How to Approximate Irrational Numbers? (+FREE Worksheet!)

Approximating irrational numbers means finding a close decimal or fraction estimate for numbers like √2 or √50 that cannot be expressed as exact fractions. This skill connects number sense to geometry (the Pythagorean theorem often produces irrational lengths) and is tested in Algebra 1 and on standardized exams. This lesson shows you exactly how to squeeze an irrational number between two consecutive integers and then narrow the estimate further.

What Are Irrational Numbers?

A rational number can be written as a fraction of two integers (e.g., \(\color{blue}{\frac{3}{4}}\), −5, 0.6). An irrational number cannot — its decimal representation goes on forever without repeating. Common irrational numbers include:

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• √2 ≈ 1.4142…
• √3 ≈ 1.7321…
• π ≈ 3.14159…

Square roots of non-perfect-square integers are always irrational.

How to Approximate an Irrational Number

Step 1 — Identify the Perfect Squares on Each Side

Find the two consecutive perfect squares that your number falls between. For example, to approximate √20:

• \(\color{blue}{4^{2} = 16}\) and \(\color{blue}{5^{2} = 25}\), and \(\color{blue}{16 < 20 < 25}\).
• So \(\color{blue}{4 < \sqrt{20} < 5}\).

Step 2 — Narrow the Estimate with Tenths

Test values between 4 and 5. Try 4.4: \(\color{blue}{4.4^{2} = 19.36}\). Try 4.5: \(\color{blue}{4.5^{2} = 20.25}\). So \(\color{blue}{4.4 < \sqrt{20} < 4.5}\).

The actual value is √20 ≈ 4.47.

Step 3 — Plot on a Number Line

Place the approximation between the two consecutive integers on the number line to visualize its location.

Step-by-Step Summary

1. Find the two consecutive perfect squares that your number (under the radical) falls between.
2. Take the square roots of those perfect squares to identify the two consecutive integers that bound the irrational number.
3. Test tenths between the two integers to narrow the estimate.
4. Round to the desired decimal place.

Watch: How to Approximate Square Roots

Khan Academy demonstrates the number-line method for approximating square roots like √32, √55, and √123:

Approximating Irrational Numbers – Worked Examples

Example 1: Approximate √20 to one decimal place.

\(\color{blue}{4^{2} = 16}\) and \(\color{blue}{5^{2} = 25}\), so \(\color{blue}{4 < \sqrt{20} < 5}\).
Test: \(\color{blue}{4.4^{2} = 19.36}\) and \(\color{blue}{4.5^{2} = 20.25}\). So \(\color{blue}{\sqrt{20} \approx 4.5}\) (to 1 d.p.).

Example 2: Approximate √50 to one decimal place.

\(\color{blue}{7^{2} = 49}\) and \(\color{blue}{8^{2} = 64}\), so \(\color{blue}{7 < \sqrt{50} < 8}\).
Test: \(\color{blue}{7.0^{2} = 49.0}\) and \(\color{blue}{7.1^{2} = 50.41}\). So \(\color{blue}{\sqrt{50} \approx 7.1}\).

Example 3: Approximate √75 to one decimal place.

\(\color{blue}{8^{2} = 64}\) and \(\color{blue}{9^{2} = 81}\), so \(\color{blue}{8 < \sqrt{75} < 9}\).
Test: \(\color{blue}{8.6^{2} = 73.96}\) and \(\color{blue}{8.7^{2} = 75.69}\). So \(\color{blue}{\sqrt{75} \approx 8.7}\).

Example 4: Between which two consecutive integers does √27 lie?

\(\color{blue}{5^{2} = 25}\) and \(\color{blue}{6^{2} = 36}\). Since \(\color{blue}{25 < 27 < 36}\), we know \(\color{blue}{5 < \sqrt{27} < 6}\). (√27 ≈ 5.20.)

More Practice: Approximating Irrational Square Roots

The Magic of Math shows how to approximate irrational square roots and place them on the number line:

Exercises for Approximating Irrational Numbers

For each irrational number, name the two consecutive integers it falls between and give a decimal approximation to one decimal place.

1. √3
2. √10
3. √27
4. √45
5. √80
6. √99

Answers

1. Between 1 and 2; √3 ≈ \(\color{blue}{1.7}\)
2. Between 3 and 4; √10 ≈ \(\color{blue}{3.2}\)
3. Between 5 and 6; √27 ≈ \(\color{blue}{5.2}\)
4. Between 6 and 7; √45 ≈ \(\color{blue}{6.7}\)
5. Between 8 and 9; √80 ≈ \(\color{blue}{8.9}\)
6. Between 9 and 10; √99 ≈ \(\color{blue}{9.9}\)

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Free Approximating Irrational Numbers Worksheet

Ready to practice on your own? Download our free Approximating Irrational Numbers worksheet below, work through each problem at your own pace, and then check your answers. If a few give you trouble, scroll back up to the worked examples and try again, steady practice is the surest way to master Approximating Irrational Numbers before a quiz or test.

Download Rational and Irrational Numbers Worksheet

Going further: hundredths and cube roots

Sometimes one decimal place is not enough. Keep the same idea and zoom in one more step.

Example: √40 to the nearest hundredth. You already know √40 is between 6.3 and 6.4. Now test hundredths: 6.32² = 39.9424 and 6.33² = 40.0689. The gap from 40 to 39.9424 is 0.0576, and the gap from 40.0689 to 40 is 0.0689. The first gap is smaller, so √40 ≈ 6.32.

Example: ∛30 to the nearest tenth. Use perfect cubes instead of perfect squares. Since 3³ = 27 and 4³ = 64, ∛30 is between 3 and 4, and much closer to 3. Test 3.1³ = 29.791 and 3.2³ = 32.768. The value 30 is much closer to 29.791, so ∛30 ≈ 3.1.

Comparing and ordering irrational numbers on a number line

To order irrational numbers, approximate each one to the same number of decimal places, then compare.

Example: Order √8, π, √10 and 3.1 from least to greatest.

• √8 ≈ 2.83 (because 2.8² = 7.84 and 2.9² = 8.41)
• 3.1 is already a decimal
• π ≈ 3.14
• √10 ≈ 3.16 (because 3.1² = 9.61 and 3.2² = 10.24)

So the order is √8 < 3.1 < π < √10. On a number line, √8 sits a little before 3, while 3.1, π and √10 all sit close together just past 3. When two values are this close, use hundredths to separate them.

Frequently Asked Questions

How do I know if a square root is irrational?

A square root √n is irrational unless n is a perfect square (0, 1, 4, 9, 16, 25, …). If n is not a perfect square, √n is irrational.

Is π also an irrational number?

Yes. π ≈ 3.14159… is irrational because its decimal expansion never terminates or repeats. It is approximated as 3.14 or \(\color{blue}{\frac{22}{7}}\) for most calculations.

Why do we need to approximate irrational numbers?

Irrational numbers cannot be written as exact decimals or fractions. Approximations allow us to compare their sizes, place them on a number line, and use them in practical calculations like measuring lengths from the Pythagorean theorem.

How do you place an irrational number on a number line?

First find the two whole numbers it sits between, using perfect squares. Then estimate to the tenths and mark that spot. For √20, you know 16 < 20 < 25, so it is between 4 and 5. Since 4.4² = 19.36 and 4.5² = 20.25, it sits just under 4.5, at about 4.47.

Can you approximate a cube root the same way?

Yes. Use perfect cubes in place of perfect squares. For ∛30, note that 3³ = 27 and 4³ = 64, so the root is between 3 and 4. Then test decimals: 3.1³ = 29.791, which is very close to 30, so ∛30 ≈ 3.1.

What fraction is a good approximation of π?

The fraction 22/7 is a common choice. It equals about 3.142857, which is close to π ≈ 3.141593. For a much closer estimate, 355/113 ≈ 3.1415929 matches π to six decimal places. Both are approximations, since π itself cannot be written exactly as a fraction.

Is 3.14 equal to π?

No. 3.14 is a rounded approximation. π is irrational, so its decimal goes on forever without repeating: 3.14159265… Using 3.14 is fine for most school problems, but the answer you get is an estimate. Some problems ask you to leave π in the answer, like 9π, to keep it exact.

Related Topics

• Integers and Absolute Value
• Order of Operations
• Exponential Growth and Decay
• The Distributive Property
• Proportional Ratios