Grades Grade 8 The Number SystemApproximating Irrational Numbers

Approximating Irrational Numbers

∞ The Number System

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Approximating Irrational Numbers

Locate irrational numbers on a number line by squeezing them between consecutive perfect squares.

Since irrational numbers cannot be written exactly as decimals, we approximate them. To estimate √n, find the two consecutive perfect squares it falls between, then narrow down the decimal.

\sqrt{2} \approx 1.414 \qquad \sqrt{3} \approx 1.732 \qquad \sqrt{5} \approx 2.236

🎯Estimating √20 step by step

Step 1: 4² = 16 and 5² = 25, so 4 < √20 < 5
Step 2: Try 4.4: 4.4² = 19.36 (too low)
Step 3: Try 4.5: 4.5² = 20.25 (too high)
Step 4: Try 4.47: 4.47² ≈ 19.98 ✓

√20 ≈ 4.47

4^2 = 16 < 20 < 25 = 5^2 \Rightarrow 4 < \sqrt{20} < 5

How to place √n on a number line:

1Find the largest perfect square less than n → this gives the whole number below

2Find the smallest perfect square greater than n → this gives the whole number above

3Use decimal approximation to place it precisely between the two integers

4Mark the point on the number line

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Remember This!

Use the "squeezing" strategy: if your guess squared is too low, try a larger value; if too high, try a smaller value. Each iteration gets you closer to the true value.

✏️ Try It!

Between which two consecutive integers does √50 fall?

Practice with CalcVerse

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