Real Numbers: Concepts, Properties, Rules and Formulas
Real Numbers include all numbers that can be represented on the number line. This topic covers natural, whole, integer, rational and irrational numbers, along with their properties, decimal expansions, ordering rules, absolute values and standard formulas used to solve quantitative aptitude questions.
What Are Real Numbers?
The set of real numbers is written as ℝ. Rational numbers can be written in the form p/q, where p and q are integers and q ≠ 0. Irrational numbers cannot be expressed in this form. Thus, ℝ = rational numbers ∪ irrational numbers. Examples of real numbers include -5, 0, 3/4, 2.6 and √2.
Real Numbers Formula & Tricks
Important Formulas
For positive integers a and b, q is the quotient and r is the remainder when a is divided by b.
For two positive integers, the product of their HCF and LCM equals the product of the numbers.
Absolute value gives the non-negative distance between two points on the number line.
A rational number has a terminating decimal expansion only when the denominator in lowest form contains no prime factors other than 2 and 5.
Quick Tricks
First reduce the fraction to its lowest form. Factor its denominator. If only 2 and 5 remain, the decimal terminates; otherwise, it is recurring.
A number farther to the right is greater. For negative numbers, the number with the smaller absolute value is greater.
When a question asks for the difference or distance between two numbers, subtract them and take the absolute value.
Real Numbers Concepts
Classification of Real Numbers
Natural numbers are {1, 2, 3, ...}. Whole numbers are {0, 1, 2, ...}. Integers include negative numbers, zero and positive numbers. Rational numbers have the form p/q, q ≠ 0. Irrational numbers include numbers such as √2, √3 and π. The set relationship is Natural ⊂ Whole ⊂ Integers ⊂ Rational ⊂ Real.
Rational and Irrational Numbers
The decimal expansion of a rational number is either terminating or non-terminating recurring. The decimal expansion of an irrational number is non-terminating and non-recurring. The sum of two rational numbers is rational, while the sum of a rational and an irrational number is irrational. The product of a non-zero rational number and an irrational number is irrational.
Properties of Real Numbers
For real numbers a, b and c: a + b and ab are real, showing closure. a + b = b + a and ab = ba, showing commutativity. (a + b) + c = a + (b + c) and (ab)c = a(bc), showing associativity. a(b + c) = ab + ac, showing distributivity. The additive identity is 0, and the multiplicative identity is 1. Every real number has an additive inverse; every non-zero real number has a multiplicative inverse.
Decimal Expansion of Rational Numbers
If the denominator is 2ᵐ5ⁿ, the decimal terminates after at most max(m, n) places when the fraction is converted to a denominator of a power of 10. If any prime factor other than 2 or 5 remains, the decimal is non-terminating recurring.
Ordering, Absolute Value and Intervals
For any real number x, |x| = x when x ≥ 0 and |x| = −x when x < 0. Therefore, |−9| = 9 and |9| = 9. The interval [a, b] includes both endpoints, while (a, b) excludes both endpoints. The inequality a < x ≤ b represents the interval (a, b].
Real Numbers Video Lessons
Watch short topic-wise lessons for quick revision.
Prime Numbers: Test and Find Them Fast
Learn how to test whether a number is prime and identify prime numbers efficiently using clear divisibility checks and practical methods.
Practice Real Numbers Questions
Practise published questions related to this topic.
Real Numbers Quick Quiz
Attempt 5 questions and check your score instantly.
Keep practising
Practice more Real Numbers questions in the PrepShots app and continue from your current topic.
Practice More Questions - Start ₹1 Trial →Quick Revision Notes
Real Numbers Quick Revision
Use these rules to revise classification, decimal expansions, properties and calculations involving real numbers.
- Real numbers consist of rational and irrational numbers.
- Natural ⊂ Whole ⊂ Integers ⊂ Rational ⊂ Real.
- A rational number has the form p/q, where p and q are integers and q ≠ 0.
- A rational decimal terminates only when the reduced denominator has prime factors 2 and/or 5.
- An irrational decimal is non-terminating and non-recurring.
- For two positive integers, HCF × LCM = product of the numbers.
- |x| is the distance of x from zero and is always non-negative.
- For real numbers, a(b + c) = ab + ac and a + 0 = a.
Real Numbers FAQs
Are all integers real numbers?
Yes. Every integer can be represented on the number line, so every integer is a real number. Integers are also rational because n = n/1.
Is zero a rational number?
Yes. Zero can be written as 0/1, so it is rational and real. It is a whole number and an integer, but it is not a natural number under the convention that natural numbers start from 1.
How can you determine whether a fraction has a terminating decimal?
Reduce the fraction first. Its decimal terminates if the reduced denominator contains only the prime factors 2 and 5; for example, 9/20 terminates because 20 = 2² × 5.
Is 0.333... rational or irrational?
It is rational because 0.333... = 1/3. Every repeating decimal represents a rational number.
Is the sum of two irrational numbers always irrational?
No. The sum can be rational or irrational. For example, √2 + (−√2) = 0, which is rational, while √2 + √3 is irrational.
What is the distance between −8 and 3?
The distance is |−8 − 3| = |−11| = 11.