Reference: John B. Fraleigh, _A First Course in Abstract Algebra_, Seventh edition, Pearson new international edition (Harlow: Pearson, 2014).
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END%%
%%ANKI
Basic
What is the horizontal axis of the complex plane typically called?
Back: The real axis.
Reference: “Complex Plane,” in _Wikipedia_, December 14, 2024, [https://en.wikipedia.org/w/index.php?title=Complex_plane](https://en.wikipedia.org/w/index.php?title=Complex_plane&oldid=1263031649).
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END%%
%%ANKI
Basic
What is the vertical axis of the complex plane typically called?
Back: The imaginary axis.
Reference: “Complex Plane,” in _Wikipedia_, December 14, 2024, [https://en.wikipedia.org/w/index.php?title=Complex_plane](https://en.wikipedia.org/w/index.php?title=Complex_plane&oldid=1263031649).
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END%%
%%ANKI
Cloze
The complex plane is formed from the {$x$}-axis and {$yi$}-axis.
Reference: John B. Fraleigh, _A First Course in Abstract Algebra_, Seventh edition, Pearson new international edition (Harlow: Pearson, 2014).
<!--ID: 1735870487472-->
END%%
%%ANKI
Basic
Which number is plotted on the complex plane below?
![[complex-plane-point.png]]
Back: $2 + i$
Reference: John B. Fraleigh, _A First Course in Abstract Algebra_, Seventh edition, Pearson new international edition (Harlow: Pearson, 2014).
<!--ID: 1735870487475-->
END%%
%%ANKI
Cloze
Real numbers are plotted on a {line} whereas complex numbers are plotted on a {plane}.
Reference: John B. Fraleigh, _A First Course in Abstract Algebra_, Seventh edition, Pearson new international edition (Harlow: Pearson, 2014).
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## Operations
Addition and multiplication of complex numbers are done in the natural way. Given complex numbers $a + bi$ and $c + di$, we have that $$\begin{align*} (a + bi) + (c + di) & = (a + c) + (b + d)i \\ (a + bi) \cdot (c + di) & = (ac -bd) + (ad + bc)i \end{align*}$$
The **absolute value** of $a + bi$, denoted $\lvert a + bi \rvert$, is defined as $$\lvert a + bi \rvert = \sqrt{a^2 + b^2}.$$
%%ANKI
Basic
Let $a + bi$ and $c + di$ be complex numbers. What is their sum?
Back: $(a + c) + (b + d)i$
Reference: John B. Fraleigh, _A First Course in Abstract Algebra_, Seventh edition, Pearson new international edition (Harlow: Pearson, 2014).
<!--ID: 1735870487481-->
END%%
%%ANKI
Basic
Let $a + bi$ and $c + di$ be complex numbers. What is their product?
Back: $(ac - bd) + (ad + bc)i$
Reference: John B. Fraleigh, _A First Course in Abstract Algebra_, Seventh edition, Pearson new international edition (Harlow: Pearson, 2014).
<!--ID: 1735870487484-->
END%%
%%ANKI
Basic
Is addition of complex numbers commutative?
Back: Yes.
Reference: John B. Fraleigh, _A First Course in Abstract Algebra_, Seventh edition, Pearson new international edition (Harlow: Pearson, 2014).
<!--ID: 1735870487487-->
END%%
%%ANKI
Basic
Is addition of complex numbers associative?
Back: Yes.
Reference: John B. Fraleigh, _A First Course in Abstract Algebra_, Seventh edition, Pearson new international edition (Harlow: Pearson, 2014).
<!--ID: 1735870487491-->
END%%
%%ANKI
Basic
What does it mean for addition of complex numbers to be commutative?
Back: For $z_1, z_2 \in \mathbb{C}$, it follows that $z_1 + z_2 = z_2 + z_1$.
Reference: John B. Fraleigh, _A First Course in Abstract Algebra_, Seventh edition, Pearson new international edition (Harlow: Pearson, 2014).
<!--ID: 1735870487494-->
END%%
%%ANKI
Basic
Is multiplication of complex numbers commutative?
Back: Yes.
Reference: John B. Fraleigh, _A First Course in Abstract Algebra_, Seventh edition, Pearson new international edition (Harlow: Pearson, 2014).
<!--ID: 1735870487498-->
END%%
%%ANKI
Basic
Is multiplication of complex numbers associative?
Back: Yes.
Reference: John B. Fraleigh, _A First Course in Abstract Algebra_, Seventh edition, Pearson new international edition (Harlow: Pearson, 2014).
<!--ID: 1735870487502-->
END%%
%%ANKI
Basic
What does it mean for multiplication of complex numbers to be associative?
Back: For $z_1, z_2, z_3 \in \mathbb{C}$, it follows that $z_1(z_2z_3) = (z_1z_2)z_3$.
Reference: John B. Fraleigh, _A First Course in Abstract Algebra_, Seventh edition, Pearson new international edition (Harlow: Pearson, 2014).
<!--ID: 1735870487506-->
END%%
%%ANKI
Basic
How is the absolute value of complex number $z \in \mathbb{C}$ denoted?
Back: As $\lvert z \rvert$.
Reference: John B. Fraleigh, _A First Course in Abstract Algebra_, Seventh edition, Pearson new international edition (Harlow: Pearson, 2014).
<!--ID: 1735870829668-->
END%%
%%ANKI
Basic
Let $z \in \mathbb{C}$. How is $\lvert z \rvert$ defined?
Back: Assuming $z = a + bi$, as $\lvert z \rvert = \sqrt{a^2 + b^2}$.
Reference: John B. Fraleigh, _A First Course in Abstract Algebra_, Seventh edition, Pearson new international edition (Harlow: Pearson, 2014).
<!--ID: 1735870829671-->
END%%
%%ANKI
Basic
Geometrically speaking, what does the absolute value of $z \in \mathbb{C}$ correspond to?
Back: $z$'s distance from the complex plane's origin.
Reference: John B. Fraleigh, _A First Course in Abstract Algebra_, Seventh edition, Pearson new international edition (Harlow: Pearson, 2014).
<!--ID: 1735870829675-->
END%%
%%ANKI
Basic
Let $a + bi$ be a complex number. How is $\sqrt{a^2 + b^2}$ more compactly written?
Back: As $\lvert a + bi \rvert$.
Reference: John B. Fraleigh, _A First Course in Abstract Algebra_, Seventh edition, Pearson new international edition (Harlow: Pearson, 2014).
<!--ID: 1735870829679-->
END%%
%%ANKI
Basic
What geometric theorem motivates the definition of complex numbers' absolute values?
Back: The Pythagorean theorem.
Reference: John B. Fraleigh, _A First Course in Abstract Algebra_, Seventh edition, Pearson new international edition (Harlow: Pearson, 2014).
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END%%
## Bibliography
* “Complex Plane,” in _Wikipedia_, December 14, 2024, [https://en.wikipedia.org/w/index.php?title=Complex_plane](https://en.wikipedia.org/w/index.php?title=Complex_plane&oldid=1263031649).
* John B. Fraleigh, _A First Course in Abstract Algebra_, Seventh edition, Pearson new international edition (Harlow: Pearson, 2014).