The Decimal Expansion of Some Irrational Numbers

Examples, solutions, and videos to help Grade 8 students learn how to use rational approximation to get the approximate decimal expansion of numbers and distinguish between rational and irrational numbers based on decimal expansions.

New York State Common Core Math Grade 8, Module 7, Lesson 11

Worksheets for Grade 8

Lesson 11 Student Outcomes • Students use rational approximation to get the approximate decimal expansion of numbers like √3 and √28
• Students distinguish between rational and irrational numbers based on decimal expansions.

Lesson 11 Summary • We know that any number that cannot be expressed as a rational number is an irrational number.
• We know that to determine the approximate value of an irrational number we must determine between which two rational numbers it would lie.
• We know that the method of rational approximation uses a sequence of rational numbers, in increments of 10⁰, 10^-1, 10^-2, and so on, to get closer and closer to a given number.
• We have a method for determining the approximate decimal expansion of the square root of an imperfect square, which is an irrational number.

Lesson 11 Classwork

Opening Exercise
Place √28 on a number line. What decimal do you think √28is equal to? Explain your reasoning.

Example 1
Recall the Basic Inequality:
Let c and d be two positive numbers, and let n be a fixed positive integer. Then c < d if and only if cⁿ < dⁿ
Write the decimal expansion of √3.
First approximation:
Second approximation:
Third approximation:

Example 2
Write the decimal expansion of √28.
First approximation:
Second approximation:
Third approximation:
Fourth approximation:

Exercise 2
Between which interval of hundredths would √14 be located? Show your work.

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