Which expression is undefined in the set of real numbers?
A. [tex]$\sqrt{-4}$[/tex]
B. [tex]$\frac{0}{-4}$[/tex]
C. [tex]$0^4$[/tex]
D. [tex]$-4 \times 0$[/tex]
Answer (1)
− 4 is undefined in the set of real numbers.
− 4 0 = 0 is defined.
0 4 = 0 is defined.
− 4 × 0 = 0 is defined.
The expression that is undefined in the set of real numbers is − 4 .
Explanation
Analyzing the Expressions We are given four expressions and need to determine which one is undefined in the set of real numbers. Let's analyze each expression:
A) − 4 : The square root of a negative number is not defined in the set of real numbers because there is no real number that, when multiplied by itself, results in a negative number. In the complex number system, − 4 = 2 i , but we are looking for an expression that is undefined in the real number system.
B) − 4 0 : Zero divided by any non-zero number is defined and equals zero. So, − 4 0 = 0 .
C) 0 4 : Zero raised to any positive integer power is defined and equals zero. So, 0 4 = 0 × 0 × 0 × 0 = 0 .
D) − 4 × 0 : Any number multiplied by zero is defined and equals zero. So, − 4 × 0 = 0 .
Identifying the Undefined Expression From the analysis above, we can see that:
− 4 is not defined in the set of real numbers.
− 4 0 = 0 is defined.
0 4 = 0 is defined.
− 4 × 0 = 0 is defined.
Conclusion Therefore, the expression that is undefined in the set of real numbers is − 4 .
Examples
Understanding which expressions are undefined in real numbers is crucial in various fields. For instance, in physics, certain equations might yield a square root of a negative number, indicating that the physical scenario described by the equation is not possible or requires a different model. Similarly, in engineering, knowing that division by zero is undefined helps prevent errors in calculations and system designs. Recognizing these undefined expressions ensures accurate and meaningful results in mathematical and real-world applications.
