Real Numbers are fundamental in mathematics and encompass almost every number you encounter in everyday life. Think of them simply as numbers on a continuous line, without any gaps.
Here are some examples of real numbers:
| 1 | 12.38 | −0.8625 | 3/4 | π | 198 |
|---|
Essentially, if you can think of a number, it’s very likely a Real Number.
Types of Real Numbers
Real numbers are a broad category that includes several familiar types of numbers:
- Whole Numbers: These are the counting numbers starting from zero: 0, 1, 2, 3, 4, and so on.
| Whole Numbers (like 0, 1, 2, 3, 4, etc) | |
|---|---|
- Rational Numbers: These numbers can be expressed as a fraction or ratio of two integers. This category includes decimals that terminate or repeat. Examples include 3/4, 0.125, 0.333…, 1.1.
| Rational Numbers (like 3/4, 0.125, 0.333…, 1.1, etc ) | |
|---|---|
- Irrational Numbers: These are numbers that cannot be expressed as a simple fraction of two integers. They are decimals that go on forever without repeating. Famous examples are π (pi) and √2 (the square root of 2).
| Irrational Numbers (like π, √2, etc ) | |
|---|---|
Real Numbers can also be positive (greater than zero), negative (less than zero), or zero.
What is NOT a Real Number?
While real numbers are extensive, there are some types of numbers that fall outside this definition:
- Imaginary Numbers: These numbers involve the square root of negative one, denoted as √−1 or i. Imaginary numbers are crucial in advanced mathematics but are not part of the real number system.
| Imaginary Numbers like √−1 (the square root of minus 1) are not Real Numbers | |
|---|---|
- Infinity: Infinity (∞) is not considered a real number. It is a concept representing a quantity without any bound or limit, rather than a specific numerical value.
| Infinity is not a Real Number | |
|---|---|
Mathematicians explore these and other special number types beyond the realm of real numbers for various theoretical and applied purposes.
The Real Number Line Explained
The best way to visualize real numbers is using the Real Number Line. Imagine a straight line that extends infinitely in both directions.
This line has a central point called the origin, which represents zero. Numbers to the right of the origin are positive real numbers, and numbers to the left are negative real numbers.
We can mark off equal distances to represent whole numbers: 1, 2, 3,… to the right, and −1, −2, −3,… to the left.
Crucially, every point on this line corresponds to a real number. This includes:
- Integers: Whole numbers and their negatives (e.g., -3, -2, -1, 0, 1, 2, 3).
- Rational numbers: Fractions that fall between integers (e.g., 20/9 is between 2 and 3).
- Irrational numbers: Numbers like π and √2, which also occupy specific, though non-fractional, positions on the line.
However, you won’t find infinity or imaginary numbers on the standard real number line.
Decimal Representation of Real Numbers
A key characteristic of real numbers is their decimal representation. A real number can have any number of digits after the decimal point.
Examples:
-
- (which is the same as 120.0)
- 0.12345
- 12.5509
- 0.0000000001
Some real numbers have a finite number of decimal places (like 0.125), while others have an infinite number of decimal places, like 1/3 = 0.333… where the 3s repeat forever.
Origin of the Name “Real” Numbers
You might wonder why they are called “Real” Numbers. The term arose historically to distinguish them from “Imaginary Numbers.”
Before mathematicians conceived of imaginary numbers, what we now call real numbers were simply “numbers.” When imaginary numbers were discovered and explored, the original numbers needed a new name to differentiate them, and “Real” Numbers became the designation. So, the name is essentially based on what they are not (imaginary).
“Real” Numbers in Mathematics vs. the “Real World”
It’s important to understand that “real” in “Real Numbers” doesn’t mean these numbers only describe things we experience in the physical “real world.”
In mathematics, we deal with pure, exact values. When we write 0.5, we mean precisely one-half.
In contrast, in the “real world,” measurements and quantities are often approximations. Trying to divide an apple “exactly” in half perfectly illustrates this difference. Mathematical real numbers are idealizations that provide a foundation for precise calculations and theoretical work.
Further Exploration:
