Which of the following is irrational number?

Question

Accepted Answer

The correct option is 4:
To solve this problem, we need to recall the definitions of rational and irrational numbers.
A rational number is any number that can be expressed as a fraction $\frac{p}{q}$, where $p$ and $q$ are integers and $q \neq 0$. In decimal form, rational numbers either terminate (end) or repeat a pattern of digits.
An irrational number is a real number that cannot be expressed as a simple fraction $\frac{p}{q}$. In decimal form, irrational numbers have infinite non-repeating decimal expansions.

Let's analyze each option:1.  $\frac{4}{5}$: This number is already in the form of $\frac{p}{q}$, where $p=4$ and $q=5$ are integers and $q \neq 0$. When converted to a decimal, $\frac{4}{5} = 0.8$, which is a terminating decimal. Therefore, $\frac{4}{5}$ is a rational number.

2.  $1.333.....$: This number has a repeating decimal pattern (the digit 3 repeats infinitely). A repeating decimal can always be expressed as a fraction. For example, $1.333... = 1\frac{1}{3} = \frac{4}{3}$. Since it can be written as a fraction, $1.333.....$ is a rational number.

3.  $3.14$: This number is a terminating decimal. It can be written as the fraction $\frac{314}{100} = \frac{157}{50}$. Since it can be expressed as a fraction, $3.14$ is a rational number. (Note: While $3.14$ is often used as an approximation for $\pi$, it is not actually $\pi$. This is a common misconception.)

4.  $\pi$: Pi ($\pi$) is a famous mathematical constant that represents the ratio of a circle's circumference to its diameter. Its decimal expansion is infinite and non-repeating ($3.14159265...$). It cannot be expressed as a simple fraction of two integers. Therefore, $\pi$ is an irrational number.The incorrect option is 1: $\frac{4}{5}$ is a rational number because it is expressed as a fraction of two integers.
The incorrect option is 2: $1.333.....$ is a rational number because it is a repeating decimal, which can be expressed as a fraction.
The incorrect option is 3: $3.14$ is a rational number because it is a terminating decimal, which can be expressed as a fraction.