Every number a student meets before senior secondary school is a real number. Counting numbers, negatives, fractions, decimals, square roots, pi. They all sit somewhere on the number line, and that is essentially what makes them real.
This guide gives the real number definition with examples, sets out the types and properties, and covers what the Class 10 chapter actually requires now, which is less than most websites still claim.
On This Page
- What are real numbers?
- Real number definition with example
- Types of real numbers
- Examples of real numbers
- What is not a real number
- Every real number is
- Properties of real numbers
- Real numbers for Class 10
- Practice questions with answers
- Frequently asked questions
What Are Real Numbers?
Real numbers are all the numbers that can be placed on the number line. They include every rational number and every irrational number, which means counting numbers, zero, negative numbers, fractions, terminating decimals, recurring decimals and non-recurring infinite decimals such as pi.
The set of real numbers is written as R.
If you can mark a number as a point on a line, it is real. If you cannot, such as the square root of a negative number, it is not.
Real Number Definition With Example
Definition. A real number is any number that can be expressed as a point on the number line. Formally, the set of real numbers R is the union of the set of rational numbers Q and the set of irrational numbers.
Examples. 5, −12, 0, 3/4, 0.75, 2.333…, √2, √7, pi and e are all real numbers.
Notice how different these look from each other. A whole number, a negative, a fraction, a repeating decimal and an endless non-repeating decimal are all in the same set, because the only test is whether the number has a definite position on the line.
Types of Real Numbers
Real numbers contain several smaller sets, each inside the next.
| Set | Symbol | What it contains | Examples |
|---|---|---|---|
| Natural numbers | N | Counting numbers from 1 | 1, 2, 3, 4 |
| Whole numbers | W | Natural numbers plus zero | 0, 1, 2, 3 |
| Integers | Z | Whole numbers plus negatives | −3, −2, 0, 5 |
| Rational numbers | Q | Numbers writable as p/q, q not zero | 1/2, −7, 0.25, 0.666… |
| Irrational numbers | — | Cannot be written as p/q | √2, √5, pi, e |
| Real numbers | R | All rational and irrational numbers | everything above |
The nesting runs one way: every natural number is a whole number, every whole number is an integer, every integer is rational, and every rational number is real. It does not run backwards. Not every real number is rational, and not every integer is natural.
Rational vs Irrational
| Feature | Rational | Irrational |
|---|---|---|
| Form | Can be written as p/q with integers p and q, q ≠ 0 | Cannot be written as p/q |
| Decimal | Terminating or recurring | Non-terminating and non-recurring |
| Examples | 3/8 = 0.375, 2/3 = 0.666… | √3 = 1.732…, pi = 3.14159… |
A useful check: if the decimal either stops or settles into a repeating block, the number is rational. If it runs on with no pattern, it is irrational.
Examples of Real Numbers
- Natural numbers: 1, 17, 250
- Zero: 0
- Negative integers: −4, −99
- Fractions: 5/8, −3/7
- Terminating decimals: 0.5, 2.75
- Recurring decimals: 0.333…, 1.272727…
- Square roots of non-perfect squares: √2, √11
- Famous irrationals: pi, e, the golden ratio
Every one of these can be marked on a number line, which is why they all qualify.
What Is Not a Real Number
Some numbers fall outside the set entirely.
- The square root of a negative number. √−1 is written as i and belongs to the imaginary numbers. No real number multiplied by itself gives a negative result.
- Complex numbers with a non-zero imaginary part. 3 + 2i is a complex number, not a real one.
- Infinity. Infinity is a concept describing unbounded growth, not a number with a fixed place on the line.
- Undefined expressions. 0/0 and 5/0 are not numbers at all.
This is the boundary most students test in exams, usually through a question asking whether √−4 is real. It is not.
Every Real Number Is
Several statements about real numbers are worth memorising, because exam questions are often phrased exactly this way.
- Every real number is either rational or irrational, and never both.
- Every real number corresponds to exactly one point on the number line, and every point corresponds to exactly one real number.
- Every real number is a complex number with an imaginary part of zero.
- Every integer is a real number, but not every real number is an integer.
- Every rational number is a real number, but not every real number is rational.
- Between any two real numbers there is another real number, however close they are.
That last property is called density, and it surprises most students the first time they meet it.
Properties of Real Numbers
These properties describe how real numbers behave under addition and multiplication.
| Property | Addition | Multiplication |
|---|---|---|
| Closure | a + b is real | a × b is real |
| Commutative | a + b = b + a | ab = ba |
| Associative | (a + b) + c = a + (b + c) | (ab)c = a(bc) |
| Identity | a + 0 = a | a × 1 = a |
| Inverse | a + (−a) = 0 | a × 1/a = 1, for a ≠ 0 |
| Distributive | a(b + c) = ab + ac |
Two further properties matter at higher levels:
- Order. Any two real numbers can be compared. For any a and b, exactly one of a < b, a = b or a > b holds.
- Completeness. The real number line has no gaps. This is what separates the reals from the rationals, since the rationals leave a hole exactly where √2 should be.
Real Numbers for Class 10
This is where most online notes are out of date, so it is worth being precise.
Under the current CBSE syllabus, the Class 10 Real Numbers chapter covers the Fundamental Theorem of Arithmetic, finding HCF and LCM by prime factorisation, and proofs that numbers such as √2, √3 and √5 are irrational. Euclid’s Division Lemma and the exercise on decimal expansions of rational numbers were removed in the rationalised syllabus and are not assessed in CBSE board exams. Students following state boards or older textbooks may still have them, so check your own syllabus rather than a website.
The Fundamental Theorem of Arithmetic states that every composite number can be written as a product of primes, and that this factorisation is unique apart from the order of the factors. For example, 36 = 2 × 2 × 3 × 3, and no other set of primes produces 36.
The HCF and LCM relation for two numbers a and b: HCF(a, b) × LCM(a, b) = a × b. This holds for two numbers only, not for three.
Proving √2 Is Irrational
The standard proof is by contradiction, and it appears in board papers regularly.
- Assume √2 is rational, so √2 = p/q where p and q are integers with no common factor and q ≠ 0.
- Square both sides: 2 = p²/q², so p² = 2q².
- Then p² is even, which means p is even. Write p = 2k.
- Substituting: 4k² = 2q², so q² = 2k². Then q² is even, so q is even.
- But p and q are both even, which contradicts the assumption that they share no common factor.
- The assumption must be wrong, so √2 is irrational.
The same structure works for √3 and √5.
Practice Questions With Answers
State whether each is rational, irrational, or not a real number.
- 0
- √16
- √17
- 22/7
- pi
- −5.25
- √−9
- 0.101001000100001…
- 3 + 2i
- 1.454545…
Answers
| # | Number | Answer | Reason |
|---|---|---|---|
| 1 | 0 | Rational, real | 0 can be written as 0/1 |
| 2 | √16 | Rational, real | Equals 4, a perfect square root |
| 3 | √17 | Irrational, real | 17 is not a perfect square |
| 4 | 22/7 | Rational, real | A fraction of integers; it approximates pi but is not pi |
| 5 | pi | Irrational, real | Non-terminating, non-recurring |
| 6 | −5.25 | Rational, real | Terminating decimal, equals −21/4 |
| 7 | √−9 | Not real | Square root of a negative number |
| 8 | 0.101001000100001… | Irrational, real | Non-terminating with no repeating block |
| 9 | 3 + 2i | Not real | Complex with a non-zero imaginary part |
| 10 | 1.454545… | Rational, real | Recurring decimal |
Question 4 catches many students. 22/7 is a rational approximation of pi, not pi itself.
Common Mistakes
Assuming every square root is irrational. √16, √25 and √81 are rational, because the numbers under the root are perfect squares.
Treating 22/7 as pi. It is a convenient fraction close to pi, and it is rational.
Thinking a long decimal must be irrational. A decimal with a thousand digits that then repeats is still rational.
Forgetting zero. Zero is a whole number, an integer, a rational number and a real number. It is not a natural number.
Applying HCF × LCM to three numbers. The relation only holds for two.
Frequently Asked Questions
What are real numbers?
Real numbers are all numbers that can be represented on the number line, including natural numbers, whole numbers, integers, rational numbers and irrational numbers. The set is written as R.
What is the real number definition with an example?
A real number is any number expressible as a point on the number line, formed by the union of rational and irrational numbers. Examples include 7, −3, 0, 2/5, 0.8, √2 and pi.
Is zero a real number?
Yes. Zero is a whole number, an integer, a rational number and therefore a real number. It is not a natural number.
Is pi a real number?
Yes. Pi is irrational, since its decimal expansion never terminates or repeats, but every irrational number is real.
Every real number is what?
Every real number is either rational or irrational, never both. Every real number also corresponds to exactly one point on the number line, and is a complex number whose imaginary part is zero.
What are the properties of real numbers?
The main properties are closure, commutativity, associativity, identity, inverse and distributivity under addition and multiplication, along with order and completeness.
Is √−1 a real number?
No. √−1 is written as i and is an imaginary number. No real number squared gives a negative result.
What is covered in real numbers for Class 10?
Under the current CBSE syllabus, the chapter covers the Fundamental Theorem of Arithmetic, HCF and LCM by prime factorisation, and proofs of irrationality for numbers such as √2, √3 and √5. Euclid’s Division Lemma and decimal expansions were removed in the rationalised syllabus.
Quick Recap
- Real numbers are every number that sits on the number line.
- R is the union of the rationals and the irrationals.
- Rational numbers are p/q; their decimals terminate or repeat. Irrational numbers do neither.
- The square root of a negative number, complex numbers with an imaginary part, and infinity are not real.
- Class 10 now needs the Fundamental Theorem of Arithmetic, HCF and LCM, and irrationality proofs.
Number systems are one of the topics where students who memorise definitions and students who understand them look identical until the exam phrases a question differently. That is why our mathematics teaching works through the reasoning behind each result rather than the statement alone, from early number work through to board level. You can read more in our guides to the Cambridge curriculum in primary and early years and the IGCSE years.
For related reading, see our guides to the importance of mathematics and to number names and how to write numbers in words.
Parents comparing schools can start with our guides to the best schools in Mysore and to Cambridge vs CBSE.
Considering a school for your child? Contact TISM Mysore to arrange a campus visit.
Further Reading
- NCERT textbooks, National Council of Educational Research and Training
- CBSE curriculum, Central Board of Secondary Education
About the Author
This guide was prepared by the mathematics faculty at The International School Mysore (TISM), a Cambridge curriculum school in Mysuru teaching students from early years through IGCSE. It reflects the questions our teachers answer most often when students first meet the real number system, and the errors that recur in tests.