Practice Maths

Estimating and Ordering Real Numbers

Key Ideas

Key Terms

benchmark
A known perfect square used to bracket a surd — e.g. 9 and 16 are benchmarks for √12.
number line
A line showing real numbers in order, used to compare and place surds and other real numbers.
estimation
Finding an approximate decimal value for a surd without a calculator, using surrounding perfect squares.
consecutive integers
Two whole numbers that differ by 1 — e.g. 3 and 4. Used to bound a surd: 3 < √12 < 4.
perfect square
An integer that is the square of another integer — e.g. 1, 4, 9, 16, 25. Used as benchmarks when estimating surds.
TaskMethod
Bound √nFind a, b: a² < n < b² ⇒ a < √n < b
Estimate to 1 d.p.Test midpoint, narrow down — e.g. √17: 4²=16, 5²=25, try 4.1: 16.81<17, try 4.2: 17.64>17 ⇒ ≈ 4.1
Compare surdsSquare both: (3√2)²=18, (2√5)²=20 ⇒ 2√5 > 3√2
Order mixed setConvert all to decimals, then sort
Hot Tip To compare two positive surd expressions without a calculator, square them both. If a > 0 and b > 0, then a > b if and only if a² > b².

Worked Example — Estimating a Surd

Question: Estimate √17 to 1 decimal place without a calculator.

Step 1: 4² = 16 and 5² = 25, so 4 < √17 < 5.

Step 2: Try 4.1: 4.1² = 16.81 < 17. Try 4.2: 4.2² = 17.64 > 17.

Step 3: 17 is between 16.81 and 17.64, closer to 17.64. Check 4.12: 16.97. Check 4.13: 17.06 > 17. So √17 ≈ 4.1 (to 1 d.p.).

Estimating Surds Using Perfect Squares

Since irrational numbers cannot be written exactly as decimals, we often need to estimate their size. The key strategy is to find the two consecutive perfect squares that bracket the number under the root.

For √20: we know 42 = 16 and 52 = 25. Since 16 < 20 < 25, we have 4 < √20 < 5. Since 20 is closer to 16 than to 25, √20 is closer to 4 (approximately 4.47).

For √50: 72 = 49 and 82 = 64. So 7 < √50 < 8. Since 50 is very close to 49, √50 ≈ 7.07.

This skill is essential for checking whether your calculator answer looks reasonable.

Rounding Irrational Numbers

When a question asks for a decimal answer, you will often need to round to a specified number of decimal places or significant figures.

Decimal places count digits after the decimal point. √7 ≈ 2.6457... Rounded to 2 decimal places: 2.65. Rounded to 3 decimal places: 2.646.

Significant figures count from the first non-zero digit. √7 to 3 significant figures: 2.65. To 4 significant figures: 2.646.

Always read the question carefully — "2 decimal places" and "2 significant figures" give different answers for most numbers.

Using Estimation to Check Calculator Answers

Before you accept a calculator result, do a quick mental estimate to check it is in the right ballpark. This catches errors from pressing the wrong button or misreading the question.

If your calculator gives √130 ≈ 1.14, something is wrong — you know √130 must be between 11 (since 112 = 121) and 12 (since 122 = 144), so the answer should be around 11.4, not 1.14. A decimal point error was made.

Scientific Notation Review

Scientific notation expresses numbers as a × 10n, where 1 ≤ a < 10 and n is an integer. It is used for very large or very small numbers.

Examples: 3 700 000 = 3.7 × 106. 0.000045 = 4.5 × 10−5. To multiply: multiply the a-values and add the exponents. (2.5 × 103) × (4 × 102) = 10 × 105 = 1.0 × 106.

Exam tip: After calculating an answer involving a surd, always re-read the question to check: (a) does it want an exact surd answer or a decimal? (b) how many decimal places or significant figures are required? (c) have you included the correct units? Write your rounding step explicitly so the marker can award method marks even if you make a small arithmetic error.

Mastery Practice

  1. State the two consecutive integers each surd lies between. Do not use a calculator. Fluency

      Surd __ < √n < __
    (a)√5 
    (b)√14 
    (c)√30 
    (d)√60 
    (e)√90 
    (f)√110 
    (g)√3 
    (h)√55 
    (i)√200 
    (j)√17 

  2. Estimate each surd to 1 decimal place without a calculator. Show your testing. Fluency

      Surd Estimate to 1 d.p.
    (a)√7 
    (b)√20 
    (c)√50 
    (d)√85 
    (e)√130 
    (f)√3 

  3. Place each set of numbers in ascending order. Show decimal estimates. Fluency

    1. √7,   2.5,   5/2,   √6,   2.6
    2. 3√2,   2√5,   √18,   4,   4.1
    3. π,   √10,   3.14,   22/7,   √9.9

  4. State whether each claim is True or False, justifying your answer. Fluency

    1. √10 is between 3 and 4.
    2. 2√3 > 3√2.
    3. √3 + √3 = √6.
    4. √49 = 7, so 7 is a surd.
    5. 4√3 > 3√5.
    6. Every irrational number lies between two consecutive integers.

  5. Ordering lengths of rope. Understanding

    Workshop Ropes. Three lengths of rope measure √20 m, 4.4 m, and 9/2 m. A worker needs to sort them from shortest to longest.
    1. Find a decimal approximation for each length.
    2. List them in ascending order.
    3. Find the total exact length of all three ropes.

  6. Identifying a student error. Understanding

    Student Work. A student claims that √3 + √3 = √6.
    1. Estimate √3 + √3 (i.e. 2√3) and √6 each to 1 decimal place without a calculator.
    2. Show algebraically that 2√3 ≠ √6 by comparing their squares.
    3. State the correct simplified form of √3 + √3.

  7. Compare each pair of surd expressions without a calculator. State which is larger. Understanding

    Squaring Technique. Square both expressions, then compare the resulting values.
    1. 3√2   versus   2√3
    2. 4√3   versus   3√5
    3. 5 − √3   versus   3
    4. 2 + √5   versus   √13

  8. Order each row from smallest to largest, showing working. Understanding

      Numbers Ascending Order
    (a)√5,   2.2,   9/4,   √6,   2.4 
    (b)π,   3.14,   22/7,   √10,   3.2 
    (c)√2,   7/5,   1.4,   √50/5 

  9. The speed of sound. Problem Solving

    Physics Context. The speed of sound in air at 20°C is approximately √142 884 m/s. A physics student needs to determine the exact value.
    1. Without a calculator, state the two consecutive integers √142 884 lies between.
    2. Check whether 142 884 is a perfect square by testing 378². What is the exact speed?
    3. Is the speed of sound in this case rational or irrational? Explain.

  10. Surds on the number line — pattern investigation. Problem Solving

    Pattern Investigation. Estimate the positions of √2, √3, √5, √7, √11 on a number line using integer benchmarks.
    1. Find a decimal estimate to 1 d.p. for each of √2, √3, √5, √7, √11.
    2. Calculate the gaps between consecutive surds in your list. Are the gaps getting larger or smaller?
    3. Explain why the gaps between consecutive surds decrease as the numbers get larger. (Hint: think about how quickly √n grows compared to n.)
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