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What are the subsets of rational numbers?

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  • Last Updated : 23 May, 2022

A number is a fundamental concept in mathematics. Numbers are used to count, measure, maintain things in order, index, and so on. Natural numbers, whole numbers, rational and irrational numbers, integers, real numbers, complex numbers, even and odd numbers, and so on are examples of different sorts of numbers. We can use the fundamental arithmetic operations of numbers to calculate the resultant number. These numbers and words can be used to express these figures and words. For example numbers such as 25 and 80 stated as figures can also be written as twenty-five and Eighty.

Number system

A number system, often known as a numeral system, is a fundamental method for expressing numbers and figures. It is a distinct method of representing numbers in arithmetic and algebraic structures. Numbers are used in various arithmetic values that may be used to do various arithmetic operations such as addition, subtraction, multiplication, and so on that are utilized in daily life for the purpose of computation. The value of a number is defined by the digit, its position in the number, and the number system’s base.

Numbers, often known as numerals, are mathematical values that are used for counting, measuring, labeling, and quantifying fundamental quantities. Numbers are mathematical values that are used to measure or calculate quantities. It is represented by numbers such as 2,4,7, and so on.  Integers, whole numbers, natural numbers, rational and irrational numbers, etc all are examples of numbers and belong to the numbers system. 

What are the subsets of rational numbers?

Answer: 

First we will understand about the rational number,

Rational Numbers: Rational numbers are those that may be stated as the ratio of two integers. It contains all integers and can be represented in fractions or decimals. It is symbolised by the letter ‘Q.’ 

Rational numbers are of the type p/q, where p and q are integers and q ≠ 0. Most people have difficulty distinguishing between fractions and rational numbers due to the underlying structure of numbers, p/q form. When a rational number is split, the result is in decimal form, which can be either ending or repeating. 7, 8, 9, and so on are instances of rational numbers since they can be written in fraction form as 7/1, 8/1, and 9/1.

The subsets of Rational numbers are whole number, natural Number and integers.

  • Whole number: Whole Numbers are the natural numbers including the number 0. In mathematics, the set of whole numbers is symbolised by the symbol W and is given as 0, 1, 2, 3,… 
    Here are some whole number facts to help comprehend them better,
    • Natural numbers are all whole numbers.
    • Every counting number is a whole number.
    • Whole numbers are all positive integer along with zero.
    • Whole numbers are all real numbers.
  • Natural Numbers: Natural numbers are numbers that are used for counting and are a subset of real numbers. Only positive integers such as 1, 2, 3, 4, 5, 6,…, are included in the set of natural numbers. Natural numbers, often known as non-negative integers (All positive integers). Some instances are 23, 56, 78, 999, 100202, and so forth.
  • Integers: An integer is a number that does not have a decimal or fractional part and is taken from the set of negative and positive numbers, including zero. Integers include the numbers -6 -5, 0, 1, 5, 8, 97, and 3043, etc.
    • Positive Integers: Integers that are positive are those which are larger than zero. As an example, 1, 2, 3,…
    • Negative Integers: Integers that are less than zero are considered negative, -1, -2, -3, etc.

Sample Questions

Question 1: Is 4.56 a rational number or an irrational number?

Answer:

Rational numbers are those that may be stated as the ratio of two integers. It contains all integers and can be represented in fractions or decimals. Rational numbers are represented by Q. Rational numbers are of the type p/q, where p and q are integers and q ≠ 0. Most people have difficulty distinguishing between fractions and rational numbers due to the underlying structure of numbers, p/q form.

Here, the given number 4.56 can be expressed in the form of p/q as 4.56 = 456/100 = 228/50 = 114/25 and has terminating digits. Hence 4.56 is a rational number.

Question 2: Determine whether -8 is a rational number?

Answer:

Rational numbers are those that may be stated as the ratio of two integers. It contains all integers and can be represented in fractions or decimals.  Rational numbers are represented by Q. Rational numbers are of the type p/q, where p and q are integers and q ≠ 0. Most people have difficulty distinguishing between fractions and rational numbers due to the underlying structure of numbers, p/q form.

Here, the given number -8 is a negative integer hence it can be represented as -8/1 therefore its an integer as well as rational number.

Question 3: Is 18 a counting number or a rational number?

Answer:

Natural numbers or Counting numbers are numbers that are used for counting and are a subset of real numbers. Positive integers are only integers included in the set of natural numbers or counting numbers. Here 18 is a counting number an dit can be represented as in the form of p/q i,e 18/1.

Therefore it’s a counting number as well as rational number.

Question 4: What are the smallest three-digit and largest three-digit counting rational numbers?

Answer:

Natural numbers or Counting numbers are numbers that are used for counting and are a subset of real numbers. Positive integers are only integers included in the set of natural numbers or counting numbers.

  1. Smallest three digit counting Rational number is 100.
  2. Largest three digit counting rational number is 999.

Question 5: Determine if 0 is a rational and counting number?

Answer:

Here given 0 is not a counting number because Natural numbers or counting numbers are numbers that are used for counting and are a subset of real numbers. It doesnot include zero. Only positive integers are included in the set of natural numbers excluding zero . Natural numbers, often known as non-negative integers (all positive integers). On the other hand 0 is a rational number as it can be expressed in form of p/q such as 0/1, 0/2 , 0/3 etc …

Question 6: Determine whether 5.5555… is a rational number?

Answer:

Rational numbers are those that may be stated as the ratio of two integers. It contains all integers and can be represented in fractions or decimals.  Rational numbers are represented by Q. Rational numbers are of the type p/q, where p and q are integers and q ≠ 0 . Most people have difficulty distinguishing between fractions and rational numbers due to the underlying structure of numbers, p/q form.

Hence, 5.5555… is a rational number as it is recurring after decimal.

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