Number Systems – ICSE Class 10 Mathematics Notes & Guide

Number systems form the foundation of all higher mathematics. In this chapter we explore how numbers are represented, how they behave under operations, and how to work with irrational quantities efficiently.

Key concepts

Real numbers and their decimal expansions

A real number includes every point on the number line. Its decimal expansion can be terminating (e.g., 0.75), non‑terminating repeating (e.g., 0.333…), or non‑terminating non‑repeating (e.g., √2 ≈ 1.414213…). A terminating or repeating decimal can always be expressed as a fraction p/q where p and q are integers and q≠0.

Rational and irrational numbers

• Rational numbers are numbers that can be written as a ratio of two integers (p/q) with q≠0. Their decimal form either ends or repeats.
• Irrational numbers cannot be expressed as a ratio of integers. Their decimal expansions go on forever without any pattern. Examples: √2, π, e. The sum of a rational and an irrational number is always irrational. The product is irrational provided the rational factor is non‑zero (e.g., r·i is irrational for any non‑zero rational r), while the product is rational if the rational factor is zero. The sum of two irrationals may be rational (e.g., √2 + (2‑√2) = 2).

Laws of exponents and indices

For any non‑zero real numbers a and b and integers m, n:

• a⁰ = 1
• a¹ = a
• aᵐ·aⁿ = a^(m+n)
• aᵐ ÷ aⁿ = a^(m‑n)
• (aᵐ)ⁿ = a^(m·n)
• (ab)ⁿ = aⁿ·bⁿ
• (a/b)ⁿ = aⁿ ÷ bⁿ (b≠0)

These laws also hold when a or b are surds, provided the radicand is positive.

Scientific notation

Large or very small numbers are written as: N = a × 10^k, where 1 ≤ a < 10 and k is an integer. This form simplifies multiplication, division and comparison of extreme values. Example: 3 200 000 = 3.2 × 10⁶.

Operations on surds

A surd is an irrational root that cannot be simplified to a rational number, e.g., √2, ∛5. Important operations:

• Addition/Subtraction: Only like surds (same radicand) can be combined: √3 + 2√3 = 3√3.
• Multiplication: √a × √b = √(ab). For powers, (√a)² = a.
• Rationalising the denominator: Multiply numerator and denominator by a suitable surd to eliminate radicals from the denominator, e.g., 1/(√2) = √2/2.
• Simplifying: Extract perfect squares/cubes from the radicand: √50 = √(25·2) = 5√2.

Important definitions and formulas

Key exponent rules: a⁰=1, aᵐ·aⁿ=a^(m+n), aᵐ÷aⁿ=a^(m‑n), (aᵐ)ⁿ=a^(m·n).

Surd simplification: √(ab) = √a·√b, √(a²b)=a√b (a≥0).

Solved examples

Example 1 – Converting a repeating decimal to a fraction

Find the fraction form of 0.7̅ (0.777…). Solution:

1. Let x = 0.777…
2. Multiply by 10 (since one digit repeats): 10x = 7.777…
3. Subtract the original equation: 10x – x = 7.777… – 0.777… → 9x = 7
4. Hence x = 7/9.

Therefore, 0.7̅ = 7/9.

Example 2 – Rationalising a denominator

Simplify frac52+√3 . Solution:

1. Multiply numerator and denominator by the conjugate (2‑√3): frac52+√3 × frac2-√32-√3
2. Numerator: 5(2‑√3) = 10 – 5√3.
3. Denominator: (2+√3)(2‑√3) = 2² – (√3)² = 4 – 3 = 1.
4. Result = 10 – 5√3.

Thus frac52+√3 = 10 - 5√3 .

Example 3 – Using exponent laws with surds

Evaluate (√2)⁴ × (2)⁻¹ . Solution:

1. (√2)⁴ = (2^(1/2))⁴ = 2^((1/2)·4) = 2² = 4.
2. 2⁻¹ = 1/2.
3. Multiply: 4 × 1/2 = 2.

Hence the expression equals 2.

Common mistakes to avoid

• Treating a non‑terminating non‑repeating decimal as rational.
• Adding unlike surds (e.g., √2 + √3) as if they were like terms.
• Forgetting to multiply both numerator and denominator when rationalising.
• Misapplying exponent rules with negative or fractional powers, especially mixing bases.
• Ignoring the condition 1 ≤ a < 10 in scientific notation; writing 0.32 × 10⁶ is incorrect.

Exam tips

• Always check if a decimal terminates or repeats before converting to a fraction.
• Write the conjugate clearly when rationalising; it saves time and avoids sign errors.
• Keep exponent rules handy; a quick rewrite often simplifies a complex expression.
• In scientific notation, adjust the coefficient to lie between 1 and 10 before finalising the answer.
• Practice simplifying surds by extracting the largest perfect square/cube first.

Frequently asked questions

How can I tell if a decimal is rational?

If the decimal terminates after a finite number of places or a block of digits repeats indefinitely, it is rational. Otherwise it is irrational.

Why do we need to rationalise the denominator?

Rationalising removes surds from the denominator, making the expression easier to compare, add, or multiply with other numbers and matches the standard form required in exams.

Can the exponent rules be used with negative bases?

Yes, but you must be careful with odd and even powers. For example, (‑2)³ = ‑8, while (‑2)² = 4. Fractional exponents with negative bases are not real numbers.