2.4 Estimating Irrational Numbers and Square Roots
Key Takeaways
- An irrational number cannot be written as a ratio of integers; its decimal expansion neither terminates nor repeats, as with √2, √5, π, and e.
- To estimate √n, locate n between the two nearest perfect squares; √n then lies between their roots, and linear interpolation refines the estimate.
- √50 lies between 7 and 8 because 49 < 50 < 64, and it is very close to 7.07 since 50 is only 1 above 49.
- The square root of a non-perfect-square integer is always irrational, so √12 cannot be written as an exact fraction even though 12 is rational.
- Comparing irrationals with rationals is done by squaring both values or by bracketing the irrational between decimal bounds.
2.4 Estimating Irrational Numbers and Square Roots
Competency 1 skill 3 asks you to estimate irrational numbers, including square roots, and compare them to rational numbers. On a test with only an on-screen scientific calculator, the calculator can give you √47 to eight decimals — but estimation items are usually written so that reasoning is faster than clicking, and comparison items often involve expressions the calculator cannot resolve for you directly.
Rational versus irrational
An irrational number cannot be expressed as a ratio of two integers. Its decimal expansion is infinite and non-repeating. The exam's standard cast:
- √n for any positive integer n that is not a perfect square — √2, √3, √5, √6, √7, √8, √10, ...
- π ≈ 3.14159265..., the ratio of a circle's circumference to its diameter
- e ≈ 2.71828..., though it appears rarely at this level
- The golden ratio (1 + √5)/2 ≈ 1.618
Watch two traps. First, √16 = 4 is rational — the radical symbol does not make a number irrational; the absence of a perfect square does. Second, 22/7 is rational, a common approximation of π but not π itself; 22/7 = 3.142857142857... repeats with period 6.
Sums and products can surprise: √2 · √2 = 2 is rational, and (3 + √5) + (3 − √5) = 6 is rational, even though each part is irrational. But a nonzero rational plus an irrational is always irrational, and a nonzero rational times an irrational is always irrational.
The perfect-square anchor method
Memorize the perfect squares through at least 20, and ideally 25.
+---------------------------------------------------------------------------+
| 1 4 9 16 25 36 49 64 81 100 |
| 11^2=121 12^2=144 13^2=169 14^2=196 15^2=225 |
| 16^2=256 17^2=289 18^2=324 19^2=361 20^2=400 |
| 21^2=441 22^2=484 23^2=529 24^2=576 25^2=625 |
+---------------------------------------------------------------------------+
To estimate √n, find the perfect squares immediately below and above n.
Estimate √136. Since 121 < 136 < 144, we get 11 < √136 < 12. Now interpolate: 136 is 15 units above 121 and the gap from 121 to 144 is 23 units, so √136 ≈ 11 + 15/23 ≈ 11.65. The true value is 11.6619..., so the estimate is accurate to two decimals.
Estimate √8. Since 4 < 8 < 9, we get 2 < √8 < 3, and 8 is 4 above 4 across a gap of 5, giving √8 ≈ 2.8. The true value is 2.8284..., close enough for any ordering item.
Interpolation slightly overestimates because the square-root curve is concave down, but the error is small in the middle of a bracket and negligible near the lower anchor. For √50, the bracket 49 < 50 < 64 gives 7 < √50 < 8 with 50 only 1 above 49 across a gap of 15, so √50 ≈ 7.07 — and indeed √50 = 7.0710....
Cube roots
The same logic applies with perfect cubes: 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000. To estimate ∛200: since 125 < 200 < 216, ∛200 lies between 5 and 6 and is much closer to 6, roughly 5.85. Unlike square roots, cube roots of negative numbers are real: ∛(−64) = −4, because (−4)³ = −64. Square roots of negatives are not real numbers.
Comparing irrationals to rationals
Three methods, in order of usefulness.
1. Bracket with decimals. Is √30 greater or less than 5.5? Since 5.5² = 30.25 and 30 < 30.25, √30 < 5.5. Squaring both sides is legitimate here because both quantities are positive.
2. Square everything. To order 2√3, 3.5, and √13: squaring gives (2√3)² = 4 · 3 = 12, 3.5² = 12.25, and (√13)² = 13. So 2√3 < 3.5 < √13. Squaring preserves order for nonnegative values, and it turns awkward radicals into integers.
3. Simplify the radical first. √72 = √(36 · 2) = 6√2 ≈ 6(1.414) = 8.49. Knowing √2 ≈ 1.414, √3 ≈ 1.732, and √5 ≈ 2.236 by heart converts many radical comparisons into one multiplication.
Where these appear in context
Estimation of irrationals shows up constantly inside geometry. A right triangle with legs 5 and 7 has hypotenuse √74; since 64 < 74 < 81, the hypotenuse is between 8 and 9, about 8.6. A square with area 40 square feet has side √40 ≈ 6.32 feet, so a fence around it needs about 25.3 feet.
A frequent stem asks which two consecutive integers an irrational lies between. Answer by anchoring, not by calculating: √150 sits between 12 and 13 because 144 < 150 < 169. And a frequent Competency 5 crossover asks what a student reveals by claiming √20 = 10 — the student is halving the radicand rather than finding a number whose square is 20, a misapplication of the "root undoes the power" idea.
Between which two consecutive integers does √183 lie, and which is it closer to?
Which of the following numbers is irrational?
Order these values from least to greatest: 3√2, √17, and 4.2