The Real Numbers
The Natural Numbers, \(\mathbb{N}\)
Our first step in building up to the real numbers is the natural numbers. These are typically the first numbers people are taught. They are the numbers you can use to count your fingers and toes and so on. All we need to do is start with \(1\) and add \(1\) over and over. The pattern looks like this:
$$\mathbb{N} = \left\{ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, \ldots \right\}$$The Whole Numbers, \(\mathbb{W}\)
There is no real clear definition of the whole numbers. There are three main interpretation of the term whole numbers that I see. The first interpretation is that they are just the integers, and the second one refers them to be to the natural numbers. The last interpretation is that the whole numbers are the natural numbers with the addition of zero. This is the interpretation presented by most textbooks used at Ivy Tech Community College where I tutor, so that is the one I will go with for now.
$$\mathbb{W} = \left\{ 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, \ldots \right\}$$The Integers, \(\mathbb{Z}\)
The integers or integer numbers is the set of numbers containing the natural numbers, the negatives of the natural numbers, and zero.
$$\mathbb{Z} = \left\{ \ldots, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, \ldots \right\}$$The Rational Numbers, \(\mathbb{Q}\)
A rational number is a number that consists of a fraction with an integer in the numerator and a non-zero integer in the denominator.
$$m \in \mathbb{Z}, n \in \mathbb{Z}, n \neq 0$$ $$\frac{m}{n} \in \mathbb{Q}$$The rational numbers contain all integers because an integer divided by \(1\) fits the definition of a rational number. When written in decimal form, a rational number will have either a terminating decimal expansion or a non-terminating, repeating decimal expansion. Here are some examples.
$$\frac{1}{3} = 0.333333333333\ldots$$ $$\frac{1}{2} = 0.5$$ $$\frac{77}{1} = 77$$ $$\frac{22}{4} = \frac{11}{2} = 5.5$$The Irrational Numbers, \(\mathbb{I}\)
An irrational number cannot be written as a fraction with an integer in the numerator and a non-zero integer in the denominator.
$$m \in \mathbb{Z}, n \in \mathbb{Z}, n \neq 0$$ $$\frac{m}{n} \notin \mathbb{I}$$Irrational numbers are not rational numbers, and rational numbers are not rational numbers. When written in decimal form, an irrational number will always have a non-terminating, non-repeating decimal expansion. For practical applications, one has to approximate an irrational number by a rational number at the desired precision. For example, you might use the approximation \(\pi \approx 3.14 \). Some examples of irrational numbers are listed below.
$$\sqrt{2}, \pi, e, \sqrt{3}, \frac{\sqrt{2}}{2}, 5^{\frac{1}{6}}, \cos{2}$$The Real Numbers, \(\mathbb{R}\)
The set of real numbers contains all rational and irrational numbers.