Learn on PengienVision, Mathematics, Grade 8Chapter 1: Real Numbers

Lesson 2: Understand Irrational Numbers

In this Grade 8 lesson from enVision Mathematics Chapter 1, students learn to identify and classify irrational numbers — numbers whose decimal expansions are nonrepeating and nonterminating and cannot be written in the form a/b. Students explore how to distinguish irrational numbers from rational numbers using examples such as non-perfect square roots like the square root of 3 and decimals like 0.24758326. The lesson builds understanding of the real number system by placing irrational numbers within the broader Venn diagram of number sets.

Section 1

Placing Irrational Numbers

Property

The set of natural numbers includes the numbers used for counting: {1, 2, 3, …}.
The set of whole numbers is the set of natural numbers plus zero: {0, 1, 2, 3, …}.
The set of integers adds the negative natural numbers to the set of whole numbers: {…, -3, -2, -1, 0, 1, 2, 3, …}.
The set of rational numbers includes fractions written as {mn∣m and n are integers and n≠0}\{\frac{m}{n} | m \text{ and } n \text{ are integers and } n \neq 0\}.
The set of irrational numbers is the set of numbers that are not rational, are nonrepeating, and are nonterminating: {hh | hh is not a rational number}.

Examples

  • Classify 64\sqrt{64}: This simplifies to 88. So, it is a natural number (N), whole number (W), integer (I), and rational number (Q).
  • Classify 143\frac{14}{3}: As a fraction of integers, it is a rational number (Q). As a decimal, it is 4.666...4.666..., which is a repeating decimal.
  • Classify 13\sqrt{13}: This cannot be simplified to a whole number or a fraction of integers, so it is an irrational number (Q').

Explanation

Think of number sets like nesting dolls. Naturals fit inside wholes, which fit inside integers, which fit inside rationals. Irrationals are a separate group, and all of them together form the real numbers.

Section 2

Irrational number

Property

An irrational number is a number that cannot be written as the ratio of two integers. Its decimal form does not stop and does not repeat.
Famous examples include π\pi and the square roots of numbers that are not perfect squares.

Examples

  • The number π\pi is a famous irrational number, beginning with 3.14159...3.14159... and continuing infinitely without repetition.
  • The square root of 3, 3\sqrt{3}, is irrational because 3 is not a perfect square. Its decimal form is 1.7320508...1.7320508....
  • A decimal that continues without a pattern, such as 67.121231234...67.121231234..., is an irrational number.

Explanation

Irrational numbers cannot be written as a simple fraction. Their decimal representations are infinite and non-repeating, meaning they go on forever without any predictable pattern. Think of them as the 'wild' numbers on the number line.

Lesson overview

Expand to review the lesson summary and core properties.

Expand

Section 1

Placing Irrational Numbers

Property

The set of natural numbers includes the numbers used for counting: {1, 2, 3, …}.
The set of whole numbers is the set of natural numbers plus zero: {0, 1, 2, 3, …}.
The set of integers adds the negative natural numbers to the set of whole numbers: {…, -3, -2, -1, 0, 1, 2, 3, …}.
The set of rational numbers includes fractions written as {mn∣m and n are integers and n≠0}\{\frac{m}{n} | m \text{ and } n \text{ are integers and } n \neq 0\}.
The set of irrational numbers is the set of numbers that are not rational, are nonrepeating, and are nonterminating: {hh | hh is not a rational number}.

Examples

  • Classify 64\sqrt{64}: This simplifies to 88. So, it is a natural number (N), whole number (W), integer (I), and rational number (Q).
  • Classify 143\frac{14}{3}: As a fraction of integers, it is a rational number (Q). As a decimal, it is 4.666...4.666..., which is a repeating decimal.
  • Classify 13\sqrt{13}: This cannot be simplified to a whole number or a fraction of integers, so it is an irrational number (Q').

Explanation

Think of number sets like nesting dolls. Naturals fit inside wholes, which fit inside integers, which fit inside rationals. Irrationals are a separate group, and all of them together form the real numbers.

Section 2

Irrational number

Property

An irrational number is a number that cannot be written as the ratio of two integers. Its decimal form does not stop and does not repeat.
Famous examples include π\pi and the square roots of numbers that are not perfect squares.

Examples

  • The number π\pi is a famous irrational number, beginning with 3.14159...3.14159... and continuing infinitely without repetition.
  • The square root of 3, 3\sqrt{3}, is irrational because 3 is not a perfect square. Its decimal form is 1.7320508...1.7320508....
  • A decimal that continues without a pattern, such as 67.121231234...67.121231234..., is an irrational number.

Explanation

Irrational numbers cannot be written as a simple fraction. Their decimal representations are infinite and non-repeating, meaning they go on forever without any predictable pattern. Think of them as the 'wild' numbers on the number line.