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Approximate
Irrational Numbers

Learn to estimate, compare, and locate irrational numbers on a number line using rational approximations.

8th Grade Math

What Are Irrational Numbers?

An irrational number is a real number that cannot be written as a fraction \(\frac{a}{b}\), where \(a\) and \(b\) are integers and \(b \neq 0\). Their decimal expansions go on forever without repeating.

Irrational
\(\sqrt{2} = 1.41421356...\) Never terminates, never repeats
Rational
\(\frac{1}{3} = 0.333...\) Repeats the digit 3 forever

Why Approximate Them?

Since irrational numbers have infinitely long, non-repeating decimals, we can never write them out exactly. But we can get as close as we need using rational approximations.

Compare
Determine which of two irrational numbers is larger or smaller
Locate
Place irrational numbers on a number line diagram
Estimate
Find approximate values of expressions like \(\pi^2\)

Irrational Numbers on the Number Line

Every irrational number lives between two rational numbers. We can zoom in to place it more precisely.

1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 \(\sqrt{2}\) between 1.4 and 1.5

The Squeezing Method

To approximate \(\sqrt{n}\), find two consecutive perfect squares that \(n\) falls between. Then narrow down using decimals.

1
Find the perfect squares. Identify integers \(a\) and \(b\) where \(a^2 < n < b^2\). Then \(a < \sqrt{n} < b\).
2
Try tenths. Square numbers like \(a.1, a.2, a.3, ...\) until you find \(a.k^2 < n < (a.k+1)^2\).
3
Try hundredths. Repeat with \(a.k1, a.k2, ...\) for even more precision.
4
Repeat as needed. Each step gives one more correct decimal place.

Example: Approximating \(\sqrt{2}\)

1
Perfect squares: \(1^2 = 1\) and \(2^2 = 4\). Since \(1 < 2 < 4\), we know \(1 < \sqrt{2} < 2\).
2
Try tenths: \(1.4^2 = 1.96\) and \(1.5^2 = 2.25\). Since \(1.96 < 2 < 2.25\), we know \(1.4 < \sqrt{2} < 1.5\).
3
Try hundredths: \(1.41^2 = 1.9881\) and \(1.42^2 = 2.0164\). Since \(1.9881 < 2 < 2.0164\), we know \(1.41 < \sqrt{2} < 1.42\).

Visualizing the Squeeze

Step 1 1 2 Step 2 1.4 1.5 Step 3 1.41 1.42 \(\sqrt{2} \approx 1.414\)

Rational vs. Irrational

Rational Numbers Irrational Numbers
Fraction form Can be written as \(\frac{a}{b}\) Cannot be written as \(\frac{a}{b}\)
Decimal Terminates or repeats Never terminates or repeats
Examples \(\frac{1}{3},\ 0.75,\ 2\) \(\sqrt{2},\ \pi,\ e\)
On number line Exact location known Approximated by squeezing

Know Your Perfect Squares

Memorizing perfect squares is the first step in approximating any square root. These are your anchors on the number line.

\(n\) 1 2 3 4 5 6 7 8 9 10
\(n^2\) 1 4 9 16 25 36 49 64 81 100

Step-by-Step Approach

1
Identify the type. Is it \(\sqrt{n}\), \(\pi\)-based, or another irrational expression?
2
Find integer bounds. For \(\sqrt{n}\), find perfect squares above and below \(n\).
3
Narrow with tenths. Square trial values at the tenths place to close in.
4
Keep refining. Go to hundredths or thousandths if needed for the problem.
5
Plot or compare. Use your approximation to place on a number line or compare with other values.

Example: Approximating \(\sqrt{5}\)

1
Integer bounds: \(2^2 = 4\) and \(3^2 = 9\). So \(2 < \sqrt{5} < 3\).
2
Tenths: \(2.2^2 = 4.84\) and \(2.3^2 = 5.29\). So \(2.2 < \sqrt{5} < 2.3\).
3
Hundredths: \(2.23^2 = 4.9729\) and \(2.24^2 = 5.0176\). So \(2.23 < \sqrt{5} < 2.24\).
2 3 2.2 2.3 \(\sqrt{5} \approx 2.236\)

Estimating Expressions: \(\pi^2\)

We can approximate expressions involving irrational numbers by using their decimal approximations.

1
Start with \(\pi \approx 3.14159\). For estimation, we can use \(\pi \approx 3.14\).
2
Compute: \(\pi^2 \approx 3.14^2 = 9.8596\).
3
Estimate: \(\pi^2\) is close to 10 but slightly less: about \(9.87\).

Comparing Irrational Numbers

To compare irrational numbers, approximate each one to enough decimal places so they differ.

1
\(\sqrt{8}\): Since \(2.8^2 = 7.84\) and \(2.9^2 = 8.41\), we get \(2.8 < \sqrt{8} < 2.9\). More precisely, \(2.82^2 = 7.9524\) and \(2.83^2 = 8.0089\), so \(\sqrt{8} \approx 2.828\).
2
\(\pi \approx 3.14159\).
3
Since \(2.828 < 3.14\), we conclude \(\sqrt{8} < \pi\).

Ordering on the Number Line

Place in order: \(\sqrt{3}\), \(\sqrt{10}\), \(\pi\)

1 2 3 4 \(\sqrt{3} \approx 1.73\) \(\pi \approx 3.14\) \(\sqrt{10} \approx 3.16\)

Common Mistakes

✕ Confusing \(\sqrt{n}\) with \(\frac{n}{2}\)
\(\sqrt{16} = 4\), not 8. The square root is the number that multiplied by itself gives \(n\).
✕ Rounding Too Early
Using \(\pi \approx 3\) when you need \(\pi^2\) gives \(9\) instead of the correct \(\approx 9.87\). Use enough decimal places.
✕ Calling \(\sqrt{4}\) Irrational
\(\sqrt{4} = 2\), a whole number! Only non-perfect-square roots (like \(\sqrt{5}\)) are irrational.

Summary of Key Approximations

Expression Integer Bounds Approximation
\(\sqrt{2}\) \(1 < \sqrt{2} < 2\) \(\approx 1.414\)
\(\sqrt{3}\) \(1 < \sqrt{3} < 2\) \(\approx 1.732\)
\(\sqrt{5}\) \(2 < \sqrt{5} < 3\) \(\approx 2.236\)
\(\pi\) \(3 < \pi < 4\) \(\approx 3.14159\)
\(\pi^2\) \(9 < \pi^2 < 10\) \(\approx 9.87\)
\(\sqrt{10}\) \(3 < \sqrt{10} < 4\) \(\approx 3.162\)

Question 1 of 5

Between which two consecutive integers does \(\sqrt{17}\) fall?

Question 2 of 5

Which is the best approximation of \(\sqrt{50}\) to the tenths place?

Question 3 of 5

Which is larger: \(\sqrt{11}\) or \(\pi\)?

Question 4 of 5

What is the best estimate for \(\pi^2\)?

Question 5 of 5

A student claims that \(\sqrt{2} = 1.41\) exactly. Is this correct?