Number System
Number system is defined as the proper understanding and usage of the numbers in the various
places. Numbers are the basic building stones of mathematics. There are different types of
numbers exist, we will learn about them in brief at number system.
It includes :-
* Ability to find the relative values of a number.
* How to use a number in different arithmetic operations like addition, subtraction, multiplication
and division
* Finding the problem solving strategies
Numbers system covers various topics. They are
* Estimating and rounding
* Rounding and addition
* Rounding and product
* Rounding and division

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In mathematics, a 'number system' is a set of numbers, (in the broadest sense of the word),
together with one or more operations, such as addition or multiplication.
Examples of number systems include: natural numbers, integers, rational numbers, algebraic
numbers, real numbers, complex numbers, p-adic numbers, surreal numbers, and hyperreal
numbers.
For a history of number systems, see number. For a history of the symbols used to represent
numbers, see numeral system.
Logical construction of number systems
Natural numbers
Simply put, the natural numbers consist of the set of all whole numbers greater than zero. The set
is denoted with a bold face capital N or with the special symbol . (In some books, the natural
numbers begin with 0. There is no general agreement on this subject.)[1][2]
Giuseppe Peano developed axioms for the natural numbers, and is considered the founder of
axiomatic number theory.
1. Axiom one :- There is a natural number 0.
2. Axiom two :- Every natural number a has a successor, denoted by S(a).
3. Axiom three :- There is no natural number whose successor is 0.
4. Axiom four :- Distinct natural numbers have distinct successors: a = b if and only if S(a) =
S(b).

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Axiom of induction :- 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.
From these five axioms, all of the properties of the natural numbers can be deduced.
The number one is defined as 1 = S(0).
Most number systems include the idea of equality. In mathematics, equality is an equivalence
relation, meaning it obeys the three axioms of equality:
Reflexive axiom: a = a.
Symmetric axiom: a = b implies b = a.
Transitive axiom: a = b and b = c implies a = c.
Integers
The natural numbers can be extended to the number system called the integers (denoted with Z)
as follows. For every non-zero natural number a, there exists an integer denoted -a, which is not a
natural number. As a special case -0 is defined as the natural number 0. The successor function
can be extended to the integers by the rule S(-a) = -(a - 1).
Addition can be defined on the integers inductively as follows. If a and b are natural numbers, then
-a + -b = -(a + b). If a is any integer, then a + 0 = a. If b is a non-zero integer, then a + b = (a - 1)
+ S(b). It is then necessary to show that addition is well-defined in the case where b is a natural
number.
The definition of subtraction extends to the integers unchanged, and now it can be proven that a -
b is defined for all integers a and b. To justify the use of - for both "minus" and "negative", one
proves that a - b = a + -b. Multiplication can be defined as follows. For all natural numbers a and


b, -a * b = a * -b = -(a * b). It follows as a theorem that -a * -b = a * b. The definition of division
extends to the integers unchanged, but division is not defined in every case.

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Rational numbers
The rational numbers (denoted with Q) are the number system that extends the integers to
include numbers which can be written as fractions. It allows division to be defined for all pairs of
numbers except for division by zero. It also allows the definition of exponents to be extended to
negative integer exponents, and to some, but not all, rational exponents.
We define a fraction a / b to be an ordered pair, where a is any integer and b is any non-zero
natural number. We define equality of fractions by a / b = c / d if and only if a * d = b * c, and define
a/1 = a, which embeds the integers in the set of all fractions.These definitions of equality partition
the set of fractions and integers into equivalence classes.
The canonical representative of an equivalence class is an element a / b where b is positive and
relatively prime to a, or the integer a if b=1.Finding the canonical representative of an equivalence
class of rational numbers is also called reducing to lowest terms. The set of rational numbers is
defined to be either the set of equivalence classes or the set of canonical representatives.
Polynomials
Polynomials are not usually called numbers, but they share many properties with numbers. All of
the axioms of operations hold for polynomials except for the axiom of multiplicative inverses.
Polynomials do not, in general, have multiplicative inverses. Thus the set of polynomials, like the
integers, is a commutative ring (with identity).
Algebraic numbers


The algebraic numbers are a number system that includes all of the rational numbers, and is
included in the set of real numbers.

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The construction of the algebraic numbers requires an understanding of the definition and
properties of an extension field. Roughly speaking, one extends the rational numbers by
appending all zeroes of polynomials with integer coefficents.
This, however, would append complex numbers, which are usually excluded from the algebraic
numbers, unless the set is called the complex algebraic numbers. It is, therefore, traditional to
construct the real numbers first, and then define the algebraic numbers as a subset of the reals.
The algebraic numbers form a field.
Real numbers
There are many ways to construct the real number (denoted R) system: equivalence classes of
Cauchy sequences, transcendental extension fields, and Dedekind cuts, to mention just three. But
the most elementary definition is that the real numbers are all numbers that can be written as
decimals.
A decimal can only have finitely many digits to the left of the decimal point, but to the right of the
decimal point there are three cases to consider. A decimal may terminate, repeat, or continue
forever without ever becoming an infinite sequence of repeating strings of digits (in brief, a non-
repeating decimal). In the first two cases, the decimal is rational, that is, it can be changed to a
fraction. In the third case, the decimal is irrational.
Examples
The number forty-two is a real number because it can be written as a decimal: 42.0. The number
o

ne half is both a rational number and a real

number because it can be written 0.5. The number
one third is both a rational number and a real number because it can be written 0.333... .

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