Irrational Numbers

Irrational numbers are fundamental in mathematics because they complete the real number system, filling gaps on the number line where rational numbers cannot reach. They are used in various fields, including geometry, calculus, and physics, for precise calculations and theoretical proofs.

Table Of Contents

What are Irrational Numbers ?
How do you distinguish between rational and irrational numbers?
Is the square root of any number an irrational number?
Why is pi considered an irrational number?
What are the properties of irrational numbers?
Is zero an irrational number?
Can the product of two irrational numbers be rational?
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What are Irrational Numbers ?

Irrational number is indeed a real number that cannot be expressed as $\frac{a}{b}$ , where ${a}$ and ${b}$ are integers, ${b} \neq {0}$ . Its decimal representation is non-terminating and non-repeating.

Example: $\sqrt{2}$
Approximate value: 1,414213562…
Decimal representation is non-terminating and non-repeating.
It cannot be expressed as a fraction $\frac{a}{b}$

Example: $\pi$
Approximate value: 3,141592653…
Decimal representation is non-terminating and non-repeating.
It is impossible to express $\pi$ as a fraction.

Example: 0,1010010001…
This number has no repeating pattern in its decimal representation.
It cannot be expressed as a fraction.

How do you distinguish between rational and irrational numbers?

Rational numbers have terminating or repeating decimal expansions, while irrational numbers do not.

Rational numbers can be expressed as $\frac{a}{b}$ , where ${a}$ and ${b}$ are integers, ${b}\neq{0}$ .

Is the square root of any number an irrational number?

The square root of non-perfect squares (e.g., $\sqrt{2}$ ) is irrational, while the square root of perfect squares (e.g., $\sqrt{9} = {3}$ ) is rational.

Why is pi considered an irrational number?

$\pi$ is an irrational number because it cannot be expressed as a fraction and its decimal representation is non-terminating and non-repeating.

What are the properties of irrational numbers?

Decimal expansions are non-terminating and non-repeating.
They cannot be written as $\frac{a}{b}$ , where ${a}$ and ${b}$ are integers, ${b}\neq{0}$ .
The sum or product of a rational and an irrational number is generally irrational, except in special cases (e.g., $\pi - \pi = {0}$ , which is rational).