Rational Number Identification

The question requires identifying the rational number from the provided mathematical options.

Understanding Rational Numbers

A rational number is defined as any number that can be expressed in the form \(\frac{p}{q}\), where '\(p\)' and '\(q\)' are integers, and the denominator '\(q\)' is not zero (\(q \neq 0\)). This means rational numbers can be written as a simple fraction.

Analyzing the Options

Let's evaluate each option to determine if it fits the definition of a rational number:

• Option 1: \(e\)
  Euler's number, \(e\), is approximately $2.71828...$. It is a well-known mathematical constant that is irrational and cannot be represented as a simple fraction \(\frac{p}{q}\).
• Option 2: \(\pi\)
  Pi, \(\pi\), is approximately $3.14159...$. Like \(e\), \(\pi\) is a transcendental number and is therefore irrational.
• Option 3: \(\sqrt{3}\)
  The square root of 3, \(\sqrt{3}\), is approximately $1.73205...$. Since 3 is not a perfect square of an integer, \(\sqrt{3}\) is an irrational number.
• Option 4: \(\sqrt{64}\)
  To determine if \(\sqrt{64}\) is rational, we calculate its value. The square root of 64 is 8, because \(8 \times 8 = 64\). The number 8 is an integer, and any integer can be expressed as a fraction by placing it over 1 (e.g., \(\frac{8}{1}\)). Thus, 8 is a rational number.

Conclusion

By evaluating each option, we find that \(\sqrt{64}\) is the only number that can be expressed as a simple fraction \(\frac{p}{q}\), making it the correct choice for the rational number.