Which one of the following is an irrational number (that is, a real number that cannot be expressed as a ratio of two integers)?

Introduction / Context:
This question checks your understanding of rational and irrational numbers. A rational number can be written as a fraction of two integers, while an irrational number cannot be expressed exactly as such a fraction and has a non terminating, non repeating decimal expansion.

Given Data / Assumptions:
Option (a): √59059.Option (b): 231/593, a fraction of two integers.Option (c): 0.454545...., a repeating decimal.Option (d): 0.121122111222, a terminating decimal with finitely many digits shown.

Concept / Approach:
All terminating decimals and repeating decimals represent rational numbers, because they can be converted into fractions. In contrast, the square root of a positive integer that is not a perfect square is known to be irrational. So we must check which option fits that pattern and which are clearly rational based on their form.

Step-by-Step Solution:
Step 1: Consider option (b), 231/593. It is explicitly written as a ratio of two integers, so it is a rational number by definition.Step 2: Consider option (c), 0.454545.... The notation with repeating 45 shows a recurring decimal. Any recurring decimal can be expressed as a fraction (for example, 0.4545.... can be converted to 45/99), so this is also rational.Step 3: Consider option (d), 0.121122111222. As written, there are a finite number of digits after the decimal point and no ellipsis. Any terminating decimal can be written as an integer divided by a power of 10, so it is rational.Step 4: Consider option (a), √59059. The integer 59059 is not a perfect square (there is no integer whose square equals 59059). The square root of a positive integer that is not a perfect square is always irrational.

Verification / Alternative check:
You can quickly check whether 59059 is a perfect square by estimating nearby squares. For example, 240^2 = 57600 and 250^2 = 62500, so 59059 lies between these, and no integer between 240 and 250 has a square equal to 59059. Therefore 59059 is not a perfect square, and its square root is indeed irrational.

Why Other Options Are Wrong:
Option (b) is clearly rational because it is already in fractional form. Option (c) is a classic repeating decimal, which always represents a rational number. Option (d) is a finite decimal, which can always be written as a rational number. None of these can be irrational.

Common Pitfalls:
Some learners mistakenly assume that any decimal with a complex looking pattern is irrational, even if it terminates. Others may think that any decimal written with dots must be irrational, without checking whether the pattern repeats. Remember: repeating or terminating decimals are always rational.

Final Answer:
The only irrational number among the options is √59059.