Understanding the Real Number System

Mathematics can be quite perplexing, especially for those who struggle to grasp its concepts, which includes many of us.

In math, mastering terminology is essential to comprehend the nature of numbers, their various types, and the unique traits each possesses. Numbers serve as mathematical symbols for counting and measuring, yet they differ significantly in their properties.

For example, consider the idea of the "real number system." If numbers can be real, does that imply the existence of fake numbers? In reality, there are real numbers and imaginary numbers. But what exactly does that signify?

Real Numbers Are All the Numbers

Real numbers encompass every number you might conceive if asked to think of one. They originate from the idea of the number line: positive numbers to the right of zero and negative numbers to the left. Any number that can be placed on this line is a real number. Examples include 27, -198.3, 0, 32/9, and even 5 billion.

Interestingly, numbers like √2 (the square root of 2, approximately 1.14142...) and the decimal form of π (3.1415...) can also be plotted on the number line, despite being nonterminating decimals. Even though their decimal representations never end, they still occupy specific positions on the line.

Real numbers comprise natural numbers, whole numbers, integers, rational numbers, irrational numbers, and real numbers themselves.

Natural Numbers

Natural numbers, often referred to as counting numbers, start from 1 and extend infinitely. These positive numbers are located to the right of 0 on the number line. Every natural number you select is also classified as a positive integer.

In mathematical terms, counting numbers are represented as: N = {1, 2, 3, 4, 5, 6, ...}.

Whole Numbers

Whole numbers include all natural numbers along with 0. Mathematically, they are denoted as: W = {0, 1, 2, 3, 4, 5, ...}.

Integers

Integers consist of whole numbers (0 and natural numbers) along with their negative counterparts. They are represented as: ℤ = {..., -3, -2, -1, 0, 1, 2, 3, ...}. Integers exclude fractions, decimals, or any fractional components, representing distinct, separate values.

Rational and Irrational Numbers

Numbers are categorized as either rational or irrational. Rational numbers can be expressed as fractions, encompassing whole numbers, which can also be written as fractions, such as 3/8, 5/1, 9/10, and so on.

Decimals can also be rational if they terminate or repeat. For example, 8.372 is a terminating decimal, and 5.2222222... is a repeating decimal. Both are rational numbers and, consequently, real numbers.

An irrational number is a real number that cannot be written as a simple fraction or ratio of two integers. These numbers have non-repeating, non-terminating decimal expansions, such as π (3.14159265...) and √2 (1.41421356...), which continue infinitely without any repeating pattern.

Absolute Value

The absolute value of a number is a mathematical concept that yields the non-negative (positive) value of a real number, disregarding its sign (whether positive or negative).

For instance, if x = 7, then |x| = 7. Similarly, if x = -7, then |x| = 7.

It essentially represents the distance of a number from zero on the number line.

Imaginary Numbers

On the other hand, an imaginary number represents the square root of a negative number. You might recall this unique mathematical principle: no real number, when squared, results in a negative value.

However, mathematicians still work with imaginary numbers, acknowledging that the results are not real. (Infinity is also considered an imaginary number.)