Irrational Numbers and Surds — Solutions

Click any answer to watch the solution video.

1. Classify rational or irrational

   1. √9: R — √9 = 3, a whole number
   2. √2: I — not a perfect square, decimal never terminates or repeats
   3. 0.3̇: R — recurring decimal = 1/3
   4. π: I — non-terminating, non-recurring decimal
   5. √16: R — √16 = 4, a whole number
   6. √5: I — not a perfect square
   7. 2/3: R — fraction of two integers
   8. √100: R — √100 = 10

2. Estimate surds between consecutive integers

   1. √15: 3 < √15 < 4  (3²=9, 4²=16)
   2. √30: 5 < √30 < 6  (5²=25, 6²=36)
   3. √50: 7 < √50 < 8  (7²=49, 8²=64)
   4. √8: 2 < √8 < 3  (2²=4, 3²=9)
   5. √45: 6 < √45 < 7  (6²=36, 7²=49)
   6. √70: 8 < √70 < 9  (8²=64, 9²=81)
   7. √3: 1 < √3 < 2  (1²=1, 2²=4)
   8. √110: 10 < √110 < 11  (10²=100, 11²=121)

3. Simplify surds

   1. √8: √(4×2) = 2√2
   2. √18: √(9×2) = 3√2
   3. √45: √(9×5) = 3√5
   4. √75: √(25×3) = 5√3
   5. √200: √(100×2) = 10√2
   6. √28: √(4×7) = 2√7
   7. √48: √(16×3) = 4√3
   8. √98: √(49×2) = 7√2

4. Order rational and irrational numbers

   1. √2≈1.41, √3≈1.73 → 1/2, √2, 1.5, √3
   2. π≈3.14, √10≈3.16 → 3, π, √10, 3.2
   3. 2√2≈2.83, √7≈2.65 → √7, 2.8, 2√2, 3
   4. √5≈2.24, √6≈2.45 → √5, 2.25, √6, 2.5
   5. 5√2=√50≈7.07, 3√3≈5.20 → 3√3, 7, 5√2=√50
   6. √0.1≈0.316, 1/3=0.3̇≈0.333 → √0.1, 1/3=0.3̇, 0.35

5. True/False about real numbers

   1. Every integer is rational: True — any integer n = n/1
   2. √4 is irrational: False — √4 = 2, which is rational
   3. Product of two irrationals is always irrational: False — e.g. √2 × √2 = 2, which is rational
   4. π = 22/7 exactly: False — 22/7 is a rational approximation; π is irrational
   5. Every real number is rational or irrational but not both: True — these categories are mutually exclusive and exhaustive
   6. A surd always lies between two consecutive integers: False — e.g. √(1/4) = 1/2 lies between 0 and 1, not between consecutive positive integers; true for most surds >1 though
   7. Irrational numbers cannot be placed on a number line: False — irrationals have exact positions on the real number line
   8. 0.9̇ is irrational: False — 0.9̇ = 1 exactly, which is rational

6. Problem solving with surds

   1. Side = √50 = √(25×2) = 5√2 m
   2. 1. h² = 3²+4² = 9+16 = 25, h = √25 = 5 cm. Rational.
      2. h² = 1+4 = 5, h = √5 cm. Irrational surd (already simplified).
   3. Diagonal = 5√2 cm = 5 × 1.41421… ≈ 7.1 cm
   4. Jamie is wrong. √2 + √2 = 2√2 (adding like surds), not √4. 2√2 ≈ 2.828, whereas √4 = 2. These are not equal.

7. Classification table

   1. Rational: √4=2, 0.6̇=2/3, √25=5, 22/7, √49=7 — Irrational: √3, π, √11, 0.101001000…, √6

8. Number line — mark surds

   √2≈1.41 (just past 1, closer to 1.5), √5≈2.24 (just past 2), √7≈2.65 (between 2.5 and 3), √10≈3.16 (just past 3)

9. Surd simplification table

   1. √12: Largest sq factor = 4; 2√3
   2. √50: Largest sq factor = 25; 5√2
   3. √72: Largest sq factor = 36; 6√2
   4. √108: Largest sq factor = 36; 6√3
   5. √?: 25 × 3 = 75, so √75

10. Extended reasoning

    1. √2 = 1.41421356… — no repeating block and no termination. A rational number must have a decimal that either terminates or eventually repeats. Since √2 does neither, it is irrational.
    2. Incorrect. √4 + √9 = 2 + 3 = 5, but √13 ≈ 3.61. The rule √a + √b ≠ √(a+b). You can only combine surds if they are like surds (same radicand).
    3. 4 < √5 < 5, so √5 < 5, meaning 2√5 < 10. More precisely: 2²=4, 3²=9, √5≈2.24, 2×2.24=4.47. So 4 < 2√5 < 5.
    4. diagonal = side×√2, so side = 10/√2 = 10√2/2 = 5√2 cm ≈ 5×1.414 = 7.1 cm