Question

fill in the blanks.

the imaginary unit \\(i\\) is defined as \\(i = \underline{\quad\quad}\\), where \\(i^2 = \underline{\quad\quad}\\).

Explanation:

<pre_analysis>
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"step_cot",
"blank_filling_formatter",
"explain_then_answer",
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],
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"concepts_used": [
"Imaginary Numbers",
"Complex Numbers"
],
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}
</pre_analysis>

<reasoning>

Define the imaginary unit

Using the Imaginary Numbers and Complex Numbers knowledge points
\[
i = \sqrt{-1}
\]

Determine the square of the imaginary unit

Using the Imaginary Numbers and Complex Numbers knowledge points
\[
i^2 = -1
\]
</reasoning>

<answer>
Fill in the blanks.

The imaginary unit \(i\) is defined as \(i =\) <blank>\(\sqrt{-1}\)</blank>, where \(i^2 =\) <blank>\(-1\)</blank>.
</answer>

<post_analysis>
{
"subject": "Mathematics",
"question_type": "Fill-in-the-blank",
"knowledge_point": [
"Mathematics",
"Algebra",
"Complex Numbers"
]
}
</post_analysis>

Answer:

<pre_analysis>
{
"quality": "clear",
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"skills_matched": [
"step_cot",
"blank_filling_formatter",
"explain_then_answer",
"tutor",
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"needs_drawing": false,
"concepts_used": [
"Imaginary Numbers",
"Complex Numbers"
],
"new_concepts": [],
"current_concepts": [
"Imaginary Numbers",
"Complex Numbers"
]
}
</pre_analysis>

<reasoning>

Define the imaginary unit

Using the Imaginary Numbers and Complex Numbers knowledge points
\[
i = \sqrt{-1}
\]

Determine the square of the imaginary unit

Using the Imaginary Numbers and Complex Numbers knowledge points
\[
i^2 = -1
\]
</reasoning>

<answer>
Fill in the blanks.

The imaginary unit \(i\) is defined as \(i =\) <blank>\(\sqrt{-1}\)</blank>, where \(i^2 =\) <blank>\(-1\)</blank>.
</answer>

<post_analysis>
{
"subject": "Mathematics",
"question_type": "Fill-in-the-blank",
"knowledge_point": [
"Mathematics",
"Algebra",
"Complex Numbers"
]
}
</post_analysis>