الأسبوع 1 · الأعداد الحقيقية: الأعداد غير النسبية وتقدير الجذور على خط الأعداد ومجموعة الأعداد الحقيقية · 3 دروس
2
Where an irrational number sits: estimating a root between two whole numbers
Mathematics Prep 2 Term 1, Unit 1, Lesson 1
GoalEstimate a square or cube root between two whole numbers, say which one it is nearer to, and refine to one decimal place.
New words
consecutive numbers · whole numbers that follow each other, such as 3 and 4estimate · a careful guess that is close to the exact valuemidpoint · the number exactly halfway between two numbersapproximately equal (≈) · nearly equal, but not exactly
Explanation
To place √11 on the number line, look for the perfect squares on either side of 11: 9 < 11 < 16, so 3 < √11 < 4. To see which whole number it is nearer to, square the midpoint 3.5: 3.5² = 12.25, which is more than 11, so √11 is less than 3.5 and nearer to 3. Now refine: 3.3² = 10.89 and 3.4² = 11.56, so √11 lies between 3.3 and 3.4. Since 3.35² = 11.2225 is more than 11, √11 is below 3.35 and to one decimal place √11 ≈ 3.3. Cube roots work in the same way with perfect cubes: 8 < 20 < 27 gives 2 < ∛20 < 3. A calculator confirms the digits, but the method tells you whether the answer makes sense.
Worked examples
Between which two whole numbers is √40, and which is it nearer to? 36 < 40 < 49, so 6 < √40 < 7. The midpoint 6.5 gives 6.5² = 42.25, more than 40, so √40 is nearer to 6. Refine: 6.3² = 39.69 and 6.4² = 40.96, so √40 ≈ 6.3 (6.35² = 40.3225 is above 40).
Estimate ∛20. 2³ = 8 and 3³ = 27, so 2 < ∛20 < 3. Try 2.7³ = 19.683 and 2.8³ = 21.952: ∛20 is between 2.7 and 2.8, and 2.75³ = 20.796875 is above 20, so ∛20 ≈ 2.7 to one decimal place.
A square flower bed has area 50 m². Its side is √50 m. 49 < 50 < 64 gives 7 < √50 < 8. Also 7.0² = 49 and 7.1² = 50.41, so the side is between 7.0 and 7.1 m. Since 7.05² = 49.7025 is below 50, the side is above 7.05 and rounds to 7.1 m.
Class activity
Square-root strip. Cut a strip of paper 12 cm long and mark 0 to 6 on it, 2 cm for each unit. With a calculator, mark √2, √5, √10 and √20, then check each mark against the perfect squares on either side of it.
Practice
1
Between which two whole numbers is √29? Which is it nearer to?
25 < 29 < 36, so 5 < √29 < 6. 5.5² = 30.25 is more than 29, so √29 is nearer to 5.
2
Which whole number is ∛50 nearer to? Cube the midpoint of 3 and 4.
27 < 50 < 64 gives 3 < ∛50 < 4. 3.5³ = 42.875 is less than 50, so ∛50 is above 3.5 and nearer to 4.
3
Use 2.2² = 4.84 and 2.3² = 5.29 to give √5 to one decimal place.
√5 is between 2.2 and 2.3. 2.25² = 5.0625 is more than 5, so √5 is below 2.25 and √5 ≈ 2.2.
4
Is √17 closer to 4 or to 5? Use the midpoint 4.5.
4.5² = 20.25 is more than 17, so √17 < 4.5 and it is closer to 4.
5
A square tile has area 30 cm². Estimate its side to one decimal place.
5.4² = 29.16 and 5.5² = 30.25, so the side is between 5.4 and 5.5 cm. 5.45² = 29.7025 is below 30, so the side rounds to 5.5 cm.
