" can be assigned to the elements of a totally ordered finite set, and also to the elements of well-ordered countably infinite sets like the set of natural numbers itself. This can be generalized to ordinal numbers which describe the position of an element in a well-ordered set in general. An ordinal number is also used to describe the "size" of a well-ordered set, in a sense different from cardinality: if there is an order isomorphism between two well-ordered sets they have the same ordinal number. The first ordinal number that is not a natural number is expressed as ; this is also the ordinal number of the set of natural numbers itself.

Many well-ordered sets with cardinal number have an ordinal number greater than ω (the latter is the lowest possible). The least ordinal of cardinality (i.e., the initial ordinal) is .

For finite well-ordered sets, there is one-to-one correspondence between ordinal and cardinal numbers; therefore they can both be expressed by the same natural number, the number of elements of the set. This number can also be used to describe the position of an element in a larger finite, or an infinite, sequence.

Hypernatural numbers are part of a non-standard model of arithmetic due to Skolem.

Other generalizations are discussed in the article on numbers.

## Formal definitions

Main article: Set-theoretic definition of natural numbers

Historically, the precise mathematical definition of the natural numbers developed with some difficulty. The Peano axioms state conditions that any successful definition must satisfy. Certain constructions show that, given set theory, models of the Peano postulates must exist.

### Peano axioms

Main article: Peano axioms

The Peano axioms give a formal theory of the natural numbers. The axioms are:

- There is a natural number 0.
- Every natural number
*a*has a natural number successor, denoted by*S*(*a*). Intuitively,*S*(*a*) is*a*+1. - There is no natural number whose successor is 0.
*S*is injective, i.e. distinct natural numbers have distinct successors: if*a*≠*b*, then*S*(*a*) ≠*S*(*b*).- If a property is possessed by 0 and also by the successor of every natural number which possesses it, then it is possessed by all natural numbers. (This postulate ensures that the proof technique of mathematical induction is valid.)

It should be noted that the "0" in the above definition need not correspond to what we normally consider to be the number zero. "0" simply means some object that when combined with an appropriate successor function, satisfies the Peano axioms. All systems that satisfy these axioms are isomorphic, the name "0" is used here for the first element (the term "zeroth element" has been suggested to leave "first element" to "1", "second element" to "2", etc.), which is the only element that is not a successor. For example, the natural numbers starting with one also satisfy the axioms, if the symbol 0 is interpreted as the natural number 1, the symbol *S*(*0*) as the number 2, etc. In fact, in Peano's original formulation, the first natural number *was* 1.

### Constructions based on set theory

#### A standard construction

A standard construction in set theory, a special case of the von Neumann ordinal construction, is to define the natural numbers as follows:

We set 0 := { }, the empty set,

and define *S*(*a*) = *a* ∪ {*a*} for every set *a*. *S*(*a*) is the successor of *a*, and *S* is called the successor function.

By the axiom of infinity, the set of all natural numbers exists and is the intersection of all sets containing 0 which are closed under this successor function. This then satisfies the Peano axioms.

Each natural number is then equal to the set of all natural numbers less than it, so that

· 0 = { }

· 1 = {0} = {{ }}

· 2 = {0, 1} = {0, {0}} = { { }, {{ }} }

· 3 = {0, 1, 2} = {0, {0}, {0, {0}}} = { { }, {{ }}, {{ }, {{ }}} }

· *n* = {0, 1, 2, ..., *n*-2, *n*-1} = {0, 1, 2, ..., *n*-2,} ∪ {*n*-1} = {*n*-1} ∪ (*n*-1) = *S*(*n*-1)

and so on. When a natural number is used as a set, this is typically what is meant. Under this definition, there are exactly *n* elements (in the naïve sense) in the set *n* and *n* ≤ *m* (in the naïve sense) if and only if *n* is a subset of *m*.

Also, with this definition, different possible interpretations of notations like **R*** ^{n}* (

*n-*tuples versus mappings of

*n*into

**R**) coincide.

Even if the axiom of infinity fails and the set of all natural numbers does not exist, it is possible to define what it means to be one of these sets. A set *n* is a natural number means that it is either 0 (empty) or a successor, and each of its elements is either 0 or the successor of another of its elements.

#### Other constructions

Although the standard construction is useful, it is not the only possible construction. For example:

one could define 0 = { }

and *S*(*a*) = {*a*},

producing

· 0 = { }

· 1 = {0} = {{ }}

· 2 = {1} ={{{ }}}, etc.

Each natural number is then equal to the set of the natural number preceding it.

Or we could even define 0 = {{ }}

and *S*(*a*) = *a* ∪ {*a*}

producing

· 0 = {{ }}

· 1 = {{ }, 0} = {{ }, {{ }}}

· 2 = {{ }, 0, 1}, etc.

The oldest and most "classical" set-theoretic definition of the natural numbers is the definition commonly ascribed to Frege and Russell under which each concrete natural number *n* is defined as the set of all sets with *n* elements.^{[5][6]} This may appear circular, but can be made rigorous with care. Define 0 as {{ }} (clearly the set of all sets with 0 elements) and define *S*(*A*) (for any set *A*) as {*x* ∪ {*y*} | *x* ∈ *A* ∧ *y* ∉ *x* } (see set-builder notation). Then 0 will be the set of all sets with 0 elements, 1 = *S*(0) will be the set of all sets with 1 element, 2 = *S*(1) will be the set of all sets with 2 elements, and so forth. The set of all natural numbers can be defined as the intersection of all sets containing 0 as an element and closed under *S* (that is, if the set contains an element *n*, it also contains *S*(*n*)). One could also define "finite" independently of the notion of "natural number", and then define natural numbers as equivalence classes of finite sets under the equivalence relation of equipollence. This definition does not work in the usual systems of axiomatic set theory because the collections involved are too large (it will not work in any set theory with the axiom of separation); but it does work in New Foundations (and in related systems known to be relatively consistent) and in some systems of type theory.

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Hope this will help you!

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Ex. - 1,2,3,4,5,............

Natural numbers - 1,2,3,.......or counting numbers are called natural numbers.

Ex- 1,2,3,4.............

Whole numbers - Natural numbers along with 0 is called whole number.

Ex - 0,1,2,3,4,...........

Integers - The collection of natural numbers, whole numbers and also negative numbers is called integer.

Ex- ......-4, -3, -2, -1, 0,1,2,3,4,.......

Rational numbers - The number which can be written in the form of p/q, where p and q are integers and q does not equal to 0, are called rational number.

Ex- 1/2, -1/3, 4/5......

Irrational numbers- The number which can't be written in the form of p/q, where p and q are integers and q does not equal to 0, are called irrational number.

Ex- ?2, ?3, 0.10100100010000......

Real number- The collection of all rational numbers and irrational numbers are called real numbers.

Ex- 1/2 , ?2, 3/6, ?5.........

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In mathematics, the natural numbers are the ordinary whole numbers used for counting("there are 6 coins onthe 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 later notion is that of a nominal number, which is used only for naming.

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.

There is no universal agreement about whether to include zero in the set of natural numbers: some define the natural numbers to be the positive integers {1, 2, 3, ...}, while for others the term designates the non-negative integers {0, 1, 2, 3, ...}. The former definition is the traditional one, with the latter definition first appearing in the 19th century. Some authors use the term "natural number" to exclude zero and "whole number" to include it; others use "whole number" in a way that excludes zero, or in a way that includes both zero and the negative integers.

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A - art of arithmetic

T - tool of knowledge

H - habit of problem solving

E - evaluation of civilization

M - magic of numbers

A - application of rules

T - tool of knowledge

I - ideas of intellect

C- creativity of algebra

S - solving the beautiful puzzle sums

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The existence of irrationality in numbers was accepted by Indian mathematicians as far back as?7th?century BC when Manava, an author of the Indian geometric text?Sulbasutras, discovered (while finding the hypotenuse of a right-angled triangle) that it is not possible to accurately calculate the square roots of numbers like 2 and 8. It is, however, the Pythagorean school of Greek mathematicians, or the Pythagoreans, who are credited with discovering irrational numbers sometime in 400 BC. In 5th?century AD, the great Indian mathematician Aryabhata suggested that the value of ? is incommensurable. Later, in the 1700s, a Swiss mathematician named Lambert and a French mathematician named Legendre proved ? to be irrational.?

In this way, a long line of mathematicians helped shed light on the concept of irrational numbers. These mathematicians questioned the rationality of those numbers that cannot be written in the form of a ratio of integers.The Pythagoreans were the first to actually prove a number to be irrational and this number was. The set of all irrational numbers is denoted by.

Go through this lesson to get a basic idea about the irrationality of numbers.

Golden ratio

Two quantities are said to be in the golden ratio if the ratio of the sum of those quantities to the larger quantity is the same as the ratio of the larger quantity to the smaller one. Let us understand this concept.

Say?a?and?b?are two line segments that are in the golden ratio.?

Therefore,

The golden ratio is represented by the Greek letter (phi), where, an irrational number.

The golden ratio is also known as ?the golden mean? and ?the golden section?. This ratio is used not only in mathematics but also in biology, art, music, architecture and in various other branches of science.

Pi?is a constant value that is equal to the ratio of the circumference of a circle to its diameter. It is an irrational number represented by the Greek letter ???. This symbol was proposed by a Welsh mathematician named William Jones in 1706. The value of pi is approximately equal to.

The Great Pyramid of Giza was constructed with a perimeter of about 1760 cubits and a height of about 280 cubits. The ratio of the perimeter to the height, i.e.,??is approximately equal to 6.285, which is almost equal to 2?. This is cited by some as the proof that the people who built the pyramid knew about the special ratio represented by ?.

Since ? is closely related to the circle, it is found in many geometric and trigonometric formulae. It is also used in many other scientific formulae such as in thermodynamics, the number theory, mechanics and electromagnetism.

A?ryabhata (476 AD?550 AD) was the first Indian mathematician and astronomer.

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The existence of irrationality in numbers was accepted by Indian mathematicians as far back as?7th?century BC when Manava, an author of the Indian geometric text?Sulbasutras, discovered (while finding the hypotenuse of a right-angled triangle) that it is not possible to accurately calculate the square roots of numbers like 2 and 8. It is, however, the Pythagorean school of Greek mathematicians, or the Pythagoreans, who are credited with discovering irrational numbers sometime in 400 BC. In 5th?century AD, the great Indian mathematician Aryabhata suggested that the value of ? is incommensurable. Later, in the 1700s, a Swiss mathematician named Lambert and a French mathematician named Legendre proved ? to be irrational.?

In this way, a long line of mathematicians helped shed light on the concept of irrational numbers. These mathematicians questioned the rationality of those numbers that cannot be written in the form of a ratio of integers.The Pythagoreans were the first to actually prove a number to be irrational and this number was?. The set of all irrational numbers is denoted by??.

Go through this lesson to get a basic idea about the irrationality of numbers.

Golden ratio

Two quantities are said to be in the golden ratio if the ratio of the sum of those quantities to the larger quantity is the same as the ratio of the larger quantity to the smaller one. Let us understand this concept.

Say?a?and?b?are two line segments that are in the golden ratio.?

?

Therefore,?

The golden ratio is represented by the Greek letter ??? (phi), where??, an irrational number.

The golden ratio is also known as ?the golden mean? and ?the golden section?. This ratio is used not only in mathematics but also in biology, art, music, architecture and in various other branches of science.

Pi?is a constant value that is equal to the ratio of the circumference of a circle to its diameter. It is an irrational number represented by the Greek letter ???. This symbol was proposed by a Welsh mathematician named William Jones in 1706. The value of pi is approximately equal to?.

The Great Pyramid of Giza was constructed with a perimeter of about 1760 cubits and a height of about 280 cubits. The ratio of the perimeter to the height, i.e.,???is approximately equal to 6.285, which is almost equal to 2?. This is cited by some as the proof that the people who built the pyramid knew about the special ratio represented by ?.?

Since ? is closely related to the circle, it is found in many geometric and trigonometric formulae. It is also used in many other scientific formulae such as in thermodynamics, the number theory, mechanics and electromagnetism.

A?ryabhata (476 AD?550 AD) was the first Indian mathematician and astronomer.

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Hope this helps.

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