Practice Maths

Comparing and Ordering Real Numbers

Key Ideas

Key Terms

real number line
a line containing every real number (integers, rationals, irrationals) with no gaps; numbers increase from left to right.
inequality
a statement comparing two values using <, >, ≤, or ≥ (e.g. √2 < 1.5).
density of real numbers
between any two distinct real numbers, there is always another real number; the real line has no gaps.
approximate value
a decimal rounded to a set number of decimal places, used when an exact value (like a surd) cannot be written as a terminating decimal.

Strategy for Comparing Mixed Types

Convert everything to a decimal approximation. For fractions, divide. For surds, estimate or simplify first then estimate. For π, use 3.14159…

Hot Tip Memorise these: √2 ≈ 1.41, √3 ≈ 1.73, √5 ≈ 2.24, √7 ≈ 2.65, π ≈ 3.14159. When comparing, convert to the same number of decimal places before deciding.

Worked Example

Compare: √5 and 7/3. Use <, >, or =.

√5 ≈ 2.236

7/3 = 2.333…

Since 2.236 < 2.333, we write √5 < 7/3.

The Strategy: Convert Everything to Decimals

Comparing integers is easy. But what do you do when a list mixes integers, fractions, decimals, and surds? The most reliable strategy is to convert everything to a decimal approximation, then compare the decimal values. Once you have decimals, ordering from smallest to largest is straightforward.

For surds like √5 or √11, use a calculator to find the decimal value. For fractions, divide the numerator by the denominator. For negative fractions, remember that more negative means smaller (e.g. −3 < −1).

Worked Example: Ordering a Mixed List

Order these numbers from smallest to largest: √3, 1.8, 74, −0.5, 1
Convert each to a decimal:
√3 ≈ 1.732
1.8 = 1.800
74 = 1.750
−0.5 = −0.500
1 = 1.000

Ordered from smallest to largest: −0.5 < 1 < √3 < 74 < 1.8

Check: 1.000 < 1.732 < 1.750 < 1.800 — yes, this is correct.

Placing Numbers on a Number Line

A number line is a visual tool for showing the order of numbers. Numbers increase from left to right. To place a mixed set on a number line:

  1. Convert all numbers to decimals.
  2. Mark a scale — decide what each grid interval represents.
  3. Place each number at its approximate position.

For example, to place −1, 0.5, √2, and 2 on a number line from −2 to 3, you would place them at approximately: −1, 0.5, 1.414, and 2. Notice that √2 sits between 1.4 and 1.5 — closer to 1.4.

Key tip: When comparing two numbers that are very close (like √3 and 1.75), use enough decimal places to tell them apart. √3 ≈ 1.7320... and 1.75 = 1.7500..., so 1.75 is larger. Do not round too early — round only at the final answer step.

Interval Notation

Interval notation is a compact way of describing a set of real numbers between two values. Instead of saying "all numbers from 2 to 5", you write an interval.

  • [a, b] — closed interval. Includes both endpoints a and b. Equivalent to a ≤ x ≤ b.
  • (a, b) — open interval. Excludes both endpoints. Equivalent to a < x < b.
  • [a, b) — half-open. Includes a, excludes b. Equivalent to a ≤ x < b.
  • (a, ∞) — all numbers greater than a (no upper limit). Equivalent to x > a.
  • (−∞, b] — all numbers up to and including b. Equivalent to x ≤ b.

Square brackets mean the endpoint is included. Round brackets mean the endpoint is excluded. Infinity is always written with a round bracket because you can never actually reach infinity.

Inequalities and Number Line Representation

Inequalities and intervals are two ways of expressing the same thing. On a number line, a filled circle (or closed dot) at an endpoint means that value is included (≤ or ≥). An open circle (hollow dot) means that value is excluded (< or >). For example, x > −1 is shown with an open circle at −1 and an arrow pointing right. The interval notation is (−1, ∞).

Mastery Practice

  1. Insert <, >, or = between each pair of numbers. Show your working (convert to decimals). Fluency

    1. √3  □  1.7
    2. 2/3  □  0.6̇
    3. π  □  3.14
    4. √10  □  π
    5. 5/4  □  √2
    6. 0.3̇  □  1/3
    7. √7  □  2.6
    8. 3/8  □  0.375

  2. Order each set of four numbers from smallest to largest. Show the decimal values you used. Fluency

    1. 1.4,   √2,   3/2,   π/2
    2. 2.5,   √6,   5/2,   √7
    3. 0.6̇,   2/3,   √(4/9),   0.67
    4. π,   √10,   22/7,   3.14
    5. √5,   9/4,   2.2,   √(19/4)
    6. 1/7,   0.14,   √(1/50),   0.143
    7. 3√2,   2√3,   √17,   √18
    8. π²/10,   √10,   3.1,   π

  3. Find one real number that lies between each pair of values. There are many correct answers — show that your number works. Fluency

    1. 1.4 and √2
    2. π and 3.15
    3. 1/3 and 0.34
    4. √5 and 2.24
    5. 0 and 0.001
    6. 2/7 and 3/7
    7. √2 and √3
    8. 0.9̇ and 1.01

  4. For each number, describe its position on the number line by stating which two integers it lies between and whether it is closer to the lower or upper integer. Understanding

    1. √11
    2. π
    3. √50
    4. 7/3
    5. √(1/4)
    6. 3√2
    7. 11/4
    8. √30

  5. Classify each number by placing it in the correct region: Integers only (I), Rational but not integer (Q), or Irrational (Irr). Understanding

    1. −5
    2. 3/4
    3. √7
    4. 0
    5. π
    6. √25
    7. 0.3̇
    8. √2
    9. 17
    10. 2.75
    11. √(9/4)
    12. √3 × √3
    13. √2 + √3
    14. 22/7
    15. −0.5

  6. Apply comparing and ordering real numbers to real-world contexts. Problem Solving

    1. A builder measures three pieces of timber: piece A = √3 m, piece B = 1.73 m, piece C = 74 m. Order them from shortest to longest. By how much does the longest exceed the shortest? (Give the difference to the nearest millimetre.)
    2. Three students estimate π. Aiko says π ≈ 3.14, Ben says π ≈ 22/7, Cass says π ≈ 3.141. Whose estimate is closest to the true value? Show working.
    3. A square has exact area A1 = 2 m2 and a second square has exact area A2 = 2.04 m2.
      1. Write the exact side lengths of each square.
      2. Which square has the longer side? By how much (to 3 d.p.)?
    4. Is there a real number between 0.9̇ and 1? Explain your answer carefully using what you know about these two values.

  7. Complete the table by converting each number to a decimal (to 3 d.p.), then rank the four numbers from 1 (smallest) to 4 (largest). Understanding

    Number Decimal (3 d.p.) Rank (1–4)
    √3??
    5/3??
    1.731.730?
    π − √2??

  8. Mark each of the following numbers on the number line. Write the letter label at the correct approximate position. Understanding

    A = 1/3,   B = √2 ≈ 1.41,   C = π/3 ≈ 1.05,   D = 5/4 = 1.25,   E = √3 ≈ 1.73

    0 1 2 3 0.5 1.5 2.5

    Scale: each unit spans 110 px (e.g. 0.1 unit ≈ 11 px). Use your decimal conversions to estimate positions.

  9. Insert <, >, or = in each box. Show the decimal values you used. Understanding

    Left < / = / > Right Working (decimals)
    3/7 √(1/5) 
    π2/10 √10 − 1 
    4√2 √32 
    2.2̇ √5 

  10. Extended comparison and ordering problems. Problem Solving

    1. List five different real numbers that lie between 2 and 3. Include at least one integer (if possible), one rational non-integer, one irrational, and one recurring decimal. Order them from smallest to largest.
    2. A student claims: “√n < n for all n > 0.” Test this claim with n = 0.25, n = 1, and n = 4. Is the claim true, false, or sometimes true? Explain fully.
    3. Four lengths are measured: √17 m, 4.1 m, 4 m, 174 m. Order them from shortest to longest and find the difference between the longest and shortest to the nearest centimetre.
    4. Explain in your own words why placing irrational numbers on a number line is valid, even though their decimal expansions never terminate or repeat.
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