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At-home math lessons: Rational and irrational numbers

Irrational numbers

By Michael Avidon, math editor

For Guardians: Performance Expectations (CCSS)

These lessons address the following Common Core State Standards (CCSS) for Grade 8:

  • 8.NS.A.1: Know that numbers that are not rational are called irrational. Understand informally that every number has a decimal expansion; for rational numbers show that the decimal expansion repeats eventually, and convert a decimal expansion that repeats eventually into a rational number.
  • 8.NS.A.2: Use rational approximations of irrational numbers to compare the size of irrational numbers, locate them approximately on a number line diagram, and estimate the value of expressions (e.g., π2). For example, by truncating the decimal expansion of √2, show that √2 is between 1 and 2, then between 1.4 and 1.5, and explain how to continue on to get better approximations.


As with any math lessons, a certain amount of basic material and vocabulary (taught prior to grade 8) is assumed. Specifically, it is assumed that the student is familiar with the following material.

  • Definition of rational numbers
  • Square roots and cube roots (This is grade 8 material.)
  • Arithmetic operations with decimals
  • Solving a simple linear equation; subtracting equations
  • Exponent notation
  • Area of a circle
More in this series
Decimal expansions of rational numbers

At-home math lesson: irrationality
of the square root of 2

Summary and Vocabulary
of Rational and Irrational Numbers

Lesson 2: Approximation of Irrational Numbers

At-home math lesson: Approximation of pi

At-home lessons: the golden ratio

At-home math lesson: doubling and dividing

General Information

The mathematics covered by these two standards is relatively abstract compared to other middle school topics. Some of the material (irrational numbers, non-terminating decimals) is of interest to mathematicians and those who like mathematics. That material may not be relevant to real-world applications, but finding good approximations of numbers is important.

A useful way of organizing information may be to divide it into categories or cases (this can be used on a large scale in studying some mathematical subjects, or on a small scale in solving a problem or creating a proof.) This is done here. All numbers, when written in decimal form, can be divided into three categories:

  • terminating decimals (have a finite number of decimal places)
  • repeating decimals (have an infinite number of decimal places and have a repeating pattern)
  • non-terminating, non-repeating decimals (have an infinite number of decimal places and do not have a repeating pattern)

Lesson 1

The lesson starts with the familiar operation of turning a simple decimal into a fraction (quotient of integers). In this context, this is showing that a simple (terminating) decimal is a rational number.

Page 1 fill-ins: 100; integers

Page 2 examines the operation of turning a fraction into a decimal. This is done with long division. If the division eventually leads to a 0 remainder, then the decimal terminates.

Page 2 fill-ins: 3, 1; 49

Page 3 re-examines the operation of turning a fraction into a decimal, but with an example that does not terminate, because the remainders repeat. This produces a decimal with repeating digits.

Page 3 fill-ins: first (or third); 1 [do the long division of 1 ¸ 37]

Page 4 explains why you obtain repeating decimals when you divide one integer by another (turn a fraction into a decimal).

The bottom half of the page explain how to take a repeating decimal and convert it into a fraction. Unlike the operation on page 1 (with a simple decimal), this is not a 1-step process.

Page 4 fill-ins: 11, 11; 9,999; 990

Page 5 has a brief description of irrational numbers (numbers that are not rational, or cannot be written as the quotient of integers). The study of these numbers is for the most part far beyond the 8th grade level. Students at this level need only know that irrational numbers exist, know that their decimal expansions are non-terminating and non-repeating, and recognize a handful of examples.

Exercises 1–3 correspond to Example 1. Exercise 4–6 employ the boxed fact on page 2. Exercises 7–12 correspond to the divisions on pages 2 and 3, and Example 2. Exercises 13–16 correspond to Example 3. The remaining exercises reinforce these ideas in different formats.

PRINTABLE VERSION: To help guardians follow along to help their child, print this lesson to make it easier

Lesson 2

The focus of this lesson is approximating irrational numbers and expressions to the nearest tenth. Most of the work looks at square roots (√n) and cube roots (∛n), but some expressions involving the constant p are also examined. Mathematicians use the word estimate when an interval of numbers is used to describe a value (the value is between a and b). Mathematicians use the word approximation when a more precise value is given (the value to the nearest tenth, for example).

The bottom of the first page and top of the second find an estimate and then an approximation for the square root of 40. Example 1 replaces 40 with 75.

The bottom of the second page goes through similar calculations for the cube root of 45.

Page 1 fill-ins: 7, 6

Page 2 fill-ins: 8, 9, 8.7;

Example 2 approximates another cube root. This is followed by a clarification of estimates versus approximations. Example 3 looks to approximate an expression rather than a simple radical. The number given (the golden ratio 𝜙) is actually an important constant in the world of mathematics, and is given its own Greek letter, just like the constant p. This number appears quite often in art, architecture, and in nature.

Page 3 fill-ins: 27, 2, 3

It is mentioned in passing that you might approximate something (a radical, say) to the nearest tenth, and substitute this into a more complicated expression to get an approximation for that, but the latter approximation may not be to the nearest tenth. This is examined with specific examples in the Challenge Problem found in the exercises (so I do not consider this necessarily standard material).

Example 4 provides a problem involving p rather than a radical.

Page 4 fill-ins: 2, 3, 2, 3.2; 36, 3, 28.8

Example 5 asks the student to compare two irrational expressions. This requires finding accurate approximations for each in order to reach a conclusion. The approximations are actually fairly close in value and if you estimated each less precisely, you would not be able to conclude which number is greater.

Page 5 fill-ins: 2, 1.4

Exercises 1–7 correspond to Examples 1 and 2. Exercises 8–10 correspond to Example 3. Exercises 11–13 and 15 correspond to Example 4. Exercise 14 corresponds to Example 5.

Exercises with answers

Lesson 1

Write the decimals as fractions to show that they are rational.

  1. 0.42   →42100
  2. 2. 0.967   →9671000
  3. 3. 1.0523  → 10,52310,000

A reduced fraction has the given number as a denominator.

Will the fraction be terminating or non-terminating?

  1. 250 → terminating
  2. 5. 30 → non-terminating
  3. 6. 88 → non-terminating

Write the fractions as decimals. Use an overbar, if necessary.

  1. 29 → .2
  2. 116 → .0625
  3. 833 → .24
  4. 11025 → .008
  5. 31303 → .1023
  6. 5033330 → .1510

Write the decimals as fractions.

  1. .779
  2. .61 16199
  3. .940940999
  4.  .382 379 ⁄ 990
  5. Which fractions with denominator 60 have a terminating decimal? Answer: The fractions whose numerator is a multiple of 3.
  6. For a simple repeating decimal expansion (one without a non-repeating part at the start), provide a simple description of how to immediately write it in fraction form, without using algebra. Answer: Use the repeating part as the numerator. Count the number of digits in the repeating part. Write that many 9s in the denominator.
  7. Suppose a decimal expansion starts with 0.12345678910… This shows the whole numbers from 0 to 10 in order. It continues by placing every whole number in order without terminating. Is this number rational or irrational? Explain. Answer: Though it has a pattern, it is non-terminating and non-repeating. Hence, it is irrational.
  8. Suppose a decimal expansion starts with 0.814259668314795… After this it is non-terminating, but you don’t know the rest of the digits. Can you determine if this number is rational or irrational? Explain. Answer: No. Even though no repeating pattern forms in the first 15 digits, there could be a repeating pattern starting at an unseen decimal place, so it could be rational. But there might be no repeating pattern, so it could be irrational.
  9. Explain why must be irrational. Answer: It is known that is irrational, so it has a non-terminating, non-repeating decimal expansion. When you add 1.5 to this, that changes only the ones and tenths places. Every digit from the hundredths place onward is unchanged, so this number has a non-terminating, non-repeating decimal expansion. Hence, it is irrational.
  10. Challenge Problem: Prove that the sum of any two rational numbers is rational. Hint: Let ab and cd represent the two numbers, where a, b, c, and d are integers with b ≠ 0 and d ≠ 0.

a / b + cd = adbd + bcbd = ad + bcbd. The integers are closed under multiplication and addition, so ad + bc and bd are integers. Also, bd ≠ 0. Hence, the sum is a rational number.

Lesson 2

Determine whole-number estimates and an approximation to the nearest tenth.

  1. √12  Answer: 3 < √12 < 4; 3..5
  2. √18  Answer: 4 < √18 < 5; 4.2
  3. √33 Answer: 5 < √33 <6; 5.7
  4. √60  Answer: 7 < √60 <8; 7.7
  5. ∛7  Answer: 1 < ∛7 < 2; 1.9
  6. ∛90  Answer: 4 < ∛90 < 5; 4.5
  7. Which point best represents √8? Explain. Answer: D; to the nearest tenth, √8 = 2.8.

Line graph

Use the estimates and approximations from the previous exercises to determine estimates and approximations for the following expressions.

    1. 10 – √12 Answer: 6 < 10 – √12 < 7; 6.5
    2. 2√18 +½ Answer: 8.5 < 2√18 +½ < 10.5; 8.9
    3. ∛90Answer: 43 < ∛903 < 53; 1.5

Use 3.1 < π < 3.2 to get estimates for the following expressions.

        1. π2 Answer: 1.55 < π2 < 1.6
        2. 1.1π Answer: 3.41 < 1.1π <3.52
        3. volume of a cylinder of radius 1 and height 10 Answer: between 31 and 32 cubic units
        4. Which expression has a greater value: 5√5 or 2√33? Answer: 2√33

      Square within a cirlce

      1. A square with side length 2 is inscribed in a cardboard circle. The square is cut out. Using the approximation 3.1 < π < 3.2, find an estimate for the remaining area, with the best possible upper and lower bounds. Answer: 2.2 < remaining area < 2.4


      1. Challenge Problem:

      (a) Use reasoning (and no calculator) to determine how accurate the approximation for 𝜙 in example 3 was. Answer: It was found that √5 was between 2.2 and 2.3 and closer to the first. This tells us 2.2 ≤

      √5, so 3.2 ≤ 1 + √5 < 3.25. Therefore, 1.6 ≤ 𝜙 <1.625. This says 𝜙 is within 0.025 = 140 of 1.6. This is accurate to the nearest twentieth.

      (b) Write another expression involving √5 such that using the approximation √5 ≈ 22 yields an approximation that is less accurate (not accurate to the nearest tenth).

      Answer: The approximation of 𝜙 was accurate to the nearest twentieth because of the division by 2. If you multiply by 2, for example, it will be less accurate. Using the reasoning above, we can say that 2√5 ≈ 4.4 is accurate to the nearest fifth.

      Enrichment Material

      Irrationality of √2

      You know that √4 is rational. Pretend that you do not and try to apply the reasoning used above to prove it is irrational. Explain where the proof “breaks down.”

      Answer: The digit 2 on the left side of all the equations is replaced by 4. You can conclude that m is a multiple of 2. The last equation would be n2 = q2 and no conclusion can be made about n.

      Approximation of π

      Provide the missing steps in solving the equation.
      1 + x24 = x2 → 1 = 3x24 → x2 = √43 = √4√3 = 2√3

      NEXT LESSON: Decimal expansions of rational numbers

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