Topics
1. Reasoning with Quantities and Expressions
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United States of America
Grades 9-12

1.01 The Real number system

Lesson
Practice

Introduction

In 7th grade, we extended our understanding of operations to include rational numbers, and in 8th grade, we were introduced to the idea that if some numbers are rational, then there must exist numbers that are not rational. Now, we will investigate the properties of irrational numbers and explore combinations of rational and irrational values. All of high school mathematics depends on the idea that we can extend the properties of rational numbers to irrational numbers.

Rational numbers

Natural, whole, and integer numbers are all examples of rational numbers. However, \dfrac{1}{2},-0.\overline{7}, and 2.123123123... are rational numbers as well. The definition of a rational number relies on our ability to rewrite it into a specific form:

Rational numbers

The set of numbers that can be expressed in the form \dfrac{a}{b} where a and b are integers and b\neq0.

Example:

-3.12,\dfrac{1}{2},2.\overline{6},5

In other words, if a number can be written as a fraction with integers for the numerator and denominator, then the number is rational.

Exploration

Complete the table by finding the sums of each row and column:

1-3\dfrac{1}{2}-0.32.\overline{2}
1
-3
\dfrac{1}{2}
-0.3
2.\overline{2}
  1. Can each sum be rewritten as a fraction of integers?
  2. What conclusions can we make about the sums of rational numbers?
  3. Complete the table again, but this time find the product of each pair of numbers.
  4. What conclusions can we make about the products of rational numbers?

The sum of two rational numbers is always rational. In the exploration, we saw that each number, including the sums, can be rewritten as a fraction where the numerator and denominator are both integers. This is also true for the product of rational numbers.

This means that the set of rational numbers is closed under addition and multiplication. We can extend this to determine that the set of rational numbers is also closed under subtraction and division.

Closure

A set of numbers is closed under an operation if performing that operation on numbers in the set produces a number that is also in the set.

Examples

Example 1

Show that the product of 3 and 0.\overline{1} is rational.

Worked Solution
Create a strategy

A rational number can be written as a fraction of integers where the denominator is not zero. If we can find the product and write it as a fraction of integers, we will prove that the product is rational.

Apply the idea

Note that 3=\dfrac{3}{1} and 0.\overline{1}=\dfrac{1}{9}.

Then we have:

\displaystyle 3\left(0.\overline{1}\right)\displaystyle =\displaystyle \dfrac{3}{1}\left(\dfrac{1}{9}\right)Substitution
\displaystyle =\displaystyle \dfrac{3}{9}Evaluate the multiplication

Since 3 and 9 are both integers, the product is rational.

Example 2

If a,b are nonzero integers, determine if \dfrac{a+b}{ab} is rational.

Worked Solution
Create a strategy

Using the fact that integers are closed under addition and multiplication, we can determine if the numerator and denominator will be integers after the operation is performed.

Apply the idea
\displaystyle a+b\displaystyle =\displaystyle \text{Integer}Closure under addition
\displaystyle ab\displaystyle =\displaystyle \text{Nonzero integer}Closure under multiplication

Therefore, \dfrac{a+b}{ab}=\dfrac{\text{Integer}}{\text{Integer}}. This is rational by the definition of rational numbers.

Example 3

If x and y are rational numbers, show that x+y is also a rational number.

Worked Solution
Create a strategy

Using the definition of rational numbers, we can rewrite x and y as fractions of integers, then use addition to simplify and verify the result.

Apply the idea

Since x and y are both rational, we know they can be rewritten so that x=\dfrac{a}{b} and y=\dfrac{c}{d} where a,b,c, and d are all integers, and b and d are nonzero.

Then we have,

\displaystyle x+y\displaystyle =\displaystyle \dfrac{a}{b}+\dfrac{c}{d}Substitution
\displaystyle =\displaystyle \dfrac{ad}{bd}+\dfrac{bc}{bd}Create a common denominator
\displaystyle =\displaystyle \dfrac{ad+bc}{bd}Add the fractions

We know that from middle school, the sum and products of integers are always integers. That means that ad, bc, and bd are all integers. We can also conclude that ad+bc is an integer. Thus, x+y=\dfrac{ad+bc}{bd}=\dfrac{\text{Integer}}{\text{Integer}}

Reflect and check

What would happen if b or d were 0?

The denominator bd, would be 0 and the number \dfrac{ad+bc}{bd} would be undefined because we cannot divide by zero.

Idea summary

We can show that a number is rational by showing that it can be rewritten as a fraction of integers where the denominator is not equal to zero.

The sums and products of rational numbers will always be rational.

Irrational numbers

Rational numbers are values that can be rewritten as a fraction of integers. But what about numbers that cannot?

Irrational numbers

The set of numbers that cannot be written in the form \dfrac{a}{b} where a and b are integers.

Example:

\dfrac{-\sqrt{102}}{5}, -\sqrt[3]{2}, \pi, e

Together, rational and irrational numbers form the real number system.

A diagram showing the Real Number System divided into Rational and Irrational. Within Rational is the subset Integers; within Integers is the subset Whole; within Whole is the subset Natural. Examples of each type of number are shown. Speak to your teacher for more information.
Real numbers

The set of rational and irrational numbers.

Example:

-3.12, \dfrac{-\sqrt{102}}{5}, \dfrac{1}{2}, \pi, 5

Exploration

Complete the tables by finding the sums and then products of the numbers in each row and column:

  1. Find each sum:

    1-3\dfrac{1}{2}0
    \sqrt{2}
    \sqrt{3}
    \pi
    • What do you think is true about the sum of a rational and irrational number? Use the table to support your reasoning.

  2. Find each product:

    1-3\dfrac{1}{2}0
    \sqrt{2}
    \sqrt{3}
    \pi
    • What do you think is true about the product of a rational and irrational number? Use the table to support your reasoning.

The sums of rational and irrational numbers are always irrational. The products of rational and irrational numbers are irrational except when the rational number is 0.

Examples

Example 4

Explain why -3\cdot \sqrt{2} must be irrational.

Worked Solution
Create a strategy

We can use what we know about sums and products of rational numbers to help support our explanation. If we can show that it can't be rational, then the number must be irrational.

Apply the idea

Suppose that -3\cdot \sqrt{2} were rational. Then -\dfrac{1}{3}\left(-3\cdot \sqrt{2}\right) would be rational because products of rational numbers are rational. But -\dfrac{1}{3}\left(-3\cdot \sqrt{2}\right)=\sqrt{2} and \sqrt{2} is irrational. Therefore, -3\cdot\sqrt{2} can't be rational.

Reflect and check

This technique of proving why something can't be true is called a proof by contradiction.

Example 5

Explain why the sum of an irrational and nonzero rational number is irrational.

Worked Solution
Create a strategy

We want this explanation to apply to any rational and irrational numbers, so we will use variables to support our explanation and use a similar approach to part (a).

Apply the idea

Let r be a rational number where r\neq 0 and let I be an irrational number. We want to show that I+r is always irrational.

Suppose that I+r was rational. Then I+r=x where x is a rational number. Then,

\displaystyle I+r\displaystyle =\displaystyle xOriginal equation
\displaystyle I\displaystyle =\displaystyle x-rSubtract r from both sides
\displaystyle I\displaystyle =\displaystyle x+(-r)Additive inverse

But this would mean I is equal to the sum of two rational numbers, and the sum of two rational numbers is always rational. Since I is irrational, x can not be rational.

Idea summary

The sum of a rational and irrational number is always irrational. The product of an irrational number and a nonzero rational number is always irrational. We can justify this relationship by assuming they are rational and showing that it is impossible.

Outcomes

N.RN.B.3

Explain why the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational.

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