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 (2)
The problem asks us to identify which of the given expressions is undefined in the set of real numbers.
Evaluate each expression: − 4 , − 4 0 , 0 4 , and − 4 × 0 .
Determine that − 4 is not a real number because it involves the square root of a negative number.
Conclude that the expression undefined in the set of real numbers is − 4 .
Explanation
Understanding the Problem We are given four expressions and asked to identify which one is undefined in the set of real numbers. An expression is undefined in the set of real numbers if it does not result in a real number when evaluated.
Evaluating the Expressions Let's evaluate each expression: A) − 4 : The square root of a negative number is not a real number. It is an imaginary number. Therefore, − 4 is undefined in the set of real numbers. B) − 4 0 : Zero divided by any non-zero real number is zero. Therefore, − 4 0 = 0 , which is a real number. C) 0 4 : Zero raised to any positive integer power is zero. Therefore, 0 4 = 0 , which is a real number. D) − 4 × 0 : Any real number multiplied by zero is zero. Therefore, − 4 × 0 = 0 , which is a real number.
Identifying the Undefined Expression Based on the evaluation, the expression − 4 is the only one that is undefined in the set of real numbers.
Final Answer 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 results involving square roots of negative numbers under specific conditions, indicating that the model is not applicable in those scenarios. Similarly, in engineering, knowing the domain of functions helps in designing systems that operate within safe and predictable limits. This concept is also vital in computer science, where algorithms must handle potential errors arising from undefined operations to ensure program stability and reliability.
The expression that is undefined in the set of real numbers is \sqrt{-4}. All other expressions either evaluate to zero or a real number. Thus, option A (\sqrt{-4}) is the correct answer.
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