Before defining what a number is we first have to look at what a set is.
A. Set
In mathematics, a set is a collection of distinct objects (generally referred to as elements) for which we can decide whether or not a given object belongs. A set can have any group of items, be it a collection of numbers, days of a week, types of vehicles, and so on. For example, the set of days of the week is a set that contains 7 objects: Mon., Tue., Wed., Thur., Fri., Sat., and Sun..
Sets are exhibited by providing a list of it's elements inside two curly brackets. Sets are named and represented using capital letters. For instance, the collection of even natural numbers less than 10 can be represented in the form of a set, A = {2, 4, 6, 8}, which is a finite set.. The order in which the objects of a set are written doesn't matter. Sets can be finite or infinite.
B. Subset
One set is a subset of another set if every object in the first set is an object of the second set as well. For instance, the set of weekdays is a subset of the set of days of the week, since every weekday is a day of the week.
C. Number/Numeral
The mental concept of number is of a mathematical object used to count, measure and label. Therefore, when we speak of numbers, we are referring to a particular set of numbers (sometimes called number systems).
A numeral is a symbol that represents a number. In common usage, number and numeral are used interchangeably.
II. Number Sets
The following are the most common sets of numbers, each increasing in complexity.
- Natural Numbers (counting numbers) - The set of numbers {1, 2, 3, 4, ...}
- Whole Numbers - The set of numbers {0, 1, 2, 3, 4, ...}
- Integers - The set of numbers {...,-3,-2,-1,0,1,2,3,…}
- Rational Numbers - A number that can be written in fraction form such as a/b where a and b are integers, but b is not equal to 0. This includes integers, terminating decimals, and repeating decimals as well as fractions.
- Irrational Numbers - A number whose decimal form is nonterminating and nonrepeating. Irrational numbers cannot be written in the form a/b, where a and b are integers (b cannot be zero). So all numbers that are not rational are irrational. π is a well know irrational number.
- Real Numbers - All the rational and irrational numbers; that is, all of the numbers that can be expressed as decimals.
- Complex Numbers - Numbers in the form of a+bi, where a is the real part of the number and bi is the imaginary part of the number.
