Grade 10 essential mathematics Numbers and Algebra – Real Numbers Notes

This note introduces the real numbers, how to classify them as rational or irrational, how to carry out combined operations on rational numbers, find reciprocals (including using a calculator), and apply these ideas to Kenyan real-life situations (money, distance, temperature, maps, commerce).

• (a) Identify and outline sub-sub-strands:
  • Operations on rational numbers
  • Reciprocal of numbers
  • Application of integers
• (b) Classify real numbers as rational or irrational.
• (c) Perform combined operations on rational numbers (fractions, decimals, negatives).
• (d) Determine the reciprocal of real numbers using calculators.
• (e) Apply rational numbers in real-life Kenyan situations (money, measurements, maps, weather).
• (f) Appreciate the use of real numbers in real life (construction, trade, science).

Real numbers are all numbers that can be found on the number line — they include both rational and irrational numbers.

Rational numbers: numbers that can be written as a fraction a/b where a and b are integers and b ≠ 0. Examples: 3, -2, 1/4, 0.75, -5/3.

Irrational numbers: numbers that cannot be written as a ratio of two integers. Their decimal expansions are non-terminating and non-repeating. Examples: √2 ≈ 1.4142135..., π ≈ 3.14159...

Use this quick checklist:

• If a number can be written exactly as a fraction a/b with integers a and b, it is rational.
• If it cannot be written as such and its decimal is non-terminating non-repeating, it is irrational.

Examples:

• 4 = 4/1 → rational
• 0.333... = 1/3 → rational
• √9 = 3 → rational (though root symbol, result integer)
• √2 → irrational
• π → irrational

Order of operations: Use BODMAS/BIDMAS — Brackets, Orders (powers/roots), Division and Multiplication (left to right), Addition and Subtraction (left to right).

Example 1 — fractions + mixed operations:

Example 2 — decimals & negatives:

The reciprocal of a non-zero number x is 1/x. Multiplying a number by its reciprocal gives 1.

• Reciprocal of 5 is 1/5 = 0.2
• Reciprocal of −3/4 is −4/3
• Reciprocal of 0.25 is 1/0.25 = 4
• Reciprocal of 0 is undefined (division by zero).

Using a calculator:

1. Scientific calculator: key in the number, then press the 1/x (x⁻¹) key. Example: press 2 then 1/x → gives 0.5.
2. Basic calculator without 1/x key: use 1 ÷ number. Example: press 1 ÷ 2 = 0.5.
3. For fractions: convert to decimal or use fraction mode if available. Example: for −3/4 → enter −0.75, then press 1/x → −1.333333... (which is −4/3).
4. For irrational numbers (e.g. √2): press √2 key (or enter approximate value 1.4142136), then press 1/x to get approximate reciprocal ≈ 0.7071068.

• Money: giving change (KES), calculating discounts in a market, splitting profits among farmers — fractions and decimals are common.
• Measurements & maps: scales on maps (1:50 000), converting km to m, using fractions for land measurements (hectares, acres).
• Temperature / altitude: temperatures above/below zero or heights above/below sea level use integers (negative and positive integers).
• Cooking & recipes: measuring ingredients as fractions (½ cup, 3/4 kg).
• Speed and time: average speed and fuel consumption often involve division and fractions.
• Construction: cutting timber to required fractional lengths; using decimals for precise measurements.

Short Kenyan example: A farmer sells 7.5 kg of maize at KES 53.50 per kg. Total revenue = 7.5 × 53.50 = KES 401.25 (use decimals and multiplication of rationals).

Real numbers allow precise description of quantities in daily life, commerce, engineering, and science. Whether planning a building, balancing accounts, reading a thermometer, or navigating with maps and GPS, real numbers are essential. Understanding their properties helps make reliable calculations and sound decisions.

1. Starter: Quick classification quiz — teacher calls numbers (e.g. 0.333..., √3, −5, 7/8), learners shout “rational” or “irrational” and explain why.
2. Group activity (market simulation): In groups, role-play traders and customers. Price goods with decimals and give change; calculate totals and discounts using rational arithmetic.
3. Calculator lab: Practice finding reciprocals. Students use their scientific calculators to find reciprocals of various numbers (integers, fractions, decimals, approximate irrationals) and record results and errors/rounding.
4. Map exercise: Using a local map scale (e.g. 1:50 000), compute real distances between towns using fractions and decimals; convert km to m and vice versa.
5. Problem-solving: Set combined operations problems (nested brackets, fractions, negatives) and solve step-by-step in pairs; exchange work for peer marking.
6. Real-life project: Each student finds one real-life Kenyan example where rational numbers are critical (e.g. cooking, construction, farming) and presents a 3–4 minute explanation including calculations.

1. Classify: State whether each is rational or irrational: 0.125, 0.101001000100001..., √16, π, −7/9.
2. Compute (show working): (5/6 + 1/4) × (−2/3).
3. Find the reciprocal (using calculator and exact form if possible): 0.2, −3/7, √5 (approx).
4. Application: A matatu travels 96 km using 12 litres of fuel. What is the fuel consumption per 100 km? (Give answer as a rational/decimal.)
5. Word problem: Farmer A sells 2/3 of his maize and has 8.4 tonnes left. How much did he start with?

• Formative: quick in-class quizzes on classification and reciprocals; observe calculator use to ensure understanding of 1/x vs 1 ÷ x.
• Summative: set a test with mixed fraction/decimal problems, classification, and a contextual problem (market or map) to test application skills.
• Differentiation: give simpler fraction operations for learners needing support; challenge fast learners with proofs that some decimals are repeating (convert to fraction) and tasks involving recurring decimals.