Number Types Explained

To find a number that is real but NOT rational, we need to understand these terms:

• Real Numbers: All numbers on the number line, including integers, fractions, terminating/repeating decimals, and irrational numbers.
• Rational Numbers: Numbers that can be written as a fraction $\frac{p}{q}$, where $p$ and $q$ are integers and $q \neq 0$. Examples include $\frac{3}{4}$, $5$ (as $\frac{5}{1}$), and $-2$ (as $\frac{-2}{1}$). Their decimal form either terminates or repeats.
• Irrational Numbers: Numbers that cannot be written as a simple fraction $\frac{p}{q}$. Their decimal representation is non-terminating and non-repeating. Examples include $\pi$ and $\sqrt{7}$. Irrational numbers are a subset of real numbers.

Real and Rational Number Analysis

Let's examine each option:

• Option 1: $\frac{3}{4}$ - This is a fraction of two integers, so it is a rational number. It is also a real number.
• Option 2: $5$ - This is an integer, which can be written as $\frac{5}{1}$. It is a rational number and a real number.
• Option 3: $\sqrt{7}$ - The number $7$ is not a perfect square. Therefore, its square root, $\sqrt{7}$, is an irrational number. Since all irrational numbers are real numbers, $\sqrt{7}$ meets the criteria of being real but not rational.
• Option 4: $-2$ - This is an integer, which can be written as $\frac{-2}{1}$. It is a rational number and a real number.

The only option that is a real number but not a rational number is $\sqrt{7}$.