Open In App

Axioms of Real Number System

Last Updated : 26 Apr, 2021
Improve
Improve
Like Article
Like
Save
Share
Report

In this article, we shall look at some very basic ideas about the Real Analysis, i.e. the study of the structure of Real Number System. We shall discuss the three axioms that are considered to be satisfied by the set of Real Numbers, R

The three axioms are : 

  1. Field Axioms
  2. Order Axioms
  3. Completeness Axiom

Field Axioms : The set R    is represented as a field (R, +, .)     where +    and .   are the binary operations of addition and multiplication respectively. It consists of 4 axioms for addition and multiplication each and one distributive law.

(i) Axioms for addition : 

  • a+b = b+a \ ∀ \ a,b \ ∈ R
  • (a+b)+c = a+(b+c) \ ∀ \ a,b,c \ ∈ R
  • R contains an element 0 such that a + 0 = a \ ∀ \ a ∈ R
  • For each a ∈ R    there corresponds an element -a ∈ R    such that a+(-a) = 0

(ii) Axioms for multiplication :

  • ab = ba \ ∀\ a,b  ∈ R
  • (ab)c = a(bc) \ ∀ \ a,b,c  ∈ R
  • R    contains an element 1    such that  1.a = a \ ∀ \ a ∈ R    and  1 ≠ 0
  • If a ∈ R \ and \ a≠0    then there exists an element \frac{1}{a} ∈ R     such that a. (\frac{1}{a} ) = 1

(iii) The distributive law :

  • a(b+c) = ab+ac \ ∀ \ a,b,c ∈ R

Order Axioms : We define >    (Greater Than) as the order relation, and it satisfies the following axioms –

  • Law of Trichotomy – For a,b∈ R    only one of the expressions can be true : a>b , a=b , b>a
  • Transitivity – For a,b,c∈R \ a>b,\ b>c ⇒ a>c
  • Monotone Property for addition – For a,b,c∈R, \ a>b ⇒ a+c > b+c
  • Monotone Property for multiplication – For a,b,c∈R,\ a>b, c>0 ⇒ ac > bc

We call >   linear order and R    is called a linearly ordered field.

Before defining the Completeness Axiom, we shall look at the concept of Boundedness. Here, we shall define a few terms before stating the Completeness Axiom.

Aggregate : Any non empty subset, say A   , of R   is known as an aggregate. For example, the set Z^+   is an aggregate. Similarly, the set B = {1,2,4,8} is also an aggregate since B ⊆ R   But, the set A = {x,y,z} and the empty set ∅   are not aggregates.

Upper bound : A subset S   of R   is said to be bounded above if ∃ \ k_1\ ∈ R   such that x ∈ S ⇒ x \leq k_1  . This number k_1   is called an upper bound of S  . For example, the set R^-    of negative real numbers is bounded above and 0   is an upper bound. Similarly, the set Z^-    of negative integers is bounded above and -1   is the upper bound. But, the set R^+   of positive real numbers is not bounded above.

Lower Bound : A subset S   of R   is said to be bounded below if ∃ \ k_2\ ∈ R   such that x ∈ S ⇒ x \geq k_2   This number k_2   is called a lower bound of S. For example, the set R^+   is bounded below and 0   is a lower bound. Similarly, the set Z^+   is bounded below and 1   is the upper bound. But, the set R^-   is not bounded below.

Least Upper Bound : Consider an upper bound u   of an aggregate S   and any real number less than u   is not an upper bound of S  , then we say u   is the least upper bound(lub) or supremum(sup) of S.

Greatest Lower bound : Consider a lower bound v   of an aggregate S   and any real number greater than v   is not a lower bound of S  , then we say v   is the greatest lower bound(glb) or infimum(inf) of S.

Example : Let S = [0,1]  . For S, we see that 1 is an upper bound and any number less than 1 is not an upper bound of S, hence, 1 is supremum of S. Also, 0 is a lower bound and any number greater than 0 is not a lower bound, so, 0 is infimum of S.

Boundedness : An aggregate S is bounded if it is both bounded above and bounded below. That is, it must have both an upper bound and a lower bound. For example, any finite set is bounded, the empty set ∅   is bounded. But, the sets Q   and R   are not bounded.

Note : An aggregate need not have a greatest and a least member to be bounded above or bounded below respectively.

Now being done with the required definition, we state the Completeness Axiom(also called the least upper bound axiom).

” Every non-empty set of Real numbers which is bounded above has a supremum.”

The set R satisfies the Field Axioms, Order Axioms, and the Completeness Axiom. Hence the set of real numbers R   is called a complete ordered field.

Also, the set of rational numbers, Q   does not satisfy the completeness axiom. Hence, Q   is not a complete field.

The completeness axiom is a really fundamental and important property of real number systems, as proofs various theorems of calculus, the concepts of maxima and minima, mean-value theorems etc. rely on the completeness property of real numbers.



Similar Reads

Peano Axioms | Number System | Discrete Mathematics
Introduction :The set of natural numbers is axiomatically defined below. G. Peano, an Italian mathematician, and J. W. R. Dedekind, a German mathematician, are credited with these axioms. The purpose of these axioms is to prove the existence of one natural number before defining a function to create the remaining natural numbers, known as the succe
4 min read
Convert the number from Indian system to International system
Given an input string N consisting of numerals and separators (, ) in the Indian Numeric System, the task is to print the string after placing separators(, ) based on International Numeric System. Examples: Input: N = "12, 34, 56, 789" Output: 123, 456, 789 Input: N = "90, 05, 00, 00, 000" Output: 90, 050, 000, 000 Approach: Remove all the separato
5 min read
Convert the number from International system to Indian system
Given string str which represents a number with separators(,) in the International number system, the task is to convert this string representation into the Indian Numeric System. Examples: Input: str = "123,456,789" Output: 12,34,56,789 Explanation: The given string represents a number in the international system. It is converted to the Indian sys
13 min read
Find the real and imaginary part of a Complex number
Given a complex number Z, the task is to determine the real and imaginary parts of this complex number.Examples: Input: z = 3 + 4i Output: Real part: 3, Imaginary part: 4Input: z = 6 – 8i Output: Real part: 6, Imaginary part: 8 Approach: A complex number can be represented as Z = x + yi, where x is real part and y is imaginary. We will follow the b
5 min read
Converting a Real Number (between 0 and 1) to Binary String
Given a real number between 0 and 1 (e.g., 0.72) that is passed in as a double, print the binary representation. If the number cannot be represented accurately in binary with at most 32 characters, print" ERROR:' Examples: Input : (0.625)10 Output : (0.101)2 Input : (0.72)10 Output : ERROR Solution: First, let's start off by asking ourselves what a
12 min read
Calculating n-th real root using binary search
Given two number x and n, find n-th root of x. Examples: Input : 5 2Output : 2.2360679768025875 Input : x = 5, n = 3Output : 1.70997594668 In order to calculate nth root of a number, we can use the following procedure. If x lies in the range [0, 1) then we set the lower limit low = x and upper limit high = 1, because for this range of numbers the n
6 min read
Real time optimized KMP Algorithm for Pattern Searching
In the article, we have already discussed the KMP algorithm for pattern searching. In this article, a real-time optimized KMP algorithm is discussed. From the previous article, it is known that KMP(a.k.a. Knuth-Morris-Pratt) algorithm preprocesses the pattern P and constructs a failure function F(also called as lps[]) to store the length of the lon
7 min read
Count triplets from an array which can form quadratic equations with real roots
Given an array arr[] consisting of N distinct positive integers, the task is to find the count of triplets (a, b, c) such that the quadratic equation aX2 + bX + c = 0 has real roots. Examples: Input: arr[] = { 2, 3, 6, 8 }Output: 6Explanation:For the triplets (a = 2, b = 6, c = 3), (a = 3, b = 6, c = 2), (a = 2, b = 8, c = 3), (a = 3, b = 8, c = 2)
15 min read
Minimum replacements with real numbers required to make given Array AP
Given an array arr[] of N integers. The task is to convert the array into an Arithmetic Progression with the minimum number of replacements possible. In one replacement any one element can be replaced by any real number. Examples: Input: N = 6, arr[] = { 3, -2, 4, -1, -4, 0 }Output: 3Explanation: Change arr[0] = -2.5, arr[2] = -1.5, arr[4] = -0.5So
7 min read
Real-life Applications of Data Structures and Algorithms (DSA)
You may have heard that DSA is primarily used in the field of computer science. Although DSA is most commonly used in the computing field, its application is not restricted to it. The concept of DSA can also be found in everyday life. Here we'll address the common concept of DSA that we use in our day-to-day lives. Application of DataStructure Appl
10 min read
Practice Tags :