A number set is a set of numbers which have a specific quality in common. They are used to categorize and understand specific types of numbers.
Natural numbers are numbers counted, denoted by the symbol \(\mathbb{N}\), or in some textbooks, by a bolded letter, N, starting from \(1\) and onward. They can be written as such:
\(1, 2, 3, 4, 5, 6, ...\)
The ellipses (or the three dots at the end) indicates that this set will continue on from there. This number set (as well as the others) is infinite, starting from \(1\) then adding \(1\) each time. This process goes on forever.
NOTE: This set does not include \(0\), negative numbers, or decimals.
Whole numbers can be summarized as the set of natural numbers but with \(0\) included. They can be denoted either by the symbol \(\mathbb{W}\), or by a bolded letter, W. They can be written as such:
\(0, 1, 2, 3, 4, 5, ...\)
Although a whole number set increases by \(1\) each time, unlike a set of natural numbers, it begins at \(0\) instead of \(1\).
Integers are sets of numbers that include whole numbers as well as their negative counterparts. They can be denoted either by the symbol \(\mathbb{Z}\), or by a bolded letter, Z. They can be written as such:
\(... , -3, -2, 1, 0, 1, 2, 3, ...\)
Much like the previous sets, each number is separated by values of \(1\). Additionally, there are no decimals or fractions involved.
Rational numbers are numbers that can be written as a fraction. It can be denoted either by the symbol \(\mathbb{Q}\) or by a bolded letter, Q.
To explain this further, a rational number is a number \(\cfrac{m}{n}\) where \(m\) and \(n\) are numbers from the set of integers but \(n \ne 0\).
When written as a decimal, there will either be a clear end after the decimal (i.e. \(1.75\)) or there will be a repeated value after the decimal point (i.e. \(0.666667\)... which is equal to \(13\) or \(0.833333...\) which is equal to \(\cfrac{5}{6}\)).
Some examples of rational numbers include:
\(-2, 0.19191919..., \cfrac{2}{5}, 85, -\cfrac{4}{9}\)
Irrational numbers are numbers that cannot be written as fractions, there is no clear end to the numbers written after the decimal, and there is no distinctive pattern when written as decimal. It can be denoted either by the symbol \(\mathbb{R} \backslash \mathbb{Q}\) or by the bolded letters R\Q. The reason why will be elaborated on soon.
Up to this point, you may have already seen a few irrational numbers such as . For reference, the value of is \(3.1415926535...\) and will continue on with no noticeable pattern to which digits will occur after the decimal.
A few other examples of irrational numbers include:
\(\sqrt{2} = 1.41421356237...\)
\(\sqrt{3} = 1.73205080757..\)
\(9.15483695763963201...\)
A real number is a number that has some sort of value. It contains the entirety of the set of rational numbers and the set of irrational numbers. It can be denoted either by the symbol \(\mathbb{R}\) or by a bolded letter, R.
The set of real numbers contains every single number set that we have gone over thus far.
NOTE: This is why the symbol \(\mathbb{R \backslash Q}\) is used for irrational numbers as it means “real numbers minus rational numbers.”
There is another set of numbers known as imaginary numbers in contrast with real numbers. This set includes the \(i\) which is equal to \(\sqrt{-1}\) but we do not need to elaborate beyond that.