RIN - Operations with Rational and Irrational Numbers (Lesson)

Operations with Rational and Irrational Numbers

Number system image explaining rational numbers, irrational numbers, whole numbers

The chart shown represents the Real Number System. It is important to understand the rules of the real number system and how to identify various numbers. For this lesson, we will focus on rational and irrational numbers.

Rational numbers are defined as any number that can be written as a ratio.

Examples: -1/2, 8, 52, 0.33333 = 1/3 equation image indicator

Irrational numbers are numbers that cannot be written as a ratio. These are decimals that go on forever without repeating. One of the most common irrational numbers is $LaTeX: \pi$ $\pi$ . Other common irrational numbers are square roots of non-perfect squares. Cube root can be irrational as well.

Examples: equation image indicator $LaTeX: \sqrt[2]{6},\sqrt[3]{14},\:\pi$ $\sqrt[2]{6},\sqrt[3]{14},\:\pi$

Operations with Rational Numbers

The set of rational numbers includes all numbers that fit into this category. Any number that can be written as a ratio would be found in the set of rational numbers. So, what happens when we add or multiply two rational numbers? Would the sum or product remain in the set? Would the result also be a rational number? Let's explore!

Sum of Two Rational Numbers

If we add two rational numbers, will the sum be rational or irrational? Let's look at a few examples.

3 + 6 = 9

4.5 - 1.2 = 3.3

$LaTeX: \frac{1}{3}+\frac{5}{6}=\frac{7}{6}$ $\frac{1}{3}+\frac{5}{6}=\frac{7}{6}$

Take a look at the results for the above equations. They all result in a rational number! When two rational numbers are added together, the sum will always be a rational number.

Product of Two Rational Numbers

How about if we multiply two rational numbers?

3 • 6 = 18

4.5 • 1.2 = 5.4

$LaTeX: \frac{1}{3}\cdot\frac{5}{6}=\frac{5}{18}$ $\frac{1}{3}\cdot\frac{5}{6}=\frac{5}{18}$

These all result in a rational number, as well! When two rational numbers are multiplied together, the product will always be a rational number.

Operations with Rational and Irrational Numbers

What do you think might happen if we instead take one number from the set of rational numbers and one number from the set of irrational numbers and add or multiply them together? Do you think the sum or product would be rational or irrational?

Sum of a Rational and an Irrational Number

In the following three equations, the first number is rational and the second number is irrational. What do you notice about the sum?

$LaTeX: 3+\sqrt{2}=4.4142135624$ $3+\sqrt{2}=4.4142135624$

$LaTeX: 4.5+\pi=7.6415926536$ $4.5+\pi=7.6415926536$

$LaTeX: 4+\sqrt[3]{10}=6.15443469$ $4+\sqrt[3]{10}=6.15443469$

These all result in an irrational number! When a rational number and an irrational number are added, the sum will always be an irrational number.

Product of a Rational (nonzero) and an Irrational Number

Why do you think we specify that the rational number has to be "nonzero" when we are working with multiplication? If we use zero in this case, it would turn anything (including irrational numbers) into zero. That wouldn't help us find our answer. So, let's explore three equations that multiply a (nonzero) rational number and an irrational number.

$LaTeX: 3\cdot\sqrt{2}=4.2426406871$ $3\cdot\sqrt{2}=4.2426406871$

$LaTeX: 4.5\cdot\pi=14.1371669412$ $4.5\cdot\pi=14.1371669412$

$LaTeX: 4\cdot\sqrt[3]{10}=8.6177387601$ $4\cdot\sqrt[3]{10}=8.6177387601$

These all result in an irrational number, as well! When a rational number and an irrational number are multiplied, the product will always be an irrational number.

Operations with Irrational Numbers

Lastly, let's explore what happens with the set of irrational numbers. What will result when we add or multiply two irrational numbers?

Sum of Two Irrational Numbers

$LaTeX: \sqrt{3}+\sqrt{2}=3.1462643699$ $\sqrt{3}+\sqrt{2}=3.1462643699$

$LaTeX: 4.5+\pi=7.6415926536$ $4.5+\pi=7.6415926536$

$LaTeX: \sqrt[3]{10}+(-\sqrt[3]{1}0)=0$ $\sqrt[3]{10}+(-\sqrt[3]{1}0)=0$

This is interesting. The first two equations definitely result in another irrational number. What happened with the last equation? The irrational number was canceled out! This goes to show that when two irrational numbers are added, the sum will sometimes be an irrational number and will sometimes be a rational number.

Product of Two Irrational Numbers

$LaTeX: \sqrt{3}\cdot\sqrt{3}=3$ $\sqrt{3}\cdot\sqrt{3}=3$

$LaTeX: 4.5\cdot\pi=14.1371669412$ $4.5\cdot\pi=14.1371669412$

$LaTeX: \sqrt[3]{10}\cdot\sqrt[3]{10}=4.6415888336$ $\sqrt[3]{10}\cdot\sqrt[3]{10}=4.6415888336$

This discovery is similar to the last one. The product of two irrational numbers will sometimes be an irrational number and will sometimes be a rational number.

Summary

Rule	Example #1	Example #2
When two rational numbers are added together, the sum will always be a rational number.	5 + 8 = 13	$LaTeX: \frac{1}{5}+\frac{1}{3}=\frac{8}{15}$ $\frac{1}{5}+\frac{1}{3}=\frac{8}{15}$
When two rational numbers are multiplied together, the product will always be a rational number.	5 • 8 = 40	$LaTeX: \frac{1}{5}+\frac{1}{3}=\frac{1}{15}$ $\frac{1}{5}+\frac{1}{3}=\frac{1}{15}$
When a rational number and an irrational number are added together, the sum will always be an irrational number.	$LaTeX: 3+\sqrt{5}=5.2360679775$ $3+\sqrt{5}=5.2360679775$	$LaTeX: 5+\sqrt[3]{4}=6.587401052$ $5+\sqrt[3]{4}=6.587401052$
When a rational number and an irrational number are multiplied together, the product will always be an irrational number.	$LaTeX: 3\cdot\sqrt{5}=6.7082039325$ $3\cdot\sqrt{5}=6.7082039325$	$LaTeX: 5\cdot\sqrt[3]{4}=7.9370052598$ $5\cdot\sqrt[3]{4}=7.9370052598$
When two irrational numbers are added together, the sum will sometimes be an irrational number and will sometimes be a rational number.	$LaTeX: \sqrt{8}+(-\sqrt{8})=0$ $\sqrt{8}+(-\sqrt{8})=0$	$LaTeX: \sqrt[3]{7}+\sqrt[3]{5}=3.6229071294$ $\sqrt[3]{7}+\sqrt[3]{5}=3.6229071294$
When two irrational numbers are multiplied together, the product will sometimes be an irrational number and will sometimes be a rational number.	$LaTeX: \sqrt{8}\cdot\sqrt{8}=8$ $\sqrt{8}\cdot\sqrt{8}=8$	$LaTeX: \sqrt[3]{7}\cdot\sqrt[3]{5}=3.2710663102$ $\sqrt[3]{7}\cdot\sqrt[3]{5}=3.2710663102$

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