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Difficulty: EasyClassification of Numbers

In the real number system, numbers are broadly classified into rational and irrational categories. Based on strict mathematical definitions, which of the following is an irrational number?

  1. A
    227\frac{22}{7}
  2. B
    25\sqrt{25}
  3. C
    3.141593.14159
  4. π\piAnswer

Answer

The mathematical constant π\pi is an irrational number.
The constant π\pi cannot be expressed exactly as a fraction of two integers. Its decimal representation is non-terminating and non-repeating, which is the defining characteristic of an irrational number.

Step-by-Step Solution

1
Recall the definition of a rational number.
A rational number can be written exactly as a fraction pq\frac{p}{q} where pp and qq are integers and q≠0q \neq 0.
Establishing the rule allows us to eliminate options that fit this criteria.
2
Evaluate the fractional and decimal options.
227\frac{22}{7} is explicitly a ratio of integers. 3.141593.14159 is a terminating decimal, which can also be written as a fraction.
Both perfectly fit the definition of a rational number and must be eliminated.
3
Evaluate the square root option.
25\sqrt{25} simplifies to the integer 55.
Since 5=515 = \frac{5}{1}, it is also a rational number.
4
Evaluate the mathematical constant.
π\pi represents a non-terminating, non-repeating decimal that cannot be written exactly as a fraction.
Numbers with non-terminating and non-repeating decimal expansions are classified as irrational numbers.

Key Concept

Distinguishing between rational numbers (exact fractions, terminating decimals, perfect roots) and irrational numbers (non-terminating, non-repeating decimals like the constant pi).
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Topics

Classification of Numbers

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