Describe the Number systems used in mathematics with the use of subsets of the real number system.

In mathematics, there are various number systems that are used to represent and manipulate different types of numbers. These number systems are typically subsets of the real number system, which is the most widely used number system. Let's explore some of the main number systems commonly used in mathematics:

1. Natural Numbers (N): The set of natural numbers includes all positive integers starting from 1 and going up infinitely. These numbers are used for counting and are denoted by the symbol N = {1, 2, 3, 4, ...}.

2. Whole Numbers (W): The set of whole numbers includes all natural numbers along with zero. It is denoted by the symbol W = {0, 1, 2, 3, 4, ...}.

3. Integers (Z): The set of integers includes all whole numbers along with their negative counterparts. It is denoted by the symbol Z = {..., -3, -2, -1, 0, 1, 2, 3, ...}.

4. Rational Numbers (Q): The set of rational numbers includes all numbers that can be expressed as a fraction, where the numerator and denominator are both integers and the denominator is not zero. It is denoted by the symbol Q. Examples of rational numbers include -3/4, 2/5, 1/3, etc.

5. Real Numbers (R): The set of real numbers includes all rational numbers and additionally includes numbers that cannot be expressed as a fraction, such as decimal numbers and irrational numbers. It is denoted by the symbol R. Examples of real numbers include 2.5, π (pi), √2 (square root of 2), etc.

6. Complex Numbers (C): The set of complex numbers includes all numbers that can be expressed in the form a + bi, where "a" and "b" are real numbers and "i" represents the imaginary unit (√-1). Complex numbers allow for the representation of numbers that are not on the real number line, and they are denoted by the symbol C.

Each number system mentioned above is a subset of the real number system. To determine which number system to use, you can consider the properties and requirements of the problem or mathematical operation at hand. For example, if you need to solve an equation involving only whole numbers, you would work within the set of whole numbers (W). Similarly, if you are dealing with a geometric calculation, you might need to work with real numbers (R) to represent decimal values.