Which statement correctly describes the number V22 =4.690415...?

Question

A. The number is rational because it can be written as a terminating decimal.
B. The number is rational because it can be written as a non-repeating, non-terminating decimal.
C. The number is irrational because it can be written as a terminating decimal.
D. The number is irrational because it can be written as a non-repeating, non-terminating decimal.

Answer 1

To determine whether the number \( V_{22} = 4.690415... \) is rational or irrational, we need to analyze its characteristics.

• A rational number is one that can be expressed as a fraction \( \frac{p}{q} \), where \( p \) and \( q \) are integers and \( q \neq 0 \). Rational numbers can either be terminating decimals or repeating decimals.
• An irrational number cannot be expressed as a fraction of integers and is characterized by non-repeating, non-terminating decimals.

The decimal \( 4.690415... \) appears to be a non-repeating and non-terminating decimal, as indicated by the ellipsis at the end. Since it cannot be expressed as a fraction of integers and continues infinitely without repeating, we can conclude that it is an irrational number.

Thus, the correct statement that describes \( V_{22} \) is:

D. The number is irrational because it can be written as a non-repeating, non-terminating decimal.