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Irrational Numbers – Definition with Examples

Jo-ann Caballes

Updated on September 29, 2026

An irrational number is a real number that cannot be written as a simple fraction (a/b) of two integers. Its decimal form goes on forever without ever repeating a pattern. Famous examples include π (pi), e (Euler’s number), and √2, the square root of 2.

Irrational numbers are usually introduced around the 8th grade, once students are comfortable with rational numbers. They round out the real number system and appear often in geometry, algebra, and higher math — a topic that high school math tutors help students master.

What Is an Irrational Number?

An irrational number is any real number that cannot be expressed as a ratio of two integers. When you write it as a decimal, it neither terminates (ends) nor repeats. For example, √2 = 1.41421356… continues forever with no repeating block of digits, which is what makes it irrational rather than one of the rational numbers.

Properties of Irrational Numbers

Irrational numbers have a few defining properties that set them apart from rational numbers. These rules explain how they behave and where they sit among the real numbers.

• They cannot be written as a fraction a/b, where a and b are integers and b ≠ 0.
• Their decimal form is non-terminating and non-repeating.
• They are dense on the number line: between any two rational numbers, there is always an irrational number.
• A nonzero rational number multiplied by an irrational number is always irrational — for example, (2/3) × √2 is irrational.
• The sum or difference of a rational and an irrational number is always irrational — for example, 3 + √2 is irrational.

List of Common Irrational Numbers

There are infinitely many irrational numbers, but a few appear so often that they have their own names and symbols. The table below lists the most common ones.

Rational vs Irrational Numbers

The key difference is simple: rational numbers can be written as a fraction, and irrational numbers cannot. Both are real numbers, so both can be placed on the number line. The table below compares them side by side.

Set of Irrational Numbers

The set of irrational numbers is often written as I. It is the collection of all real numbers that are not rational, so it is a subset of the real numbers (R). In set notation, this is written I = {x ∈ R | x ∉ Q}, which reads: the irrational numbers are all real numbers x that do not belong to the set of rational numbers Q.

Sum and Product of Two Irrational Numbers

The sum or product of two irrational numbers can be either rational or irrational — it depends on the numbers. These examples show both outcomes:

• √2 + √3 is irrational.
• √2 × √3 = √6, which is irrational.
• √2 + (−√2) = 0, which is rational.
• √2 × √2 = 2, which is rational.

Irrational Number Theorem and Proof

The irrational number theorem states that the square root of any positive integer that is not a perfect square is irrational. A classic way to prove it is by contradiction — assuming the opposite and showing it leads to something impossible. Here is the proof that √2 is irrational:

1. Assume √2 is rational, so √2 = a/b, where a and b are integers with no common factor other than 1, and b ≠ 0.
2. Square both sides: 2 = a²/b², so 2b² = a². This means a² is even, so a must be even. Let a = 2k.
3. Substitute: 2b² = (2k)² = 4k², so b² = 2k². This means b² is even, so b must be even too.
4. But if a and b are both even, they share a factor of 2, which contradicts step 1. So √2 cannot be rational, which proves it is irrational.

How to Find an Irrational Number Between Two Numbers

A reliable way to find an irrational number between two values is to use square roots. Between the numbers a and b, look for a non-perfect square whose square root falls in that range. You can also write a non-terminating, non-repeating decimal by hand.

1. To find an irrational number between 2 and 3: since 2² = 4 and 3² = 9, any √n with 4 < n < 9 (and n not a perfect square) works. For example, √5 ≈ 2.236 lies between 2 and 3.
2. To find an irrational number between 1 and 2: since 1² = 1 and 2² = 4, √2 ≈ 1.414 and √3 ≈ 1.732 both lie between 1 and 2.
3. Another method is to write a decimal that never repeats, such as 2.1010010001…, which is irrational and lies between 2 and 3.

Solved Examples on Irrational Numbers

These worked examples show common operations with irrational numbers.

• Example 1: √2 + √3 ≈ 1.4142 + 1.7321 = 3.1463 (the sum of two irrational numbers).
• Example 2: √2 × √3 = √6 ≈ 2.4495 (the product of two irrational numbers).
• Example 3: 2 + √3 ≈ 3.7321 (the sum of a rational and an irrational number, which is irrational).
• Example 4: 2 × √3 ≈ 3.4641 (the product of a nonzero rational and an irrational number, which is irrational).

Practice Questions on Irrational Numbers

Try these problems, then check the answers below.

1. Simplify (√5 + √2)².
2. Prove that √3 is irrational.
3. Find an irrational number between 1 and 2.
4. Calculate the product of √7 and √14.

Answers: 1) (√5 + √2)² = 5 + 2√10 + 2 = 7 + 2√10 ≈ 13.32. 2) Use the same proof-by-contradiction method as for √2. 3) √2 ≈ 1.414 (or √3 ≈ 1.732). 4) √7 × √14 = √98 = 7√2 ≈ 9.90.

Irrational Numbers: Practice Test

Try these questions, then check the marked answers.

Irrational Numbers Worksheets

Put this into practice with related Brighterly worksheets:

• Square Root Worksheets
• Square Roots and Cube Roots Worksheet With Answers PDF
• Adding and Subtracting Rational Numbers Worksheets
• Rational Exponents Worksheets

Frequently Asked Questions About Irrational Numbers

What Is an Irrational Number?

An irrational number is a number that cannot be written as a simple fraction (a/b) where a and b are integers and b ≠ 0. Its decimal form is non-terminating and non-repeating, meaning it goes on forever without settling into a repeating pattern. Well-known examples are π, e, and √2.

Are Irrational Numbers Real Numbers?

Yes, irrational numbers are real numbers. The set of real numbers is made up of both rational numbers and irrational numbers. Every irrational number can be placed on the number line, sitting among the rational numbers, which is why it counts as a real number.

Can Irrational Numbers Be Written as Decimals?

Yes, but their decimals never end and never repeat. For example, √2 = 1.41421356… and π = 3.14159265… both continue forever without a repeating block of digits. Any decimal you actually write down is only an approximation of the true irrational value.

Can Irrational Numbers Be Written as Fractions?

No. By definition, an irrational number cannot be expressed as a fraction a/b where a and b are integers. This is exactly what separates irrational numbers from rational numbers, which can always be written as such a fraction.

Can the Sum of Two Irrational Numbers Be Rational?

Yes. Although two irrational numbers often add to another irrational number, they can also add to a rational number. For example, √2 and −√2 are both irrational, but their sum, √2 + (−√2), equals 0, which is rational.

How Do You Find an Irrational Number Between Two Numbers?

Use square roots: look for a non-perfect square whose root falls between the two numbers. For example, between 1 and 2, you can use √2 ≈ 1.414, since 1² = 1 and 2² = 4. You can also write a non-repeating decimal, such as 1.1010010001…, that lies in the range.

Jo-ann Caballes

30 articles

As a seasoned educator with a Bachelor’s in Secondary Education and over three years of experience, I specialize in making mathematics accessible to students of all backgrounds through Brighterly. My expertise extends beyond teaching; I blog about innovative educational strategies and have a keen interest in child psychology and curriculum development. My approach is shaped by a belief in practical, real-life application of math, making learning both impactful and enjoyable.