Which one of the following is an irrational number?
0.12112211122211112222.......
Identifying the Irrational Number
This solution explains how to identify an irrational number from a list of options. We will analyze each option based on the definitions of rational and irrational numbers.
Understanding Rational and Irrational Numbers
Before diving into the options, let's clarify what rational and irrational numbers are:
- Rational Numbers: These are numbers that can be expressed as a fraction \(\frac{p}{q}\), where p and q are integers, and q is not zero (\(q \neq 0\)). Examples include integers (like 5, written as \(\frac{5}{1}\)), terminating decimals (like 0.5, written as \(\frac{1}{2}\)), and repeating decimals (like 0.333..., written as \(\frac{1}{3}\)).
- Irrational Numbers: These are numbers that cannot be expressed as a simple fraction \(\frac{p}{q}\). Their decimal representation is non-terminating (goes on forever) and non-repeating (no pattern of digits repeats indefinitely). Famous examples include \(\pi\) (pi) and \(\sqrt{2}\).
Analysis of Options
Option 1 Analysis: \(\sqrt {59049}\)
To determine if this is rational or irrational, we need to check if 59049 is a perfect square.
Let's find the square root of 59049.
We can test potential integer roots. We find that:
\(243 \times 243 = 59049\)
Therefore, \(\sqrt {59049} = 243\).
Since 243 is an integer, it can be written as the fraction \(\frac{243}{1}\). Thus, this is a rational number.
Option 2 Analysis: \(\frac{{231}}{{593}}\)
This option is presented as a fraction where the numerator is 231 and the denominator is 593.
Both 231 and 593 are integers, and the denominator (593) is not zero.
By the definition of rational numbers, any number that can be written in the form \(\frac{p}{q}\) (where p and q are integers, \(q \neq 0\)) is rational.
Therefore, \(\frac{{231}}{{593}}\) is a rational number.
Option 3 Analysis: 0.45454545......
This is a decimal number where the digits '45' repeat indefinitely.
Let's represent this repeating decimal as x:
\(x = 0.454545...\)
To convert this to a fraction, we can use the following steps:
- Multiply by 100 (since 2 digits repeat): \(100x = 45.454545...\)
- Subtract the original equation from this new one: \(100x - x = (45.454545...) - (0.454545...)\)
- Simplify: \(99x = 45\)
- Solve for x: \(x = \frac{45}{99}\)
- Simplify the fraction: \(x = \frac{5}{11}\)
Since 0.454545... can be expressed as the fraction \(\frac{5}{11}\), it is a rational number.
Option 4 Analysis: 0.12112211122211112222.......
Let's examine the pattern of this decimal:
- It starts with '1'.
- Followed by '2'.
- Then '11'.
- Followed by '22'.
- Then '111'.
- Followed by '222'.
- Then '1111'.
- Followed by '2222', and so on.
The pattern consists of blocks of '1's and '2's, where the number of digits in each block increases (one '1', one '2', two '1's, two '2's, three '1's, three '2's, etc.).
Crucially, there is no finite block of digits that repeats indefinitely. The decimal is non-terminating and non-repeating.
Numbers with decimal representations that are non-terminating and non-repeating are, by definition, irrational numbers.
Conclusion: The Irrational Number
Based on the analysis of each option:
- Option 1 is rational (\(\sqrt {59049} = 243\)).
- Option 2 is rational (\(\frac{{231}}{{593}}\)).
- Option 3 is rational (0.454545... = \(\frac{5}{11}\)).
- Option 4 is irrational (0.121122111222...).
Therefore, the number 0.12112211122211112222....... is the irrational number among the choices.
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