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Real Numbers — Definition, Examples & Number Line

Real Numbers

All numbers on the number line. This includes (but is not limited to) positives and negatives, integers and rational numbers, square roots, cube roots , π (pi), etc. Real numbers are indicated by either Blackboard bold capital R symbol (ℝ), used to denote the set of all real numbers. or Blackboard bold letter ℝ, the mathematical symbol used to denote the set of all real numbers..

 

Nested diagram showing number sets: natural, whole, integers, rationals, algebraic, and reals (ℝ), alongside complex (ℂ) and...

 

 

See also

Natural numbers, whole numbers, algebraic numbers, imaginary numbers, complex numbers

Key Formula

R={x∣x is a rational or irrational number}\mathbb{R} = \{ x \mid x \text{ is a rational or irrational number} \}
Where:
  • R\mathbb{R} = The symbol for the set of all real numbers
  • xx = Any number that can be located on the number line

Worked Example

Problem: Classify each of the following numbers as real or not real: 5, −3.7, √2, 0, π, √(−4).
Step 1: Check whether 5 is on the number line. It is a positive integer, so it is a real number.
5∈R5 \in \mathbb{R}
Step 2: Check −3.7. It is a negative decimal (rational number), so it lies on the number line and is real.
−3.7∈R-3.7 \in \mathbb{R}
Step 3: Check √2. Its decimal expansion is 1.41421… (non-repeating, non-terminating), making it irrational — but it still has a definite position on the number line, so it is real.
2≈1.4142…∈R\sqrt{2} \approx 1.4142\ldots \in \mathbb{R}
Step 4: Check 0 and π. Zero is a whole number and π ≈ 3.14159… is irrational. Both are on the number line, so both are real.
0∈R,π∈R0 \in \mathbb{R}, \quad \pi \in \mathbb{R}
Step 5: Check √(−4). No real number squared gives −4. This equals 2i, an imaginary number, so it is NOT a real number.
−4=2i∉R\sqrt{-4} = 2i \notin \mathbb{R}
Answer: 5, −3.7, √2, 0, and π are all real numbers. √(−4) is not a real number.

Another Example

This example focuses on the nested subset structure of the real numbers (natural ⊂ whole ⊂ integer ⊂ rational ⊂ real), rather than simply classifying a number as real or not real.

Problem: Determine where the number 7/3 fits in the real number system by identifying all subsets it belongs to.
Step 1: Is 7/3 a natural number? Natural numbers are 1, 2, 3, … Since 7/3 ≈ 2.333… is not a counting number, the answer is no.
73∉N\frac{7}{3} \notin \mathbb{N}
Step 2: Is it a whole number or an integer? No — it is not a whole number (0, 1, 2, …) nor an integer (…, −2, −1, 0, 1, 2, …) because it falls between 2 and 3.
73∉Z\frac{7}{3} \notin \mathbb{Z}
Step 3: Is it a rational number? Yes — it can be written as a ratio of two integers, 7 and 3, with a non-zero denominator.
73∈Q\frac{7}{3} \in \mathbb{Q}
Step 4: Since every rational number is a real number, 7/3 is also real.
73∈R\frac{7}{3} \in \mathbb{R}
Answer: 7/3 is a rational number and a real number. It is not a natural number, whole number, or integer.

Frequently Asked Questions

What is the difference between real numbers and imaginary numbers?
Real numbers are all values on the standard number line — they can be positive, negative, or zero. Imaginary numbers involve the square root of a negative number and are written with the unit i, where i = √(−1). For example, 5 and −3.7 are real, while 2i and √(−9) are imaginary. Together, a real part and an imaginary part form a complex number.
Is zero a real number?
Yes. Zero sits on the number line between the positive and negative numbers. It is a whole number, an integer, a rational number (0/1), and therefore also a real number.
Are all fractions and decimals real numbers?
All ordinary fractions and decimals — whether terminating like 0.75 or repeating like 0.333… — are rational numbers and therefore real. Non-repeating, non-terminating decimals like π or √2 are irrational but still real. The only numbers that are not real are those involving i (the imaginary unit).

Real Numbers vs. Complex Numbers

Real NumbersComplex Numbers
DefinitionAll numbers on the number line (rational and irrational)Numbers of the form a + bi, where a and b are real and i = √(−1)
Symbolℝℂ
Includes imaginary part?No (b = 0)Yes (b can be any real number)
Examples−7, 0, 2/5, √3, π3 + 2i, −i, 4 + 0i (which equals 4)
RelationshipSubset of the complex numbersSuperset that contains all real numbers

Why It Matters

Real numbers form the foundation for nearly every topic you study in algebra, geometry, trigonometry, and calculus. Whenever you solve an equation, measure a distance, or graph a function on the coordinate plane, you are working with real numbers. Understanding where real numbers end — and where imaginary and complex numbers begin — is essential when you encounter equations like x² + 1 = 0 that have no solution on the real number line.

Common Mistakes

Mistake: Thinking that irrational numbers are not real numbers.
Correction: Numbers like √2, π, and e are irrational (they cannot be written as fractions), but they still have exact positions on the number line and are fully real. "Irrational" is a subset of "real," not a separate category.
Mistake: Confusing "real" with "positive" and assuming negative numbers or zero are not real.
Correction: The real number line extends infinitely in both directions. Negative numbers like −100 and zero itself are real numbers just as much as positive numbers are.

Related Terms