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Approximating Irrational Numbers

  • Grade 8
  • Mathematics
  • 8.NS.A.2
  • The Number System
  • Free lesson

Aligned to 8.NS.A.2 — Common Core State Standards for Mathematics.

What this lesson teaches

Irrational numbers like √2 or π have decimals that never end or repeat. You can approximate one by finding the two whole numbers it falls between, then narrowing down with decimals to compare sizes and place it on a number line.

Worked examples

Example 1

Estimate √10. Since 3^2 = 9 and 4^2 = 16, √10 is between 3 and 4. Try 3.1^2 = 9.61 and 3.2^2 = 10.24, so √10 is between 3.1 and 3.2. Because 10 is close to 9.61, √10 ≈ 3.16 — place it just past 3.1 on the number line.

Example 2

Estimate √50. Since 7² = 49 and 8² = 64, √50 is just past 7 — about 7.07.

Example 3

√64 = 8 exactly, so it is rational. Only roots that do not come out even, like √50, are irrational.

Practice questions

  1. Between which two whole numbers does √20 lie?
  2. Approximate √7 to one decimal place.
  3. Order √5, 2.5, and 7/3 from least to greatest on a number line.

Watch the lesson

Every lesson comes with a video taught in English and Spanish — the same video the QR code in the printed workbook opens.

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En español

Aproximación de Números Irracionales

Los números irracionales como √2 o π tienen decimales que nunca terminan ni se repiten. Puedes aproximar uno encontrando entre qué dos números enteros está, y luego afinando con decimales para comparar tamaños y ubicarlo en una recta numérica.

Ejemplo 1: Estima √10. Como 3^2 = 9 y 4^2 = 16, √10 está entre 3 y 4. Prueba 3.1^2 = 9.61 y 3.2^2 = 10.24, así que √10 está entre 3.1 y 3.2. Como 10 está cerca de 9.61, √10 ≈ 3.16 — ubícalo justo después de 3.1 en la recta numérica.

Ejemplo 2: Estima √50. Como 7² = 49 y 8² = 64, √50 está apenas después del 7 — como 7.07.

Ejemplo 3: √64 = 8 exacto, así que es racional. Solo las raíces que no salen exactas, como √50, son irracionales.

More The Number System lessons

Rational and Irrational Numbers8.NS.A.1

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