Question

which of the following statements is not true?

choose the incorrect statement below.

a. the imaginary unit, \\(i\\), is defined as \\(i = \sqrt{-1}\\).
b. because the imaginary unit \\(i\\) is not a real number, it is impossible to evaluate an expression such as \\(i^n\\) if \\(n\\) is a negative integer.
c. the imaginary unit, \\(i\\), is a solution to the equation \\(x^2 + 1 = 0\\).
d. if \\(i\\) is the imaginary unit, then \\(i^2 = -1\\).

Explanation:

Analyze definition of imaginary unit

Using the Imaginary Unit knowledge point

$$ i = \sqrt{-1} \quad \text{and} \quad i^2 = -1 $$

This confirms that statement A and statement D are mathematically true.

Evaluate solution to quadratic equation

Using the Imaginary Unit knowledge point

$$ x^2 + 1 = 0 \implies x^2 = -1 \implies x = \pm i $$

Thus, \(i\) is indeed a solution, which makes statement C true.

Analyze negative integer powers of i

Using the Powers of i knowledge point

$$ i^{-n} = \frac{1}{i^n} $$

Since \(i^n\) is always non-zero (\(\pm 1\) or \(\pm i\)), negative powers are always evaluable. Thus, statement B is false.

Answer:

• A. The imaginary unit, \(i\), is defined as \(i = \sqrt{-1}\).
• B. Because the imaginary unit \(i\) is not a real number, it is impossible to evaluate an expression such as \(i^n\) if \(n\) is a negative integer. (Correct answer)
• C. The imaginary unit, \(i\), is a solution to the equation \(x^2 + 1 = 0\).
• D. If \(i\) is the imaginary unit, then \(i^2 = -1\).