Rational and Irrational Numbers Explained Simply

Let’s talk about rational numbers and irrational numbers, two big categories of numbers that you’ll see all throughout math. Don’t worry, it’s actually a very simple idea!

In this article, WuKong Math will dive into the definition of rational numbers, explore their key properties, and walk through various examples to help you better understand them.

What is a Rational Number?

Let’s start with the word rational.
Notice that it contains the word ratio. That’s a big hint!

A rational number is any number that can be written as a fraction — a ratio of two whole numbers.

For example:

• The number 4 is rational, because you can write it as 4/1.
• The number 10 is rational, because you can write it as 10/1.
• Even negative numbers like –2 are rational, because we can write them as –2/1.

So, all whole numbers and their negatives are rational.

Decimals Can Be Rational Too

Now, what about decimals like 1.25?

It might not look like a fraction, but it actually is!
1.25 can be written as 5/4 — that’s a fraction.
So, any decimal that stops (also called a terminating decimal) is rational.

Here are a few examples:

• 0.5 = 1/2
• 0.75 = 3/4
• 2.68 = 268/100

Even decimals that go on forever but have a repeating pattern are still rational.

For instance:

• 0.3333… = 1/3
• 0.6666… = 2/3
• 0.123123123… = 123/999

So here’s the rule:

If the decimal stops or repeats with a pattern, it’s rational.

Key Properties of Rational Numbers

Rational numbers have several important properties that make them distinct from other types of numbers.

Expressed as a Fraction

A rational number is any number that can be expressed as the ratio of two integers, and this includes both positive and negative numbers. A rational number’s denominator (the bottom number of the fraction) cannot be zero. Division by zero is undefined, which means any fraction with q = 0 is not a rational number.

Decimal Form

When rational numbers are written in decimal form, they can either be terminating decimals (ending after a certain number of digits) or repeating decimals (where the digits repeat in a pattern). For example, 0.75 is a terminating decimal, where the digits stop after the decimal point. On the other hand, 0.333… (where the digit 3 repeats indefinitely) is a repeating decimal, and the repeating pattern after the decimal point is 3.

Closure Property

Rational numbers are closed under addition, subtraction, multiplication, and division (except division by zero). This means that performing any of these operations on rational numbers will always result in another rational number. For example, adding 2/3 and 3/5 will give you a rational number.

Negative and Positive Numbers

Rational numbers can be both positive and negative. Negative rational numbers are simply those where either the numerator or the denominator (but not both) is negative. For example, -2/3 and 3/-4 are both negative rational numbers.

Equivalent Fractions

Two fractions are equivalent if they represent the same value. For example, 1/2 is equivalent to 2/4, because they both represent the same ratio.

Types of Rational Numbers

Rational numbers are a broad category of numbers that can be expressed as the ratio of two integers, where the denominator is not zero. These numbers can take various forms, each with distinct characteristics. Let’s explore the different types of rational numbers:

1. Integers

Integers are the simplest form of rational numbers. They include all positive whole numbers, zero, and negative whole numbers. Integers can always be expressed as a ratio of two integers, with the denominator being 1.

For example:

• 5 can be written as 5/1.
• -3 can be written as -3/1.
• 0 can be written as 0/1.

Since these numbers have a denominator of 1, they are classified as rational numbers.

2. Proper Fractions

A proper fraction is a rational number where the numerator is smaller than the denominator. The value of a proper fraction is always less than 1.

For instance:

• 2/5 is a proper fraction because the numerator (2) is smaller than the denominator (5).
• -3/8 is also a proper fraction because the numerator (-3) is smaller than the denominator (8).

Proper fractions are rational because they represent a ratio of two integers, with a non-zero denominator.

3. Improper Fractions

An improper fraction is a rational number where the numerator is greater than or equal to the denominator. The value of an improper fraction is either equal to or greater than 1.

For example:

• 7/4 is an improper fraction because the numerator (7) is greater than the denominator (4).
• 5/5 is also an improper fraction because the numerator (5) equals the denominator (5).

Improper fractions can often be converted into mixed fractions or decimals, but they are still rational numbers because they can be expressed as a ratio of two integers.

4. Terminating Decimals

A terminating decimal is a decimal that comes to an end after a finite number of digits. All terminating decimals are rational because they can be written as fractions with integers in the numerator and denominator.

For example:

• 0.75 is a terminating decimal, which is equivalent to the fraction 3/4.
• -2.5 is a terminating decimal, which is equivalent to the fraction -5/2.

Terminating decimals occur when the denominator of the fraction (in its lowest terms) is a power of 2, 5, or both. For example, 1/8 gives 0.125, which is a terminating decimal.

5. Repeating Decimals

A repeating decimal is a decimal in which one or more digits repeat infinitely. These decimals are rational numbers because they can be expressed as fractions.

For example:

• 0.333… (where 3 repeats) is equivalent to the fraction 1/3.
• 0.666… (where 6 repeats) is equivalent to the fraction 2/3.

Such decimals occur when the denominator of a fraction (in its simplest form) has prime factors other than 2 and 5. For instance, 1/3 gives the decimal 0.333…, which can be written as a ratio of two integers.

6. Negative Rational Numbers

Negative rational numbers are those where either the numerator, the denominator, or both are negative. These numbers can appear as negative integers, fractions, or decimals. Essentially, they are rational numbers with a negative value.

For example:

• -5/7 is a negative rational number because the numerator is negative.
• -1.5 is a negative rational number because it can be written as -3/2.

Negative rational numbers behave the same way as positive rational numbers in arithmetic operations, with the additional consideration of signs.

What Is an Irrational Number?

Now, what about irrational numbers?

An irrational number is one that cannot be written as a fraction of two whole numbers.
Their decimal expansions go on forever and never repeat.

The most famous one?
π (pi).

People often say π = 22/7, but that’s only an approximation.
In reality, π = 3.141592… and the digits go on forever with no pattern at all.
Even with computers calculating millions of digits, there’s still no pattern found — which means π is irrational.

More Examples of Irrational Numbers

Other irrational numbers include:

• √2 = 1.4142135… (no pattern)
• √3, √5, √6, √7, etc. (as long as the square root doesn’t give a whole number)
• The constant e = 2.71828…, which shows up in growth and finance calculations

Basically:

If you take the square root of a number and don’t get a nice whole number, it’s irrational.

Rational Numbers vs. Irrational Numbers

While rational numbers are numbers that can be expressed as the ratio of two integers, irrational numbers cannot. Irrational numbers include numbers like √2, π, and e, which cannot be written as a fraction of two integers.

Unlike rational numbers, irrational numbers have non-terminating and non-repeating decimal expansions.

Definition of Irrational Numbers

Irrational numbers are real numbers that cannot be expressed as p/q, where p and q are integers. These numbers have decimal expansions that never terminate or repeat in a regular pattern.

Difference Between Rational and Irrational Numbers

The primary difference between rational and irrational numbers is that rational numbers can be written as fractions, whereas irrational numbers can not.

• Rational Numbers: Can be written as p/q, where p and q are integers, with q ≠ 0. Their decimal expansions are either terminating or repeating.
• Irrational Numbers: Cannot be written as p/q, and their decimal expansions never terminate or repeat. Examples include π and √2.

A Quick Summary

Here’s how you can tell the difference:

Practice Problems about Rational Numbers

To deepen your understanding of rational numbers, try solving the following practice problems. Each one is designed to test your ability to identify, simplify, and work with rational numbers in various forms. The solutions will help you grasp the concepts more clearly.

Problem 1: Identifying Rational Numbers

Which of the following numbers are rational?

A) 0.75
B) √5
C) -3
D) π
E) 1/2

Hint: A rational number can be written as a fraction of two integers.

Problem 2: Writing Decimals as Fractions

Convert the following repeating decimal into a fraction:
0.666…

Hint: Recognize that 0.666… is a repeating decimal and can be written as 2/3.

Problem 3: Simplifying Fractions

Simplify the following fractions:
A) 12/18
B) 45/60
C) 100/400

Hint: Find the greatest common divisor (GCD) of the numerator and denominator and divide both by it.

Problem 4: Adding Rational Numbers

Add the following rational numbers:
A) 2/5 + 3/10
B) -4/9 + 7/9
C) -1/3 + 2/5

Hint: For adding fractions, make sure the denominators are the same, or find a common denominator first.

Problem 5: Subtracting Rational Numbers

Subtract the following rational numbers:
A) 5/8 – 3/4
B) -7/12 – 5/6
C) 1/2 – 2/3

Hint: Remember that subtracting fractions requires finding a common denominator, then subtracting the numerators.

Solutions:

Here are the solutions for you to check your answers:

1.

• A) 0.75 is rational (can be written as 3/4).
• B) √5 is irrational (it cannot be written as a fraction of two integers).
• C) -3 is rational (it can be written as -3/1).
• D) π is irrational (it cannot be expressed as a fraction).
• E) 1/2 is rational (it’s already in fraction form).

2.

• 0.666… = 2/3

3.

• A) 12/18 simplifies to 2/3.
• B) 45/60 simplifies to 3/4.
• C) 100/400 simplifies to 1/4.

4.

• A) 2/5 + 3/10 = 7/10
• B) -4/9 + 7/9 = 3/9 = 1/3
• C) -1/3 + 2/5 = 7/15

5.

• A) 5/8 – 3/4 = -1/8
• B) -7/12 – 5/6 = -17/12
• C) 1/2 – 2/3 = -1/6

FAQs about Rational Numbers

Demi | WuKong Math Teacher

I am an educator from Yale University with ten years of experience in this field. I believe that with my professional knowledge and teaching skills, I will be able to contribute to the development of Wukong Education. I will share the psychology of children’s education and learning strategies in this community, hoping to provide quality learning resources for more children.