Certainly! Let's analyze the two given numbers: [tex]\(\frac{22}{7}\)[/tex] and [tex]\(\pi\)[/tex].
a) Which one is the rational number? Give reason.
The number [tex]\(\frac{22}{7}\)[/tex] is a rational number. This is because a rational number is defined as any number that can be expressed as a fraction, where both the numerator and the denominator are integers (and the denominator is not zero). In this case, [tex]\(\frac{22}{7}\)[/tex] clearly fits this definition, since 22 and 7 are both integers. Therefore, [tex]\(\frac{22}{7}\)[/tex] is rational.
b) Which one is the irrational number? Give reason.
The number [tex]\(\pi\)[/tex] (pi) is an irrational number. This is because an irrational number is defined as any number that cannot be expressed as a simple fraction - that is, it cannot be written as the ratio of two integers. Additionally, the decimal representation of an irrational number is non-repeating and non-terminating. The number [tex]\(\pi\)[/tex] fits this description perfectly. Its decimal representation starts with 3.14159... and continues infinitely without repeating. Consequently, [tex]\(\pi\)[/tex] is an irrational number.
To summarize:
- a) Rational: [tex]\(\frac{22}{7}\)[/tex], because it can be expressed as a fraction of two integers.
- b) Irrational: [tex]\(\pi\)[/tex], because it cannot be expressed as a fraction and its decimal is non-repeating and non-terminating.
