Estimating Irrational Numbers
An irrational number never ends, so we estimate it. Squares tell you where a square root sits.
What you'll be able to do
- Find the two whole numbers a root is between.
- Estimate a root to a tenth or a hundredth.
- Place roots and π on a number line.
- Compare and order numbers like 3√2 and √17.
The big idea
Start with squares. 49 < 60 < 64. So √60 is between 7 and 8.
Go to tenths. Try 7.7² = 59.29 and 7.8² = 60.84. Now √60 is between 7.7 and 7.8.
Pick the closer one. 60 is 0.71 above 59.29 and 0.84 below 60.84. It is closer to 7.7², so √60 ≈ 7.7.
Hundredths work the same way. Square two numbers like 3.60 and 3.61. See which square is closer.
π. π ≈ 3.14. It is a bit less than 22/7, which is about 3.143.
Compare by squaring. To compare 3√2 and √17, square both. (3√2)² = 9 × 2 = 18 and (√17)² = 17. So 3√2 is bigger.
Negative roots. √30 is between 5 and 6. So −√30 is between −6 and −5.
Squares spread out. 6² = 36, 7² = 49 and 8² = 64. The gaps grow: 13, then 15. So roots grow more slowly.
Words you need
- Square root
- √n is the number that times itself makes n. √49 = 7.
- Irrational number
- A number whose decimal never ends or repeats, like √2 or π.
- Perfect square
- A whole number times itself, like 49 = 7 × 7.
- Integer
- A whole number or its negative: …, −2, −1, 0, 1, 2, ….
- Estimate
- A close value, not the exact one.
- Cube root
- ∛n times itself 3 times makes n. ∛64 = 4.
- Product
- The answer when you multiply.
Squares to know
| n | n² |
|---|---|
| 4 | 16 |
| 5 | 25 |
| 6 | 36 |
| 7 | 49 |
| 8 | 64 |
| 9 | 81 |
| 10 | 100 |
Worked examples
Example 1
Find √60 to the nearest tenth.
- 49 < 60 < 64, so √60 is between 7 and 8.
- 7.7² = 59.29 is 0.71 below 60. 7.8² = 60.84 is 0.84 above.
- 60 is closer to 7.7², so √60 ≈ 7.7.
Answer: 7.7
Example 2
Place √3, π − 1 and 7/4 on a number line.
- √3 ≈ 1.73, because 1.73² ≈ 2.99.
- 7/4 = 1.75. π − 1 ≈ 3.14 − 1 = 2.14.
- From left to right: √3, 7/4, π − 1.
Answer: √3 < 7/4 < π − 1
Example 3
Order 2√2, 3 and √7 from least to greatest.
- Square each one: (2√2)² = 4 × 2 = 8.
- 3² = 9 and (√7)² = 7.
- 7 < 8 < 9, so √7 < 2√2 < 3.
Answer: √7 < 2√2 < 3
Common mistakes
Placing a root at the wrong end. √5 is near 2, not 3. 5 is much closer to 4 than to 9.
Thinking 22/7 is π. 22/7 is about 3.143. π is about 3.142.
Saying 2√3 is more than 3.5 because it rounds to 3.5. (2√3)² = 12 is less than 3.5² = 12.25. So 2√3 is less.
Counting the ends. 36 and 49 have roots of exactly 6 and 7. They are not between.
Practice: 20 questions
5 of 20 are word problems. For a word problem, sketch it first and mark what you know.
Tier 1 Foundations Use the rule directly
- A TV screen is 30 inches wide and 20 inches tall. Its diagonal is √1300 inches. Between which two whole numbers is it? Which is it closer to? Word problem
- Between which two integers is -√30?
- A number line shows A at 2.2, B at 2.4, C at 2.6 and D at 2.8. Which point is closest to √7?
- Order π, 3.1, 22/7 and √10 from least to greatest.
- Tom marks √5 just left of 3 on a number line. Find his mistake.
Tier 2 Intermediate One more step to think through
- Find √13 to the nearest hundredth.
- Find 2√3 to the nearest tenth. Is 2√3 more or less than 3.5?
- Place √2 + 1 and 4 − √3 on a number line from 2 to 3. Which is farther right?
- A clock's minute hand is 10 cm long. How far does its tip travel in one hour? Round to the nearest cm. Word problem
- How many whole numbers have a square root between 6 and 7?
Tier 3 Advanced Several steps - find the hidden structure
- The square root of a whole number n is between 4.5 and 4.6. Find n.
- A field is 40 m long and 25 m wide. Will a 47 m rope reach from one corner to the opposite corner? Word problem
- Order 3√2, 2√5, √17 and 13/3 from least to greatest.
- Max says √30.5 = 5.5, because 30.5 is halfway between 25 and 36. Find his mistake.
- A bike wheel has a diameter of 60 cm. How many full turns does it make in 100 m? Word problem
Tier 4 Challenge Stretch problems, some above grade level
- Name a fraction that lies between √10 and √11.
- Which whole number n makes √n closest to 7.4?
- A cube-shaped box holds 100 cm³. How long is each edge, to the nearest tenth of a cm? Word problem
- Which is greater: √50 − 7 or 7 − √48?
- √72 = 6√2, because 72 = 36 × 2 and √36 = 6. Write √50 and √98 the same way. Then add them. Above grade level
Answers
Tier 1 Foundations
- Between 36 and 37, closer to 36. 36² = 1296 and 37² = 1369. 1300 is between them.
- Between −6 and −5. 5² = 25 and 6² = 36. 30 is between them. So √30 is between 5 and 6, and −√30 is between −6 and −5.
- C (at 2.6). 2.6² = 6.76 and 2.7² = 7.29, so √7 ≈ 2.65. C is the closest point.
- 3.1 < π < 22/7 < √10. π ≈ 3.142, 22/7 ≈ 3.143, √10 ≈ 3.162.
- Between 2 and 3, closer to 2. 5 is 1 more than 4 but 4 less than 9. So √5 is near 2, about 2.24. 2² = 4 and 3² = 9. 5 is between them.
Tier 2 Intermediate
- 3.61. 3.60² = 12.96 is 0.04 below 13. 3.61² = 13.0321 is 0.0321 above. 13 is closer to 3.61², so √13 ≈ 3.61.
- 3.5. It is less than 3.5. (2√3)² = 12. 3.4² = 11.56 is 0.44 below 12. 3.5² = 12.25 is 0.25 above. 12 is closer to 3.5², so 2√3 ≈ 3.5. 12 is below 3.5² = 12.25, so it is less than 3.5.
- √2 + 1 is farther right. 4 − √3 ≈ 2.27, √2 + 1 ≈ 2.41.
- About 63 cm. 20π ≈ 62.83 (π ≈ 3.14).
- 12 whole numbers: 37 to 48. 6² = 36 and 7² = 49. The whole numbers between them have roots between 6 and 7. 36 and 49 have roots of exactly 6 and 7, so they don't count.
Tier 3 Advanced
- 21. 4.5² = 20.25 and 4.6² = 21.16. The whole numbers between them have roots between 4.5 and 4.6.
- No, it is too short. Diagonal² = 40² + 25² = 2225. 47² = 2209, which is less than 2225. So the diagonal is √2225 m ≈ 47.17 m.
- √17 < 3√2 < 13/3 < 2√5. Square each one: 3√2 → 18, 2√5 → 20, √17 → 17, 13/3 → 169/9 ≈ 18.8. Bigger squares mean bigger numbers.
- 5.52. Halfway between two squares is not halfway between their roots. 5.5² = 30.25, not 30.5. 5.52² = 30.4704 is 0.0296 below 30.5. 5.53² = 30.5809 is 0.0809 above. 30.5 is closer to 5.52², so √30.5 ≈ 5.52.
- 53 full turns. One turn is 60π ≈ 188.5 cm. 10,000 ÷ (60π) ≈ 53.05. Only full ones count, so round down.
Tier 4 Challenge
- 16/5 works. Many answers work. Look for a decimal whose square is between 10 and 11. 3.2² = 10.24, and 3.2 = 16/5.
- n = 55. 7.4² = 54.76, which is closer to 55 than to 54. Check: √55 ≈ 7.416 and √54 ≈ 7.348.
- About 4.6 cm. 4.6³ = 97.336 is 2.664 below 100. 4.7³ = 103.823 is 3.823 above. 100 is closer to 4.6³, so ∛100 ≈ 4.6.
- 7 − √48 is greater. Squares spread out as numbers grow. 36 to 49 is 13 apart, but 49 to 64 is 15 apart. So 1 below 49 moves the root more than 1 above. 7 − √48 is greater.
- 12√2. 50 = 25 × 2, so √50 = 5√2. 98 = 49 × 2, so √98 = 7√2. Add: 5√2 + 7√2 = 12√2 ≈ 16.97.
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