Real numbers on the number line: √5 by Pythagoras, 0.93̄ as p/q

Every real number has a point

You already place fractions on a number line. Irrational numbers need a construction, because you cannot count out their decimal places.

To place √5: mark OA = 2 units from O, draw AB = 1 unit upright at A. Pythagoras gives OB² = 2² + 1² = 5, so OB = √5.

With centre O and radius OB, draw an arc down to the line. It meets the line at P, and |OP| = √5 ≈ 2.236.

A recurring decimal can be turned back into p/q. Let x be the decimal, multiply by 10 or 100 to shift one block, then subtract.

Multiply by 10 when one digit repeats and by 100 when two digits repeat. The repeating tails then cancel.

Key terms

Pythagoras theorem

Building a length of √5 from 2 and 1.

Arc

Carrying the length OB down onto the line.

p/q form

Writing a recurring decimal as a fraction.

Average

Finding a rational number between two others.

Short questions with model answers

1. Q1. Example 3 (i): express 0.5̄ as p/q.

   • x = 0.5555… (i)
   • 10x = 5.5555… (ii)
   • 10x − x = 5 → 9x = 5 → x = 5/9

   0.5̄ = 5/9.

2. Q2. Example 4: find two rational numbers between 2 and 3.

   • (2 + 3)/2 = 5/2
   • (5/2 + 3)/2 = (11/2)/2 = 11/4

   5/2 and 11/4.

Common mistakes

• ✗ Multiplying 0.93̄ by 10.

  ✓ Two digits repeat, so use 100; then 100x − x = 93.

• ✗ Writing 0.5̄ = 5/10.

  ✓ 5/10 = 0.5 stops. The recurring one is 5/9.

• ✗ Drawing the arc with centre A instead of O.

  ✓ The length √5 is OB, measured from O, so the arc's centre is O.

MCQs

1. 1. In Example 2, OA = 2 and AB = 1. Then OB² is:

   1. (a) 3
   2. (b) 5
   3. (c) 4
   4. (d) 1
   Show answer

   (b) 2² + 1² = 4 + 1 = 5 (p. 5).

2. 2. 0.5̄ as p/q is:

   1. (a) 5/10
   2. (b) 1/2
   3. (c) 5/9
   4. (d) 55/100
   Show answer

   (c) 9x = 5, so x = 5/9 (p. 6).

3. 3. To convert 0.93̄, multiply x by:

   1. (a) 100
   2. (b) 10
   3. (c) 9
   4. (d) 1000
   Show answer

   (a) Two digits repeat (p. 6).

4. 4. Which is a rational number between 2 and 3 from Example 4?

   1. (a) √5
   2. (b) 7/2
   3. (c) 5/3
   4. (d) 11/4
   Show answer

   (d) (5/2 + 3)/2 = 11/4 (p. 6).

5. 5. By your book's Remember box, 2 + √5 is:

   1. (a) rational
   2. (b) an integer
   3. (c) irrational
   4. (d) a natural number
   Show answer

   (c) Rational + irrational = irrational (p. 5).

Quick revision

• √5 is placed with a right triangle of sides 2 and 1 and an arc from O.
• Recurring decimals become p/q: multiply by 10 or 100, subtract, divide.
• Averaging gives as many rationals between two numbers as you want.