Rational Numbers

Understanding numbers that can be expressed as ratios (fractions).

Introduction to Rational Numbers

Rational numbers include integers and fractions. The word "rational" comes from the word "ratio". These numbers can be expressed as a ratio of two integers.

Definition and Notation

The set of rational numbers is denoted by the letter Q (from "Quotient").

A number 'r' is called a rational number if it can be written in the form p/q, where p and q are integers and q ≠ 0.

Examples

  • 1/2 is a rational number (p=1, q=2)
  • -3/4 is a rational number (p=-3, q=4)
  • 5 is a rational number because it can be written as 5/1
  • 0 is a rational number because it can be written as 0/1, 0/2, etc.

Decimal Expansion

Rational numbers have two types of decimal expansions:

  • Terminating: The division ends after a finite number of steps (remainder becomes zero).
    Example: 1/2 = 0.5, 7/8 = 0.875
  • Non-Terminating Repeating (Recurring): The division never ends, but digits repeat.
    Example: 1/3 = 0.333..., 1/7 = 0.142857142857...

Equivalent Rational Numbers

Rational numbers do not have a unique representation in the form p/q.
Example: 1/2 = 2/4 = 10/20 = 25/50.
However, when we say p/q is a rational number, we typically assume that p and q are Co-prime (they share no common factors other than 1).

Finding Rational Numbers Between Two Numbers

There are infinitely many rational numbers between any two given rational numbers.

Method 1 (Mean Method): Find (a+b)/2.

Method 2 (Equivalent Fractions): Make denominators same and pick numbers between numerators.

Practice Questions

Free Preview - 10 Questions

Test your understanding of Rational Numbers with these questions.

1 Which symbol denotes the set of rational numbers?
  • A R
  • B N
  • C Q
  • D Z
Explanation:
Rational numbers are denoted by Q, which stands for Quotient.
2 A rational number is defined as a number that can be expressed in the form p/q, where p and q are integers and:
  • A q = 0
  • B p = 0
  • C q ≠ 0
  • D p > q
Explanation:
The definition of a rational number requires the denominator q to be non-zero (division by zero is undefined).
3 Which of the following is NOT a rational number?
  • A 2/3
  • B -5
  • C 0
  • D √2
Explanation:
√2 cannot be expressed as a simple fraction p/q. Its decimal expansion is non-terminating and non-recurring. Hence, it is irrational.
4 The decimal representation of a rational number is:
  • A Always terminating
  • B Always non-terminating repeating
  • C Either terminating or non-terminating repeating
  • D Neither terminating nor repeating
Explanation:
Rational numbers have decimal expansions that either stop (terminate) or repeat a pattern endlessly (recurring).
5 Can 0 be written in the form p/q?
  • A Yes
  • B No
  • C Only if q = 1
  • D Undefined
Explanation:
Yes, 0 can be written as 0/1, 0/5, 0/-2, etc., where p=0 and q is any non-zero integer.
6 Find a rational number between 1 and 2.
  • A 1.1
  • B 1.5
  • C 3/2
  • D All of the above
Explanation:
1.1 (11/10), 1.5 (3/2), and 3/2 are all rational numbers lying between 1 and 2.
7 Which fraction yields a terminating decimal?
  • A 1/3
  • B 2/7
  • C 5/8
  • D 1/6
Explanation:
A fraction produces a terminating decimal if the prime factorization of the denominator contains only powers of 2 and/or 5. 8 = 2³, so 5/8 = 0.625 (terminating).
8 Every integer is a rational number. This statement is:
  • A True
  • B False
  • C True only for positive integers
  • D True only for zero
Explanation:
Any integer 'z' can be written as z/1, which fits the form p/q. So every integer is a rational number.
9 0.333... can be written as:
  • A 3/10
  • B 1/3
  • C 3/99
  • D 33/100
Explanation:
0.333... is a repeating decimal. Let x = 0.333..., then 10x = 3.333... Subtracting: 9x = 3, so x = 3/9 = 1/3.
10 The number 3.142678 is:
  • A Rational
  • B Irrational
  • C Whole Number
  • D Integer
Explanation:
Since the decimal expansion terminates (it has a finite number of digits), it is a rational number. It can be written as 3142678/1000000.

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