Table of Contents
Classifying numbers in the Real Number System
The set of all real numbers, also referred to as the real number system, is made up of multiple subsets.
The subsets of the set of all real numbers are:
- natural numbers
- whole numbers
- integers
- rational numbers
- irrational numbers
Every number in the real number system can be classified into one or more of these subsets. Use the interactive diagram below to explore more about each number set.
Recognizing rational and irrational numbers
Can numbers from different subsets be combined?
Numbers from different subsets can be combined by using different operations (add, subtract, multiply, divide, etc.). For example, we can add the natural number 5 with the rational number 2/3.
\[5+\frac{2}{3}=\frac{17}{3}\]As you can see, adding the natural number 5 and the rational number 2/3 resulted in another rational number, 17/3.
For our purposes here, we will focus on performing operations solely on rational and irrational numbers.
Relationships that are ALWAYS true
A note on closed number sets and operations
A closed operation in mathematics refers to an operation that, when performed on any elements within a given set, results in an output that is also within the same set.
For example, because adding two rational numbers together results in another element from the set of rational numbers, we can say that the set of rational numbers is “closed under addition.”
Likewise, because multiplying two rational number together results in another element from the set of rational numbers, we can say that the set of rational numbers is “closed under multiplication.”
If you’d like to learn more about what it means for a set to be closed, watch the following video on mathematical closure.
The sum of two rational numbers is rational.
1/4+\frac{3}{4}=\frac{4}{4}
\\\text{ }\\
6 + 7 = 13
\\\text{ }\\
2.25 + 7.15 = 9.4\]
If you want to see the proof demonstrating why the sum of two rational numbers is always a rational number, watch this video.
The product of two rational numbers is rational.
\\\text{ }\\
-2 \cdot 13 = -26
\\\text{ }\\
1.5 \cdot 1.5 = 2.25\]
If you want to see the proof demonstrating why the product of two rational numbers is always a rational number, watch this video.
The sum of a rational and an irrational number is irrational.
\frac{1}{2} + \sqrt{3} = \frac{1+2\sqrt{3}}{2}
\\\text{ }\\
3 + \sqrt{5} = 3+\sqrt{5}
\]
If you want to see the proof demonstrating why the sum of a rational number and an irrational number is irrational, watch this video.
Relationships that are ALMOST ALWAYS true
The product of a rational and an irrational number is irrational.
The product of a rational and an irrational number almost always results in an irrational number.
\[\frac{1}{3} \cdot \sqrt{5} = \frac{\sqrt{5}}{3}
\\\text{ }\\
12 \cdot \pi = \frac{\pi}{12}
\]
However, if the rational number is zero, the product is zero, which is a rational number.
\[0 \cdot \sqrt{7} = 0
\\\text{ }\\
0 \cdot \pi = 0
\]
If you want to see the proof demonstrating why the product of a rational number and an irrational number is irrational (except when the rational number is zero), watch this video.
Relationships with ambiguous results
The sum of two irrational numbers can be irrational or rational.
The sum of many irrational numbers results in an irrational number.
\[\sqrt{3} + \sqrt{3} = 2\sqrt{3}
\\\text{ }\\
\sqrt{7} + \pi = \sqrt{7} + \pi
\\\text{ }\\
\pi + 2\pi = 3\pi
\]
However, there are instances when the sum of two irrational numbers results in a rational number (for instance, the sum of additive inverses, as shown below).
\[-\sqrt{5} + \sqrt{5} = 0
\\\text{ }\\
-\pi + \pi = 0
\]
The product of two irrational numbers can be irrational or rational.
The product of many irrational numbers results in an irrational number.
\[\sqrt{3} \cdot \sqrt{5} = \sqrt{15}
\\\text{ }\\
\pi \cdot \sqrt{2} = \pi\sqrt{2}
\\\text{ }\\
\pi \cdot \pi = \pi^2
\]
However, there are instances when the product of two irrational numbers results in a rational number (for instance, the product of multiplicative inverses).
\[\sqrt{5} \cdot \sqrt{5} = \sqrt{25} = 5
\\\text{ }\\
\pi \cdot \frac{1}{\pi} = \frac{\pi}{\pi} = 1
\\\text{ }\\
\sqrt{7} \cdot \frac{1}{\sqrt{7}} = \frac{\sqrt{7}}{\sqrt{7}} = 1
\]
What about subtraction and division?
Addition and subtraction, as well as multiplication and division, are pairs of operations that are essentially the same operation viewed from different perspectives. This is because subtraction can be considered as the addition of a negative number, and division can be considered as the multiplication by a reciprocal.
Subtraction and division are just a different way of looking at addition and multiplication.
Subtraction is adding a negative
\[5-3=2
\\\text{ }\\
5+(-3)=2
\]
Division is multiplying by the reciprocal
\[15 \div 3 = 5
\\\text{ }\\
15 \cdot \frac{1}{3}=5
\]
Because of this relationship, all of the relationships between the number sets discussed previously for addition and multiplication also apply to subtraction and division, respectively.