Real Numbers System (Definition - Properties and Examples)

Real Numbers System (Definition, Properties and Examples)

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Real number system

Real numbers are the set of numbers containing all of the rational numbers and all of the irrational numbers. The real numbers are “all the numbers” on the number line. There are infinitely many real numbers just as there are infinitely many numbers in each of the other sets of numbers. However, it can be proved that the infinity of the real numbers is a bigger infinity. The "smaller", or countable infinity of the integers and rationales is sometimes called ℵ0(alef-naught), and the uncountable infinity of the reals is called ℵ1(alef-one). The set of real numbers also satisfies the closure property, the associative property, the commutative property, and the distributive property.

What are Real Numbers?

The real numbers are the set of numbers containing all of the rational numbers and all of the irrational numbers. The real numbers are “all the numbers” on the number line. Any number that we can think of, except complex numbers, is a real number. Real numbers include rational numbers like positive and negative integers, fractions, and irrational numbers. The set of real numbers, which is denoted by R, is the union of the set of rational numbers (Q) and the set of irrational numbers (Q). So, it can be written as the set of real numbers as, R = Q ∪ Q. This indicates that real numbers include natural numbers, whole numbers, integers, rational numbers, and irrational numbers. For example, 3, 0, 1.5, 3/2, √5, and so on are real numbers.

There are certain numbers that are not real numbers. The numbers that are neither rational nor irrational are non-real numbers, like, √-1, 2 + 3i, and -i. These numbers include the set of complex numbers, C.

What is types of Real Numbers?

Real numbers include rational numbers and irrational numbers. Thus, there does not exist any real number that is neither rational nor irrational. It simply means that if we pick up any number from R, it is either rational or irrational.

Rational Numbers

Any number which can be defined in the form of a fraction p/q is called a rational number. The numerator in the fraction is represented as 'p' and the denominator as 'q', where 'q' is not equal to zero. A rational number can be a natural number, a whole number, a decimal, or an integer. For example, 1/2, -2/3, 0.5, 0.333 are rational numbers.

Irrational Numbers

Irrational numbers are the set of real numbers that cannot be expressed in the form of a fraction p/q where 'p' and 'q' are integers and the denominator 'q' is not equal to zero (q≠0.). For example, π (pi) is an irrational number. π = 3.14159265...In this case, the decimal value never ends at any point. Therefore, numbers like √2, -√7, and so on are irrational numbers.

Symbol of Real Numbers

Real numbers are represented by the symbol R. Here is a list of the symbols of the other types of numbers.

N - Natural numbers

W - Whole numbers

Z - Integers

Q - Rational numbers

-Q- Irrational numbers

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Difference Between Real Numbers and Integers

Scope of Real Numbers and Integers

Real numbers include integers, rational, irrational, natural, and whole numbers. On the other hand, integers’ scope is mainly concerned with whole numbers which are negative and positive. Hence, real numbers are more general.

Fractions

Real numbers can include fractions such as rational and irrational numbers. However, fractions cannot be integers.

Countable

As a set, real numbers are uncountable while integers are countable.

Field

Real numbers are a kind of field which is an essential algebraic structure where arithmetic processes are defined. On the contrary, integers are not considered as a field.

Archimedean Property

The Archimedean Property, which is the assumption that there is a natural number that is equal to or greater than any real number, can be applied to real numbers. On the contrary, the Archimedean Property cannot be applied to integers.

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What are the Subsets of Real Numbers?

All numbers except complex numbers are real numbers. Therefore, real numbers have the following five subsets:

Natural numbers: All positive counting numbers make the set of natural numbers, N = {1, 2, 3, ...}

Whole numbers: The set of natural numbers along with 0 represents the set of whole numbers. W = {0, 1, 2, 3, …}

Integers: All positive counting numbers, negative numbers, and zero make up the set of integers. Z = {..., -3, -2, -1, 0, 1, 2, 3, ...}

Rational numbers: Numbers that can be written in the form of a fraction p/q, where 'p' and 'q' are integers and 'q' is not equal to zero are rational numbers. Q = {-3, 0, -6, 5/6, 3.23}

Irrational numbers: The numbers that are square roots of positive rational numbers, cube roots of rational numbers, etc., such as √2, come under the set of irrational numbers. (¯Q) = {√2, -√6}

Among these sets, the sets N, W, and Z are the subsets of Q.