Number Sets
We will be discussing four important sets of numbers: natural numbers, integer numbers, rational numbers, and real numbers. In future weeks of the class we will dive deeper into the meaning, properties, and operations regarding general sets, but for the purpose of this section we can just think of a set as a collection of numbers.
Each of the following sets are traditionally represented with the double struck capital letters $\mathbb{N}, \mathbb{Z}, \mathbb{Q}, \mathbb{R}$ for natural numbers, integers, rational numbers and real numbers. We will give a few examples of their elements that encompass these sets.
Natural Numbers
We begin with natural numbers, which are the counting numbers we use to count objects. Depending on who you ask, Natural numbers start from 0 or from 1. Regardless they continue indefinitely. You should always make sure that you are in accord with the text, video, class or person that you are interacting with on whether or not you consider 0 to be part of the natural numbers.
For the purpose of this class the Natural Numbers start at 0. They are denoted by the symbol $\mathbb{N}={0, 1, 2, 3, 4, ...}$.
Integer Numbers
Building upon natural numbers, we introduce integers. Integers include all the natural numbers and their negatives. In other words, integers are positive and negative whole numbers, along with zero. We use the symbol $\mathbb{Z}={..., -3, -2, -1, 0, 1, 2, 3, ...}$ to represent the set of integers.
Rational Numbers
Next, we move on to rational numbers. Rational numbers are numbers that can be expressed as fractions, where the numerator and denominator are both integers. These numbers can be written in the form $p/q$, where $p$ and $q$ are integers, and $q$ is not equal to zero. Rational numbers include both terminating decimals and recurring decimals. Some examples of rational numbers are 1/2, -3/4, and 0.25. The set of rational numbers is denoted by the symbol $\mathbb{Q}$.
Real Numbers
Finally, we come to real numbers, which encompass all rational and irrational numbers. Real numbers are essentially the numbers we use in everyday life. They include fractions, decimals, and even numbers that cannot be expressed as fractions, such as the square root of 2 (√2) or pi (π). Real numbers are represented by the symbol $\mathbb{R}$.
Summary
Natural numbers ($\mathbb{N}$) are the counting numbers starting from 0.
Integer numbers ($\mathbb{Z}$) include natural numbers, and their negatives.
Rational numbers ($\mathbb{Q}$) are numbers that can be expressed as fractions.
Real numbers ($\mathbb{R}$) include both rational and irrational numbers.
Understanding these number systems is fundamental in mathematics and various other fields. They allow us to perform calculations, solve equations, measure quantities, and explore the vastness of mathematical concepts.