- In NCERT's Reprint 2026-27 Class 10 mathematics textbook, the chapter Real Numbers covers only the Fundamental Theorem of Arithmetic and proofs by contradiction that the square roots of 2 and 3 are irrational
- NCERT replaced the Class 9 mathematics textbook in April 2026 with Ganita Manjari, whose first chapter is on coordinates rather than number systems
- The mathematics itself has not changed: the real numbers still form a complete ordered field, and the set of real numbers remains uncountable.
Any number you can mark as a point on the number line is a real number. That single sentence covers natural numbers, whole numbers, integers, fractions, terminating decimals, repeating decimals and non-repeating decimals such as pi. What it leaves out matters just as much: imaginary numbers are not real, and infinity is not a real number at all.
What Are Real Numbers?
A real number is any number that can be located as a point on the number line. The real number system, written R, is the union of the rational numbers and the irrational numbers, so it contains natural numbers, whole numbers, integers, fractions, terminating decimals, repeating decimals and non-repeating decimals such as pi.
NCERT states the same idea from the other direction: a number s is called irrational if it cannot be written in the form p by q, where p and q are integers and q is not zero. The rationals and the irrationals together make up the real numbers, and neither set alone is enough.
The correspondence runs both ways. Every real number sits at exactly one point on the line, and every point on the line is exactly one real number. That is why the line has no gaps, and it is the informal version of the completeness property described further down this page.
Why Are They Called Real Numbers?
They are called real only to separate them from imaginary numbers, which are built on i, the symbol NCERT uses for the square root of minus one. The name makes no claim about the physical world. The set is written R, with the positive reals written R plus and the negative reals written R minus.
The label is historical rather than descriptive. NCERT records that Bhaskara wrote in Bijaganita in 1150 that there is no square root of a negative quantity, and that Cardan in 1545 dismissed such roots as useless. W. R. Hamilton later treated a complex number as an ordered pair of real numbers, which removed the need to call anything imaginary at all.
Which Numbers Belong to the Real Number System?
Naturals, whole numbers, integers, rationals and irrationals all belong to R. Imaginary numbers such as the square root of minus three do not, and neither does infinity, which is a symbol for unbounded behaviour rather than a number. The table below sets out each family with an example and its status.
| Number Type | Symbol | Example | In R (NCERT 2026-27) |
|---|---|---|---|
| Natural numbers | N | 1, 2, 3 | Yes |
| Whole numbers | W | 0, 1, 2 | Yes |
| Integers | Z | minus 3, 0, 7 | Yes |
| Rational numbers | Q | 1 by 2, 2.5, minus 4 by 7 | Yes |
| Irrational numbers | R without Q | square root of 2, pi | Yes |
| Imaginary numbers | i | square root of minus 3 | No |
| Infinity | not applicable | unbounded growth | No |
One clarification that trips students up: 22 by 7 is a rational approximation of pi, not pi itself. Pi is irrational, so it has a non-terminating, non-repeating decimal expansion and cannot be written as one integer divided by another.
What Properties Do Real Numbers Have?
The reals form a complete ordered field. Every non-zero real is positive or negative with no third option, sums and products of positive reals stay positive, the rationals are dense in R, every non-empty set bounded above has a supremum, every Cauchy sequence converges, and the whole set is uncountable.
- Order: every non-zero real number is either positive or negative, and there is no value in between.
- Closure: the sum and the product of two positive real numbers is again a positive real number.
- Density: between any two distinct real numbers there are infinitely many rationals, and the irrationals are dense in R as well.
- Dedekind completeness: every non-empty set of real numbers that is bounded above has a least upper bound, called its supremum. Rational numbers fail this test.
- Cauchy completeness: every Cauchy sequence of real numbers converges to a real number. The same is not true inside the rationals.
- Uncountability: the reals cannot be listed one by one, so they are strictly larger than the naturals in cardinality.
Two of these are worth stating carefully, because popular explanations garble them. Completeness does not say that every sequence of real numbers has a limit, which is false; the sequence 1, 2, 3 and so on has none. It says that every Cauchy sequence has a limit inside R. Similarly, R is separable, not inseparable, precisely because the countable set of rationals is dense in it.
Number System ExplainedRead →Is Zero a Real Number?
Yes. Zero lies on the number line, so it is real, and it is rational because it can be written as zero divided by any non-zero integer. Zero is neither positive nor negative, so it belongs to neither R plus nor R minus. In the complex system NCERT writes it as 0 plus i0.
A common error is to argue that zero is imaginary because imaginary numbers are square roots of negative numbers and zero is not positive. The premise is wrong. Imaginary numbers come from square roots of strictly negative reals, and the square root of zero is zero. Zero is the additive identity of both the real and the complex number systems.
How Are Real Numbers Related to Complex Numbers?
Every real number is a complex number, so R is a subset of the complex numbers C. NCERT Class 11 defines a complex number as a plus ib, where a and b are real and i squared equals minus one. When the imaginary part b is zero the number is real, which is why 3 plus 4i is not.
For the complex number written as a plus ib, NCERT calls a the real part and b the imaginary part. Engineers often write j instead of i to avoid a clash with electric current, so the same number may appear as 3 plus 4j in an engineering text and as 3 plus 4i in a mathematics text.
Even Numbers and Their PropertiesRead →Why Are Real Numbers Uncountable?
Because no list can ever contain them all. Cantor's diagonal argument of 1891 shows that any proposed enumeration of the reals misses at least one number, so the reals cannot be paired one to one with the natural numbers. Their cardinality is called the cardinality of the continuum.
This is often stated loosely as there being more reals than naturals, which is correct, but the two sets are certainly not equal in size. Both are infinite; only one of them is countable. The rationals are countably infinite even though they are dense in the line, which is exactly why the diagonal argument targets the reals rather than the rationals.
How Do Computers Store Real Numbers?
Only approximately. Hardware follows the IEEE 754 standard and stores each value in a fixed number of bits, so a 32-bit float can represent at most 2 to the power 32 distinct values while the reals are infinite. Values such as pi, e and one third are rounded rather than stored exactly.
A 32-bit float uses an 8-bit exponent and a 23-bit mantissa, giving roughly six to seven significant decimal digits. A 64-bit double uses an 11-bit exponent and a 52-bit mantissa and reaches roughly 14 to 15 significant digits. Both are finite grids laid over an infinite line, which is why floating-point arithmetic carries rounding error and why equality tests on computed floats are unreliable.
Famous Mathematicians and Their WorkRead →What Does the NCERT 2026-27 Syllabus Cover on Real Numbers?
The Class 10 chapter Real Numbers in the Reprint 2026-27 textbook now focuses on two things: the Fundamental Theorem of Arithmetic, which says every composite number factorises into primes uniquely apart from order, and proofs by contradiction that the square roots of 2 and 3 are irrational. Its summary lists only those points.
- Theorem 1.1, the Fundamental Theorem of Arithmetic: every composite number can be expressed as a product of primes, uniquely apart from the order of the factors.
- Theorem 1.2: if a prime p divides a squared, then p divides a, for any positive integer a.
- Theorem 1.3: the square root of 2 is irrational, proved by contradiction; the same method is used for the square root of 3.
- Exercise 1.2 extends the method to the square root of 5, to 3 plus 2 times the square root of 5, and to expressions such as 7 times the square root of 5.
- Stated results used throughout: the sum or difference of a rational and an irrational is irrational, and the product or quotient of a non-zero rational and an irrational is irrational.
NCERT also replaced the Class 9 mathematics textbook in April 2026. The new Grade 9 book, Ganita Manjari, opens with a chapter on coordinates, so students arriving at the Class 10 real numbers chapter in 2026-27 meet the material in a different order from earlier cohorts.