Rational & Irrational Numbers
Rational or Irrational?
Sort each number into the correct column.
Surd Approximations
Draw a line from each surd to its best decimal approximation (to 2 decimal places).
Classify the Number
Circle the correct classification for each number.
0.666...
sqrt(16)
sqrt(11)
pi / pi
Simplifying Surds
Simplify each surd by finding the largest perfect-square factor. Example: sqrt(12) = sqrt(4x3) = 2sqrt(3)
sqrt(18) =
sqrt(50) =
sqrt(72) =
sqrt(200) =
Estimating Surds
Without a calculator, circle the two consecutive integers that each surd lies between.
sqrt(20)
sqrt(50)
sqrt(90)
sqrt(150)
Circle Problems with Pi
Use pi approx 3.14 to solve these problems. Show all working.
A circle has a radius of 7 cm. Calculate its circumference (C = 2 pi r).
A circle has a diameter of 10 cm. Calculate its area (A = pi r squared).
A pizza has a circumference of 62.8 cm. What is its radius?
Ordering Surds
Order these surds from smallest to largest. Use: sqrt(2) approx 1.41, sqrt(3) approx 1.73, sqrt(5) approx 2.24, sqrt(8) approx 2.83.
Write sqrt(8), sqrt(2), sqrt(5), sqrt(3) in order from smallest to largest:
Real-World Surds Challenge
Answer these questions using your knowledge of irrational numbers.
A square paddock has an area of 50 m squared. What is the exact length of one side? Write as a simplified surd.
Explain in your own words why sqrt(2) is irrational.
Simplifying Surds -- Adding and Subtracting
Simplify by collecting like surds. Example: 3 sqrt(2) + 5 sqrt(2) = 8 sqrt(2).
3 sqrt(5) + 2 sqrt(5) =
7 sqrt(3) - 4 sqrt(3) =
sqrt(8) + sqrt(2) = (Hint: simplify sqrt(8) first)
sqrt(50) - sqrt(18) = (Hint: simplify both surds first)
Multiplying Surds
Use sqrt(a) x sqrt(b) = sqrt(ab) to simplify each product. Simplify fully.
sqrt(3) x sqrt(3) =
sqrt(2) x sqrt(8) =
3 sqrt(5) x 2 sqrt(5) =
sqrt(6) x sqrt(6) =
Real Number Classification
Sort each number into the MOST specific category it belongs to. (Every natural number is also an integer, rational, and real -- place it in 'Natural Numbers'.)
Which Simplification Is Correct?
Circle the correct simplified form of each surd.
sqrt(8)
sqrt(75)
sqrt(12)
sqrt(45)
Surds and Irrational Numbers at Home
Explore irrational numbers in everyday life.
- 1Measure the diagonal of a square tile using a ruler. Calculate what the exact diagonal should be using surds. How close is your measurement?
- 2Look up the first 20 decimal places of pi online. Explain to a family member why pi is irrational and how it differs from 22/7.
- 3Use a calculator to check that sqrt(2) x sqrt(2) = 2 exactly. Now try: is sqrt(2) + sqrt(3) equal to sqrt(5)? Why or why not?
Simplify Larger Surds
Simplify each surd fully. Show the factor step clearly.
sqrt(98) =
sqrt(128) =
sqrt(180) =
sqrt(300) =
Surds on the Number Line
Estimate the position of each surd on the number line below. Mark with an X and write the surd label.
Number line from 0 to 5. Mark: sqrt(2) approx 1.41, sqrt(3) approx 1.73, sqrt(5) approx 2.24, sqrt(7) approx 2.65, sqrt(10) approx 3.16, sqrt(15) approx 3.87, sqrt(20) approx 4.47.
Between which two surds does sqrt(6) lie? Estimate sqrt(6) to 1 decimal place.
Simplifying Surds with Coefficients
Simplify each expression fully. Coefficients outside the surd multiply with the simplified surd.
2 sqrt(18) = 2 x ___ =
3 sqrt(50) = 3 x ___ =
5 sqrt(12) = 5 x ___ =
4 sqrt(75) = 4 x ___ =
Adding Unlike Surds
Simplify each expression. First simplify any surds that can be simplified, then collect like surds.
sqrt(12) + sqrt(27) =
sqrt(20) + sqrt(45) =
sqrt(32) - sqrt(8) =
sqrt(75) + sqrt(48) - sqrt(12) =
Multiplying Surds with Integers
Expand and simplify each expression. Multiply surd parts together and simplify.
sqrt(5) x sqrt(20) =
sqrt(7) x sqrt(7) =
2 sqrt(3) x 3 sqrt(12) =
sqrt(6) x sqrt(24) =
Squaring Expressions with Surds
Expand and simplify each expression by squaring.
(sqrt(5))^2 =
(3 sqrt(2))^2 =
(sqrt(7) + 1)^2 =
(sqrt(3) - sqrt(2))^2 =
Rationalising Simple Denominators
Rationalise the denominator of each fraction by multiplying by the surd / surd.
1 / sqrt(3) = (multiply by sqrt(3)/sqrt(3)) =
5 / sqrt(5) =
4 / sqrt(2) =
sqrt(3) / sqrt(6) =
Expanding Brackets with Surds
Expand each expression using the distributive law.
sqrt(2)(sqrt(2) + 3) =
sqrt(3)(2 sqrt(3) - sqrt(12)) =
2 sqrt(5)(sqrt(5) + sqrt(20)) =
Surd or Not a Surd?
Circle whether each expression is a surd (irrational square root) or not.
sqrt(49)
sqrt(50)
sqrt(100)
sqrt(48)
sqrt(121)
Using Pythagoras Theorem with Surds
Use a^2 + b^2 = c^2 to find the missing side. Leave answers in surd form if they are irrational.
a = 1, b = 1: c =
a = 2, b = 3: c =
a = 1, b = sqrt(3): c =
a = sqrt(5), c = sqrt(14): b =
Perimeter Problems with Surds
Calculate the exact perimeter of each shape. Simplify all surds.
An equilateral triangle with side length sqrt(12) cm. Perimeter = 3 x sqrt(12) =
A rectangle with length 3 sqrt(2) cm and width sqrt(2) cm. Perimeter =
A square with area 50 cm^2. Side length = ___, Perimeter =
Comparing Surds Without a Calculator
Decide which is larger in each pair by squaring both expressions. Show your working.
3 sqrt(2) or 2 sqrt(3)? Square both to decide.
4 sqrt(3) or 3 sqrt(5)? Square both to decide.
5 sqrt(2) or 4 sqrt(3)? Square both to decide.
Surds in Geometric Formulas
Use the formula for the area of an equilateral triangle: A = (sqrt(3) / 4) x s^2, where s is the side length.
Find the exact area of an equilateral triangle with side length 4 cm. Leave your answer in surd form.
Find the exact area of an equilateral triangle with side length 6 cm.
If the area of an equilateral triangle is 9 sqrt(3) cm^2, find the side length.
Decimal Expansion Investigation
Investigate the decimal expansions of these numbers and classify them.
Use a calculator to find the first 8 decimal places of sqrt(2). Write them here. Does the decimal terminate or repeat? What does this tell you?
Use long division (or a calculator) to find 1 / 7 to 8 decimal places. Does it terminate or repeat? Classify 1/7.
Write a decimal that you know is rational (not terminating). Explain how you know it is rational.
Irrational Number Proofs and Reasoning
Answer these conceptual questions about irrational numbers.
Give an example of two irrational numbers whose sum is rational. Explain why this works.
Give an example of two irrational numbers whose product is rational. Explain why.
Is sqrt(2) + sqrt(3) rational or irrational? Justify your answer.
Simplify and Match
Simplify each surd on the left, then draw a line to its simplified form on the right.
Adding Surds — Identify the Correct Result
Circle the correct simplified result.
sqrt(8) + sqrt(18)
sqrt(12) + sqrt(27)
sqrt(50) - sqrt(8)
Area Problems with Surds
Calculate the exact area of each shape. Leave answers in simplified surd form.
A rectangle with sides sqrt(6) cm and sqrt(24) cm. Area = sqrt(6) x sqrt(24) =
A right-angled triangle with legs 2 sqrt(3) cm and sqrt(3) cm. Area =
A square with side (1 + sqrt(2)) cm. Area = (1 + sqrt(2))^2 =
Rationalising with Conjugates
Rationalise each denominator by multiplying by the conjugate.
1 / (sqrt(2) + 1) = multiply by (sqrt(2) - 1)/(sqrt(2) - 1) =
3 / (sqrt(5) - 2) =
Irrational Numbers in Technology
Investigate where irrational numbers appear in technology and design.
- 1Look up the A4 paper size standard. The sides of an A4 sheet are in the ratio 1 : sqrt(2). Measure an A4 page and verify this ratio approximately. Why is this ratio useful for printing?
- 2The musical frequency ratio between notes in equal temperament is the twelfth root of 2 (a surd!). Research what this means and why it makes tuning instruments mathematically elegant.
- 3Research the Fibonacci sequence (1, 1, 2, 3, 5, 8, 13, 21...). Calculate the ratio of consecutive terms. What irrational number does this ratio approach?
Simplify Complex Surd Expressions
Simplify each expression fully. Show all steps.
3 sqrt(8) + 2 sqrt(18) - sqrt(2) =
sqrt(75) - 2 sqrt(12) + sqrt(27) =
(sqrt(5) + sqrt(2))(sqrt(5) - sqrt(2)) =
(2 sqrt(3) + 1)^2 =
Challenge: Nested Surds
Evaluate each expression step by step. Start from the innermost surd.
sqrt(sqrt(16)) = sqrt(___) =
sqrt(sqrt(81)) = sqrt(___) =
Estimate sqrt(sqrt(10)). First approximate sqrt(10), then take the square root again.
Finding Exact Lengths in Coordinate Geometry
Use the distance formula and leave answers in simplified surd form.
Find the exact distance between A(0, 0) and B(3, 5).
Find the exact distance between P(1, 2) and Q(4, 6).
Find the exact distance between C(-2, 1) and D(3, 7). Simplify your surd.
Surd Form in Trigonometry
Use exact values: sin 45 = cos 45 = sqrt(2)/2, sin 60 = sqrt(3)/2, cos 30 = sqrt(3)/2, tan 60 = sqrt(3).
In a right triangle, the hypotenuse is 8 cm and one angle is 45 deg. Find the exact length of the opposite side.
In a right triangle, the hypotenuse is 10 cm and one angle is 60 deg. Find the exact length of the adjacent side.
Simplify: 6 x sin 60 = 6 x sqrt(3)/2 =
Mixed Surd Problem Set
Show full working for each problem.
Simplify: sqrt(2) x sqrt(18) + sqrt(8)
A right-angled triangle has legs sqrt(7) cm and sqrt(7) cm. Find the exact hypotenuse and the exact perimeter.
Show that (sqrt(5) - 1)(sqrt(5) + 1) is a whole number.
Rationalise and simplify: (sqrt(6) + sqrt(2)) / sqrt(2)
Write and Solve: Real-World Surd Problems
Read each context, write an equation involving surds, and solve it.
A square window has area 32 m^2. What is the exact side length? What is the exact diagonal of the window?
A 5 m ladder leans against a vertical wall with its foot 2 m from the wall. How far up the wall does the ladder reach? Give the exact answer as a simplified surd.
Two parks are located at coordinates A(0, 0) and B(6, sqrt(3)). Calculate the exact distance AB.
Extending Surds: Cube Roots
Just as sqrt(n) is the square root, cube_root(n) is the number that cubed gives n. Evaluate each cube root.
cube_root(8) = (because ___ cubed = 8)
cube_root(27) =
cube_root(64) =
Is cube_root(10) rational or irrational? Explain.
Surds and Exact Perimeter
Each shape has side lengths involving surds. Find the exact perimeter in simplest form.
A triangle with sides sqrt(18), sqrt(8) and sqrt(50). Perimeter =
A rectangle with length 3 sqrt(5) cm and width 2 sqrt(5) cm. Perimeter =
A rhombus with all sides equal to sqrt(20) cm. Perimeter =
Reasoning with the Real Number System
Answer each conceptual question with a complete sentence and supporting example or calculation.
Is there a rational number between sqrt(2) and sqrt(3)? If yes, give one. If no, explain why.
Can two different surds simplify to the same value? Give an example or explain why not.
Why is it useful to write answers in surd form rather than as decimals in geometry and trigonometry?
Investigation: Surds and the Unit Circle
The unit circle (radius 1) is a key tool in mathematics. Answer these questions using exact values.
A point on the unit circle at 45 degrees has coordinates (cos 45, sin 45). Write these coordinates in exact surd form.
A point on the unit circle at 60 degrees has coordinates (cos 60, sin 60). Write these in exact form.
Verify using Pythagoras: for the 45-degree point, does x^2 + y^2 = 1? Show the calculation.
Challenge: Surd Equations
Solve each equation involving surds. Check your solution by substituting back.
sqrt(x) = 5. Find x and verify.
sqrt(2x) = 6. Find x and verify.
sqrt(x + 1) = 3. Find x and verify.
Comprehensive Review: Rational and Irrational Numbers
This final review covers all key skills in this worksheet. Show full working for each question.
Classify each number as rational or irrational: sqrt(49), sqrt(50), 0.121212..., pi, 3/7, sqrt(3) + sqrt(3)
Simplify: sqrt(48) + sqrt(75) - sqrt(12)
Rationalise the denominator: 10 / (2 sqrt(5))
A square garden has a diagonal of 10 m. Find the exact side length and the exact perimeter.
Simplify Surd Expressions with Mixed Operations
Combine addition, subtraction, and multiplication of surds in each expression. Simplify fully.
sqrt(2)(sqrt(8) + sqrt(18)) =
sqrt(3)(2 sqrt(3) + sqrt(27)) =
(sqrt(5) + 2)(sqrt(5) - 2) =
(sqrt(7) + sqrt(2))^2 =
Rationalisation Check
A student rationalised each fraction. Circle the correct result.
3 / sqrt(3)
10 / sqrt(5)
sqrt(8) / sqrt(2)
6 / sqrt(6)
Surd Form vs Decimal — When to Use Which
Decide whether to give an exact surd answer or a decimal approximation for each context.
A builder needs to know the length of a hypotenuse to cut wood. The legs are 3 m and 4 m. Which is more useful — sqrt(25) = 5 m, or 5.00 m? Explain.
A student needs to find the diagonal of a square paddock to fence it. The paddock is 50 m x 50 m. Give both exact and decimal answers (to 1 decimal place).
Why do mathematicians prefer exact surd form when the result will be used in a further calculation?
Match Surds to Equivalent Rational Results
Draw a line from each surd expression to its rational (whole number or fraction) equivalent.
Applying DOTS with Surds
Use the difference of two squares pattern to evaluate each expression without a calculator.
(sqrt(3) + 1)(sqrt(3) - 1) =
(sqrt(10) + sqrt(2))(sqrt(10) - sqrt(2)) =
(sqrt(6) - sqrt(5))(sqrt(6) + sqrt(5)) =
(2 + sqrt(7))(2 - sqrt(7)) =
Surd Patterns Investigation
Investigate the pattern in these expressions and predict the next two.
(sqrt(1 + 1) - sqrt(1)) x (sqrt(1 + 1) + sqrt(1)) = 1 x 2 - 1 = 1 (sqrt(2 + 1) - sqrt(2)) x (sqrt(2 + 1) + sqrt(2)) = ___ (sqrt(3 + 1) - sqrt(3)) x (sqrt(3 + 1) + sqrt(3)) = ___ Describe the pattern:
Exact Answers in Trigonometry
Use sin 30 = 1/2, cos 30 = sqrt(3)/2, tan 30 = 1/sqrt(3), sin 60 = sqrt(3)/2, cos 60 = 1/2, tan 60 = sqrt(3) to give exact answers.
In a right triangle, angle = 30 deg, hypotenuse = 12 cm. Find the exact length of the side opposite to 30 deg.
In a right triangle, angle = 60 deg, adjacent = 4 cm. Find the exact length of the opposite side.
Verify: sin^2(30) + cos^2(30) = 1. Show using exact values.
Sort Expressions: Rational or Surd Result?
Sort each expression into the correct column based on whether its value is rational or irrational.
Surd Application: Geometry of a Regular Hexagon
A regular hexagon with side length s has area A = (3 sqrt(3) / 2) s^2.
Find the exact area of a regular hexagon with side length 2 cm.
Find the exact area of a regular hexagon with side length 4 cm.
If the area is 6 sqrt(3) cm^2, find the side length.
Surds as Exact Lengths — Rectangular Diagonal
A rectangle has integer dimensions. Find the exact length of the diagonal using Pythagoras, leaving in surd form.
Rectangle 5 cm by 7 cm. Diagonal = sqrt(___ + ___) = sqrt(___) =
Rectangle 3 cm by 8 cm. Diagonal =
Rectangle 6 cm by 6 cm. Diagonal = (Simplify your surd)
A 4 cm by 10 cm rectangle. Diagonal = (Simplify your surd)
Proving Irrationality by Contradiction
Follow the steps to understand WHY sqrt(2) is irrational.
Step 1: Assume sqrt(2) = p/q (a fraction in lowest terms, so p and q share no common factors). Then square both sides: 2 = p^2 / q^2. So p^2 = ___.
Step 2: Since p^2 = 2q^2, p^2 is even. This means p itself is even. Write p = 2k for some integer k. Then (2k)^2 = 2q^2, so 4k^2 = 2q^2, so q^2 = ___. This means q is also even.
Step 3: Both p and q are even. But we said the fraction was in lowest terms (no common factors). This is a CONTRADICTION. What does this mean about our assumption?
Speed and Distance with Exact Values
Solve each problem giving an exact answer in surd form. Then provide a decimal approximation to 2 decimal places.
A car travels sqrt(50) km in 1 hour. At the same speed, how far does it travel in sqrt(2) hours?
A runner covers sqrt(72) km in 2 hours. What is their speed in km/h? Simplify the surd.
Extending Rationality: Rational Operations on Irrationals
Determine whether each result is rational or irrational. Give a reason.
sqrt(2) x sqrt(18) — is the result rational or irrational? Why?
sqrt(3) + (1 - sqrt(3)) — is the result rational or irrational? Why?
pi - pi — is the result rational or irrational? Why?
pi + 1 — is the result rational or irrational? Why?
Investigation: Surds Between Integers
Without using a calculator, find two consecutive integers that each expression lies between. Then estimate to 1 decimal place.
sqrt(30) lies between ___ and ___. Estimate:
sqrt(60) lies between ___ and ___. Estimate:
2 sqrt(5) lies between ___ and ___. Estimate: (Hint: square 2 sqrt(5))
3 sqrt(2) lies between ___ and ___. Estimate:
Surd Scavenger Hunt
Look for irrational numbers in your home, nature, and technology.
- 1Measure the side of any square object (tile, book, frame). Calculate the diagonal using sqrt(2) x side. Measure the actual diagonal and compare — are they equal?
- 2Find the frequency of the note A4 on a piano (440 Hz). The note A#4 is 440 x 2^(1/12) Hz. Use a calculator to find this frequency and research which note it corresponds to.
- 3Look up the aspect ratio of your television or computer screen. Is the ratio of the diagonal to the shorter side close to a recognisable surd?
Final Challenge: Multi-Step Surd Problem
This problem requires combining several skills. Show all working clearly.
A right-angled triangle has legs of length sqrt(18) cm and sqrt(8) cm. (a) Find the exact length of the hypotenuse in simplified surd form. (b) Find the exact perimeter of the triangle. (c) Find the exact area of the triangle.
Simplify Surds with Large Radicands
Simplify each surd. Find the largest perfect-square factor each time.
sqrt(252) =
sqrt(338) =
sqrt(432) =
sqrt(500) =
Estimating Surds to 1 Decimal Place
Circle the best estimate for each surd (to 1 decimal place).
sqrt(17)
sqrt(40)
sqrt(73)
sqrt(110)
Surds and Algebraic Identities
Use the identity (a + b)(a - b) = a^2 - b^2 to evaluate each expression without expanding manually.
(sqrt(11) + sqrt(3))(sqrt(11) - sqrt(3)) = ___ - ___ = ___
(sqrt(13) + 2)(sqrt(13) - 2) =
(3 + sqrt(5))(3 - sqrt(5)) =
Match the Surd to Its Decimal
Draw a line from each simplified surd to its approximate decimal value (to 2 decimal places).
Surds in Architecture: The Golden Ratio
The golden ratio is phi = (1 + sqrt(5)) / 2.
Use sqrt(5) approx 2.236 to calculate phi to 3 decimal places.
Show that phi^2 = phi + 1 using exact surd values. (Hint: square (1 + sqrt(5)) / 2 and simplify.)
A golden rectangle has length phi cm and width 1 cm. What is its exact perimeter?
Proving Irrationality by Contradiction
Follow the steps to understand WHY sqrt(2) is irrational.
Step 1: Assume sqrt(2) = p/q in lowest terms. Square both sides: p^2 = ___.
Step 2: p^2 = 2q^2 means p^2 is even, so p is even. Write p = 2k. Then 4k^2 = 2q^2, so q^2 = 2k^2. This means q is also ___.
Step 3: Both p and q are even — contradiction! What does this tell us about our assumption that sqrt(2) = p/q?
Connecting Surds to the Number Line
Place the numbers in order from smallest to largest. Show your reasoning.
Order: 1.4, sqrt(2), 1.42, 3/2, sqrt(3) - 0.3. Justify your ordering.
Name a rational number between sqrt(5) and sqrt(6). Show it lies in that interval.
Surd Scavenger Hunt
Look for irrational numbers in your home, nature, and technology.
- 1Measure the side of any square object (tile, book, frame). Calculate the diagonal using sqrt(2) x side. Measure the actual diagonal — are they equal?
- 2Find the frequency of the note A4 on a piano (440 Hz). The note A#4 is 440 x 2^(1/12) Hz. Use a calculator to find this frequency.
- 3Look up the aspect ratio of your television or computer screen. Is the ratio of the diagonal to the shorter side close to a recognisable surd?
Comprehensive Surd Mastery Assessment
Complete all parts without notes. This covers every skill in the worksheet.
Classify as rational or irrational and give a reason: (a) sqrt(144) (b) sqrt(144 + 1) (c) sqrt(2) x sqrt(2) (d) sqrt(2) + sqrt(2)
Simplify fully: 2 sqrt(50) + 3 sqrt(32) - sqrt(18)
Expand and simplify: (3 + sqrt(5))^2
Rationalise the denominator: (2 + sqrt(3)) / sqrt(3)
A square has diagonal 6 sqrt(2) cm. Find its exact side length and area.
Surds and Exact Answers in Science
Scientists often need exact answers before rounding for a final result. Use surd form where appropriate.
The speed of a wave is v = sqrt(T/m) m/s where T is tension (N) and m is mass per metre (kg/m). Find the exact speed when T = 50 N and m = 2 kg/m.
The period of a pendulum is T = 2 pi sqrt(L/g) seconds, where L is the length in metres and g = 10 m/s^2. Find the exact period when L = 2.5 m. Use pi approx 3.14 for a decimal approximation.