Rational and Irrational Numbers
Rational and Irrational Numbers
This topic sits in the class 8 syllabus and is the one place where the paper tests reasoning rather than computation. Expect 2 to 3 questions on properties of rational numbers, representation on the number line, or identifying an irrational number among options.
What Makes a Number Rational
A rational number can be written as p/q, where p and q are integers and q ≠ 0
Every integer, every terminating decimal and every recurring decimal is rational. The number 0.333... is rational because it equals 1/3, and 7 is rational because it equals 7/1.
An irrational number cannot be written as p/q. Its decimal expansion is non-terminating and non-recurring. The square root of any number that is not a perfect square is irrational, as are π and e.
✗ π equals 22/7, so π is rational | ✓ 22/7 is only an approximation; π is irrational
Properties of Rational Numbers
| Property | Addition | Subtraction | Multiplication | Division |
|---|---|---|---|---|
| Closure | Yes | Yes | Yes | No (division by 0) |
| Commutative | Yes | No | Yes | No |
| Associative | Yes | No | Yes | No |
| Identity | 0 | — | 1 | — |
Rational numbers are closed under subtraction, unlike natural numbers — that expansion is exactly why the set exists. The additive inverse of a/b is −a/b, and the multiplicative inverse is b/a, which does not exist for zero.
Density: Numbers Between Numbers
Between any two rational numbers there are infinitely many rational numbers. This is called the density property and it is a standard one-mark question.
To find a rational number between two given ones, take their mean.
A rational number between a and b is (a + b) ÷ 2
Between 1/2 and 3/4 the mean is 5/8. Repeat the process and you can produce as many as you like — which is the proof of density in a form a class 8 student can follow.
On the Number Line
- Every rational number has a definite point on the number line.
- So does every irrational number, which is why the line is called the real number line.
- √2 can be located exactly using a right triangle with both legs of length 1, since its hypotenuse is √2.
- Rationals and irrationals together make up the real numbers.
The sum of a rational and an irrational number is always irrational, but the product of two irrationals may be rational: √2 × √2 = 2. Questions are built on exactly this asymmetry.
परिमेय = p/q रूप में लिखा जा सके; अपरिमेय का दशमलव न समाप्त होता है न आवर्ती।
TRE pointer: Two questions recur in almost every teaching paper: whether π is rational, and how many rational numbers lie between two given ones. The answer to the second is infinitely many, and the wrong options usually offer a finite count. Since you will teach this chapter, the paper may also ask why the set of rationals had to be invented — because natural numbers and integers were not closed under division. Reasoning questions like this cannot be guessed when a wrong answer costs −1/3.
60-Second Recap
- Rational means expressible as p/q with q not zero; integers and recurring decimals qualify.
- Irrational numbers have non-terminating, non-recurring decimals: √2, √3, π, e.
- 22/7 is an approximation of π, not its value.
- Rationals are closed under addition, subtraction and multiplication, but not division by zero.
- Between any two rationals lie infinitely many rationals — the density property.
- Rational plus irrational is always irrational; irrational times irrational may be rational.
Frequently Asked Questions
Is pi a rational number?
No. Pi is irrational, meaning its decimal expansion never terminates and never repeats. The value 22 by 7 is only a convenient approximation used in calculation, not the exact value. Treating that fraction as pi itself is the most common error in this topic.
How many rational numbers lie between two rational numbers?
Infinitely many. Taking the average of the two gives one, and averaging again gives another, and the process never ends. This is the density property of rational numbers, and the options in such questions usually offer one, two or ten to catch a hurried answer.
Is the product of two irrational numbers always irrational?
No. The square root of two multiplied by itself gives exactly two, which is rational. However, adding a rational number to an irrational one always produces an irrational result. This asymmetry between sum and product is exactly what statement-based questions test.
Why were rational numbers needed at all?
Because integers are not closed under division: three divided by two has no answer inside the integers. Extending the number system to fractions of the form p over q fixed that gap. The same reasoning explains every extension, from natural numbers to whole to integers to rationals.