183r. Classifying Number Types

Learning Intentions

• Identify rational and irrational numbers in familiar examples.
• Identify numbers as integers, fractions, decimals, rational or irrational.
• Explain why terminating and recurring decimals are rational.

Pre-requisite Summary

• A rational number can be written in the form , where and are integers and .
• An irrational number cannot be written in the form , where and are integers and .
• Integers include positive whole numbers, negative whole numbers and .
• A number can belong to more than one type. For example, is an integer and also rational because .
• Terminating decimals have a finite number of decimal places.
• Recurring decimals have a digit or block of digits that repeats forever.

Worked Examples

Worked Example 1

Classify Each Number as Rational or Irrational.

a.

b.

c.

Worked Example 2

Classify Each Number as Rational or Irrational.

a.

b.

c.

Worked Example 3

For each number, identify all number types that Apply: integer, fraction, decimal, rational or irrational.

a.

b.

c.

Worked Example 4

For each number, identify all number types that apply: integer, fraction, decimal, rational or irrational.

a.

b.

c.

Worked Example 5

Explain why each terminating decimal is rational by writing it as a fraction.

a.

b.

c.

Worked Example 6

Explain why each recurring decimal is rational by setting up an equation.

a.

b.

c.

Problems

Problem 1

Classify each number as rational or irrational.

a.

b.

c.

Problem 2

Classify each number as rational or irrational.

a.

b.

c.

Problem 3

For each number, identify all number types that apply: integer, fraction, decimal, rational or irrational.

a.

b.

c.

Problem 4

For each number, identify all number types that apply: integer, fraction, decimal, rational or irrational.

a.

b.

c.

Problem 5

Explain why each terminating decimal is rational by writing it as a fraction.

a.

b.

c.

Problem 6

Explain why each recurring decimal is rational by setting up an equation.

a.

b.

c.

Exercises

Understanding and Fluency

Exercise 1.

Classify each number as rational or irrational.

a.

b.

c.

Exercise 2.

Classify each number as rational or irrational.

a.

b.

c.

Exercise 3.

For each number, identify all number types that apply: integer, fraction, decimal, rational or irrational.

a.

b.

c.

Exercise 4.

For each number, identify all number types that apply: integer, fraction, decimal, rational or irrational.

a.

b.

c.

Exercise 5.

Write each terminating decimal as a fraction to show that it is rational.

a.

b.

c.

Exercise 6.

Write each recurring decimal as a fraction to show that it is rational.

a.

b.

c.

Exercise 7.

Copy and complete each statement Use the words rational or irrational.

a. Every integer is __________.

b. A non-terminating, non-recurring decimal is __________.

c. A recurring decimal is __________.

Exercise 8.

Classify each number as rational or irrational.

a.

b. where the digits do not repeat

c.

Exercise 9.

For each number, identify all number types that apply: integer, fraction, decimal, rational or irrational.

a.

b.

c.

Exercise 10.

Write each number in the form , where and are integers and .

a.

b.

c.

Reasoning

Exercise 11.

Explain why is both an integer and a rational number.

Exercise 12.

A student says, “ is irrational because it has a square root symbol.” Explain why this is incorrect.

Exercise 13.

A student says, “All decimals are irrational.” Explain why this is incorrect using two examples.

Exercise 14.

Explain why is rational without using a calculator.

Exercise 15.

Explain why is rational by describing how it can be changed into a fraction.

Exercise 16.

Decide whether each statement is always true, sometimes true or never true. Explain your reasoning.

a. Every fraction is rational.

b. Every rational number is an integer.

c. Every irrational number is a decimal.

Problem-solving

Exercise 17.

Create a set of five numbers that includes:

• one positive integer
• one negative integer
• one terminating decimal
• one recurring decimal
• one irrational number

Then classify each number as rational or irrational.

Exercise 18.

Place the following numbers into a Venn diagram with the categories rational and irrational.

, , , , , ,

Exercise 19.

Three students are classifying the number .

• Aria says it is irrational because it goes forever.
• Ben says it is rational because it repeats.
• Chen says it is an integer.

Decide who is correct and explain why.

Exercise 20.

A number has the following properties:

• It is rational.
• It is a decimal.
• It is not an integer.
• It does not terminate.

Give two possible numbers and explain why they match all the properties.

Potential Misunderstandings

• Students may think that all square roots are irrational, but square roots such as and are integers and rational.
• Students may think that a number can only have one classification, but a number such as is both an integer and rational.
• Students may think that fractions are a separate type from rational numbers, but any fraction of the form , where and are integers and , is rational.
• Students may think that all decimals are irrational, but terminating decimals and recurring decimals are rational.
• Students may think that a decimal is irrational just because it continues forever, but recurring decimals continue forever and are still rational.
• Students may think that is less than , but , so it is rational.
• Students may think that is rational because it is often approximated by , but is only an approximation of .

Next: 184. Surds and Decimals