Rational, Irrational & Recurring Decimals
Rational or Irrational?
Sort each number into the correct column.
Match Fraction to Decimal
Draw a line to match each fraction to its decimal form.
Terminating or Recurring?
Sort each decimal into the correct column.
Estimate Irrational Numbers
Circle the best estimate.
√2 is approximately
√3 is approximately
√5 is approximately
π is approximately
Identify Recurring Decimal Notation
Circle the correct answer.
Which is a recurring decimal?
1/7 = 0.142857142857... How many digits recur?
0.333... expressed as a fraction:
Number Type Vocabulary
Draw a line to match each term to its correct definition.
Place Surds on the Number Line
Each row gives approximate decimal values. Write the values in order from smallest to largest.
Order on the Number Line
Sort each set of numbers from smallest to largest.
Recurring Decimals to Fractions
Draw a line to match each recurring decimal to its fraction equivalent.
Compare Irrational Numbers
Circle the larger value in each pair.
√2 ≈ 1.414 or √3 ≈ 1.732
π ≈ 3.142 or √10 ≈ 3.162
√5 ≈ 2.236 or 2.3
√8 ≈ 2.828 or √7 ≈ 2.646
Between Which Two Integers?
Sort each surd into the correct pair of consecutive integers it lies between.
Decimal Expansion Quick Check
Use a calculator. Circle the correct decimal expansion.
1/9 =
5/11 =
7/8 =
1/12 =
Explain Your Reasoning
Answer in complete sentences.
Explain why √2 is irrational. What does irrational mean?
Is 0.999... equal to 1? Explain your thinking.
Converting Recurring Decimals to Fractions
Show full working for each conversion.
Let x = 0.777... Multiply both sides by 10. Subtract x from 10x to find x as a fraction.
Use the same method to convert 0.363636... to a fraction. (Hint: multiply by 100.)
Fractions — Terminating or Recurring?
A fraction terminates if, after simplifying, the denominator has only 2 and/or 5 as prime factors. Sort each fraction.
Proof that 0.999... = 1
Follow these steps to show that 0.999... equals exactly 1.
Let x = 0.999... Write 10x. Then subtract: 10x − x. What does 9x equal? Solve for x.
Does this prove 0.999... = 1? Explain in one or two sentences why mathematicians accept this result.
Locate on the Number Line
Choose the correct position description for each irrational number.
√10 is located between:
√20 is located between:
√50 is located between:
√2 is located between:
Ordering Mixed Number Types
Write each set in ascending order (smallest to largest). Show any conversions.
Order: √7, 2.5, 5/2, √6, 2.6
Order: π, 22/7, 3.1, √10, 3.15
Order: √18, 4.2, 4¼, √17, 4.1
Match to the Number Set
Sort each number into the most specific set it belongs to.
Real Number Line Density
Answer each question about the real number line.
Between any two rationals, there is:
Between 1 and 2 there are:
The set of irrationals is:
Find a Rational Between Two Surds
Find a rational number that lies strictly between each pair. Show your method.
Find a rational between √2 (≈ 1.414) and √3 (≈ 1.732). Write it as both a decimal and a fraction.
Find a rational between √5 (≈ 2.236) and √6 (≈ 2.449). Write it as both a decimal and a fraction.
Reverse: Fraction to Recurring Decimal
Divide to find the decimal expansion. Label each as terminating or recurring.
Perform the long division 5 ÷ 11. Write the decimal expansion and identify the repeating block.
Perform the long division 7 ÷ 12. Write the decimal expansion and identify the repeating block.
Which Conversion is Correct?
Check each working and circle whether it is correct or incorrect.
0.222...: 10x = 2.222..., 9x = 2, x = 2/9. Is this correct?
0.12121212...: 100x = 12.1212..., 99x = 12, x = 12/99 = 4/33. Is this correct?
0.15: This terminates, so x = 15/100 = 3/20. Is this correct?
Diagonal of a Square
Use Pythagoras' theorem (a² + b² = c²) to find the diagonal of each square.
A square has side length 5 cm. Find the exact length of its diagonal (leave as a surd). Then give a decimal approximation to 2 decimal places.
A square has side length 3 m. Find its diagonal exactly and as a decimal.
Is the Sum Rational or Irrational?
Adding two irrationals can give a rational! Sort each sum by whether the result is rational or irrational.
Products of Irrationals
Is the product rational or irrational?
√2 × √2 =
√3 × √3 =
√2 × √3 =
π × (1/π) =
Simplify the Surd
Draw a line to match each surd to its simplified form.
Simplify Surds
Circle the correctly simplified form.
√8 =
√18 =
√50 =
√75 =
√48 =
Surd Arithmetic
Add or subtract surds by treating them like like terms. Show all working.
Simplify: 3√2 + 5√2
Simplify: 7√3 − 2√3
Simplify: √8 + √2 (hint: simplify √8 first)
Simplify: √18 − √8 (hint: simplify both surds first)
Rationalising the Denominator (Introduction)
Multiply the numerator and denominator by the surd to remove it from the denominator.
Simplify 1/√2 by multiplying top and bottom by √2. What do you get?
Simplify 3/√3 by multiplying top and bottom by √3. What do you get?
Simplify 6/√2. Show all working.
Rationalise the Denominator
Circle the rationalised form.
1/√5 =
4/√2 =
√3/√6 =
Problem Solving with Surds
Use surd arithmetic to solve each problem. Give exact (surd) answers where possible.
A square has area 50 cm². Find the exact side length in simplified surd form, then round to 2 decimal places.
A right triangle has legs √3 cm and √6 cm. Find the exact hypotenuse length.
Simplify: √72 + √18 − √8. Write your final answer in the form a√2.
Square Root of a Fraction
Simplify each square root of a fraction.
√(4/9) =
√(25/36) =
√(1/4) =
√(3/4) =
Proof: √2 is Irrational (by Contradiction)
Work through the classic proof step by step.
Assume √2 = p/q in lowest terms (no common factors). Square both sides. Show that p² = 2q². What does this say about p? (Hint: if p² is even, then p is even — why?)
Write p = 2k. Substitute into p² = 2q². Show that q² = 2k², meaning q is also even. Explain why this contradicts our assumption and what conclusion we reach.
Nested Surds
Simplify each nested (surd within a surd) expression. Show all working.
Simplify: √(√16). First evaluate √16, then take the square root of the result.
Simplify: √(2√8 + √18). Start by simplifying √8 and √18, then combine under the outer root.
Advanced Surd Expressions
Expand and simplify each expression.
(1 + √2)² =
(√5 + √3)(√5 − √3) =
(2√3)² =
(√7 + 1)(√7 − 1) =
Surds in Quadratic-Style Expressions
Expand and simplify each expression. Use (a + b)² = a² + 2ab + b² or (a + b)(a − b) = a² − b².
Expand and simplify: (√3 + 2)²
Expand and simplify: (√5 + √2)(√5 − √2)
Expand and simplify: (2 + √3)(4 − √3)
Rationalising Two-Term Denominators
Multiply by the conjugate to rationalise each denominator. Show full working.
Simplify: 1/(√3 + 1) by multiplying by (√3 − 1)/(√3 − 1). Expand both numerator and denominator.
Simplify: 2/(√5 − √2) by multiplying by (√5 + √2)/(√5 + √2).
Ordering Surds — Advanced
Order each group of irrational numbers from smallest to largest without using a calculator. Show your reasoning.
Connecting Rational Numbers to Decimal Patterns
Investigate the decimal expansions of ninths.
Find the decimal expansions of 1/9, 2/9, 3/9, ..., 8/9 using a calculator. Describe the pattern you notice.
Predict the decimal expansion of 1/99 and 1/999 without using a calculator. Then check. Explain why the pattern works.
Real-World Investigation: Pi
Use a calculator and real measurements.
Measure the circumference and diameter of 5 circular objects at home (cups, plates, tins). For each, calculate C ÷ d. Record all results in a table. How close is each to 3.14159?
Calculate the average of your C ÷ d results. What is the percentage error from the true value of π? Why might your measurements be slightly off?
Irrational Number Hunt
Explore irrational numbers in everyday life.
- 1Use a calculator to find √2, √3, √5, √6, √7 and √8. Write down the first 10 decimal places of each. Do any repeat? Do any terminate?
- 2Measure the circumference and diameter of three round objects (plate, cup, tin). Divide circumference by diameter each time. How close to π do you get? Calculate your percentage error.
- 3Research the golden ratio φ ≈ 1.618... Find two real-life examples where it appears (art, nature, architecture). Verify that φ satisfies φ² = φ + 1 using your calculator.