Card 1 of 20
EasyStudy card 1
Prompt
How is the imaginary unit \(i\) defined?
Answer
\(i=\sqrt{-1}\), so \(i^2=-1\).
MathChapter 13: Polynomial and Radical Functions
Flashcards
Complex Numbers Flashcards
Twenty concise cards on imaginary units, standard form, operations, conjugates, division, and equality.
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Card 1 of 20. Answer hidden.
Card 2 of 20
EasyStudy card 2
Prompt
What is standard form for a complex number?
Answer
\(a+bi\), where \(a\) and \(b\) are real.
Card 3 of 20
EasyStudy card 3
Prompt
In \(7-3i\), what is the real part?
Answer
\(7\).
Card 4 of 20
EasyStudy card 4
Prompt
In \(7-3i\), what is the imaginary coefficient?
Answer
\(-3\).
Card 5 of 20
EasyStudy card 5
Prompt
What is \(i^2\)?
Answer
\(-1\).
Card 6 of 20
EasyStudy card 6
Prompt
What is \(i^3\)?
Answer
\(-i\).
Card 7 of 20
EasyStudy card 7
Prompt
What is \(i^4\)?
Answer
\(1\).
Card 8 of 20
MediumStudy card 8
Prompt
How often do powers of \(i\) repeat?
Answer
Every four powers.
Card 9 of 20
MediumStudy card 9
Prompt
How do you add complex numbers?
Answer
Combine real parts and imaginary coefficients separately.
Card 10 of 20
MediumStudy card 10
Prompt
What must happen before subtracting a complex number?
Answer
Distribute the negative sign to both its real and imaginary parts.
Card 11 of 20
MediumStudy card 11
Prompt
What identity simplifies complex-number products?
Answer
Replace every \(i^2\) by \(-1\).
Card 12 of 20
MediumStudy card 12
Prompt
What is the conjugate of \(a+bi\)?
Answer
\(a-bi\).
Card 13 of 20
MediumStudy card 13
Prompt
What is \((a+bi)(a-bi)\)?
Answer
\(a^2+b^2\), a real number.
Card 14 of 20
MediumStudy card 14
Prompt
How do you divide by \(c+di\)?
Answer
Multiply numerator and denominator by the conjugate \(c-di\).
Card 15 of 20
MediumStudy card 15
Prompt
When are \(a+bi\) and \(c+di\) equal?
Answer
When \(a=c\) and \(b=d\).
Card 16 of 20
MediumStudy card 16
Prompt
Simplify \(\sqrt{-20}\).
Answer
\(2i\sqrt5\).
Card 17 of 20
HardStudy card 17
Prompt
How do you reduce \(i^{57}\)?
Answer
Use \(57\bmod4=1\), so \(i^{57}=i\).
Card 18 of 20
HardStudy card 18
Prompt
Why is the conjugate useful in complex division?
Answer
It turns the denominator into the real value \(c^2+d^2\).
Card 19 of 20
HardStudy card 19
Prompt
Simplify \((2+3i)(4-i)\).
Answer
\(11+10i\).
Card 20 of 20
HardStudy card 20
Prompt
If \((a+2)+(3b-1)i=7+8i\), find \(a+b\).
Answer
\(a=5\), \(b=3\), so \(a+b=8\).
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