Have you heard the term “rational numbers?” Are you wondering, “What is a rational number?” If so, you’re in the right place!
In this article, we’ll discuss the rational number definition, give rational numbers examples, and offer some tips and tricks for understanding if a number is rational or irrational.
What Is A Rational Number?
In order to understand what rational numbers are, we first need to cover some basic math definitions:
- Integers are whole numbers (like 1, 2, 3, and 4) and their negative counterparts (like -1, -2, -3, and -4).
- Fractions are numbers that are expressed as ratios. A fraction is a part of a whole.
- Fractions have numerators, which are the numbers on the top of the fraction that show the parts taken from the whole.
- Fractions also have denominators, which are the numbers on the bottom of the fraction that show how many parts are in the whole.
Okay! Now that we know those terms, let’s turn to our original question.
What is a rational number?
A rational number is a number that can be expressed as a fraction where both the numerator and the denominator in the fraction are integers. The denominator in a rational number cannot be zero.
Expressed as an equation, a rational number is a number
a/b, b≠0
where a and b are both integers.
This equation shows that all integers, finite decimals, and repeating decimals are rational numbers. In other words, most numbers are rational numbers.
Here’s a hint: if you’re working with a number with a long line of different decimals, then your number is irrational! If you’re working with an integer or a number with terminal or repeating decimals (like 1.333333), then your number is rational!
Rational Number Examples
Now that we know the rational number definition, let’s use that definition to examine some numbers and see if they’re rational or not.
Let’s start with the number 6.
The number 6 is an integer. It’s also a rational number. Why?
Because 6 can also be expressed as 6/1.
When expressed as 6, both the numerator and the denominator are integers. The denominator doesn’t equal 0.
What about the number -6?
-6 can be written as -6/1. Or 6/-1.
Either way, -6 is a rational number, because it can be expressed as a fraction where the numerator and denominator are integers and the denominator doesn’t equal 0.
What’s an Irrational Number?
The opposite of rational numbers are irrational numbers.
In simple terms, irrational numbers are real numbers that can’t be written as a simple fraction like 6/1.
Take π.
π is a real number. But it’s also an irrational number, because you can’t write π as a simple fraction:
π = 3.1415926535897932384626433832795 (and counting)
There’s no way to write π as a simple fraction, so it’s irrational.
The same goes for √2.
The √2 equals 1.4142135623730950...(etc).
You can’t make √2 into a simple fraction, so it’s an irrational number.
Famous Irrational Numbers
There aren’t any famous rational numbers, because the vast majority of numbers are rational. There are a few famous irrational numbers. Here are some ones you might have seen:
- e: The number e (Euler's Number) is another famous irrational number. People have also calculated e to lots of decimal places without any pattern showing. The first few digits look like this: 2.7182818284590452353602874713527.
- π: People have calculated Pi to over a quadrillion decimal places and still there is no pattern. The first few digits look like this: 3.1415926535897932384626433832795
- √: Many square roots, cube roots, etc are also irrational numbers. Examples:
- √3 = 1.7320508075688772935274463415059 (etc)
- √99 = 9.9498743710661995473447982100121 (etc)
Not all square roots are irrational numbers, though! If your square root results in a whole number (like √4 or √9), then you actually are working with a rational number!
That’s not the only thing you have to be careful about! Sometimes, multiplying two irrational numbers will result in a rational number. For example,
√2 * √2 = 2
2 is a rational number.
Main Takeaways
Rational numbers are numbers that can be expressed as simple fractions.
Irrational numbers are numbers that can’t be expressed as simple fractions.
What's Next?
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