Number Systems Notes
This chapter extends the idea of number from natural numbers to real numbers. It covers rational and irrational numbers, representing numbers on the number line, decimal expansions, operations on real numbers, laws of exponents for real bases, and rationalising denominators.
- 3 min read
- 12 practice questions
- Aligned to CBSE 2025–26 syllabus
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- Updated Sept 2026
Use this map for a first read, then return to the checks and formulas during revision.
- 1.Rational and Irrational Numbers
- 2.Decimal Expansions
- 3.Operations on Real Numbers and Rationalising
- 4.Laws of Exponents for Real Numbers
- 3 key formulas
- 2 concept checks
Rational and Irrational Numbers
A rational number can be written as p/q where p and q are integers and q ≠ 0. Between any two rational numbers there are infinitely many rational numbers, so the rationals are 'dense'.
An irrational number cannot be written in the form p/q. Examples are √2, √3, √15, π, and 0.101001000100001… (a non-terminating, non-recurring decimal). The square root of any positive integer that is not a perfect square is irrational.
Rational and irrational numbers together make up the real numbers, and every real number corresponds to exactly one point on the number line, and vice versa.
Decimal Expansions
The decimal expansion of a rational number is either terminating (e.g. 7/8 = 0.875) or non-terminating recurring (e.g. 1/3 = 0.333… = 0.3̄).
A rational number p/q in lowest terms has a terminating decimal expansion exactly when the prime factorisation of q contains only 2s and/or 5s. Otherwise the expansion is non-terminating recurring.
The decimal expansion of an irrational number is non-terminating and non-recurring. To convert a recurring decimal like x = 0.6̄ to a fraction: 10x = 6.6̄, so 10x − x = 6, giving x = 6/9 = 2/3.
Operations on Real Numbers and Rationalising
The sum, difference and product of two rational numbers are rational. Their quotient is rational when the divisor is non-zero. The sum of a rational and an irrational number is irrational. Their product is irrational only when the rational factor is non-zero: 5 times the square root of 2 is irrational, but zero times any irrational number is zero. The sum of two irrational numbers can be rational or irrational.
To rationalise a denominator of the form 1/√a, multiply numerator and denominator by √a. To rationalise 1/(a + √b), multiply by the conjugate (a − √b), using (a + √b)(a − √b) = a² − b.
These identities are used constantly: (√a)² = a for a ≥ 0, √(ab) = √a·√b, and √a/√b = √(a/b) for a ≥ 0, b > 0.
Laws of Exponents for Real Numbers
For a positive real number a and rational numbers m and n, the exponent laws hold: aᵐ · aⁿ = aᵐ⁺ⁿ, (aᵐ)ⁿ = aᵐⁿ, aᵐ / aⁿ = aᵐ⁻ⁿ, and aᵐ · bᵐ = (ab)ᵐ.
Rational exponents extend roots: a^(1/n) means the positive nth root of a, and a^(m/n) = (a^(1/n))ᵐ = (aᵐ)^(1/n).
For example, 8^(2/3) = (8^(1/3))² = 2² = 4, and 5^(1/2) · 5^(1/2) = 5^(1) = 5.
Number Systems: revision sheet
Close the notes and explain each idea in your own words. Then check the formulas and attempt the examples before opening their solutions.
Recall checklist
- Rational and Irrational Numbers
- Decimal Expansions
- Operations on Real Numbers and Rationalising
- Laws of Exponents for Real Numbers
For each topic: state the rule, explain when it applies, give an example and identify one mistake to avoid. Return to any topic you cannot explain without looking.
Formula recap (3)
Rational and Irrational Numbers
Operations on Real Numbers and Rationalising
Laws of Exponents for Real Numbers
Explain your method
Write the given information, choose a method, show the steps, then check units and whether the answer makes sense.
Check worked solution 1
Check worked solution 2
Turn mistakes into revision
Record the question, the step that went wrong and the corrected reason. Try it again tomorrow without the solution, then revisit it after a week. A correct guess still needs an explanation.
Practice and Revision
Test your understanding with quick chapter-level practice.
Chapter Q&A
Is every real number rational?
No. Every rational number is real, but numbers like √2 and π are real and irrational. Real numbers are the union of rational and irrational numbers.
Are the square roots of all positive integers irrational?
No. √4 = 2, √9 = 3, √25 = 5 are rational. Only the square root of a positive integer that is not a perfect square is irrational.
What does it mean to rationalise a denominator?
It means to rewrite the fraction so that its denominator has no surd (irrational root). We multiply the numerator and denominator by a suitable factor (often the conjugate).
Why is 0.101001000100001… irrational?
Its decimal expansion never ends and never settles into a repeating block (the number of zeros keeps increasing). A decimal that is non-terminating and non-recurring represents an irrational number.