246 lines
8.7 KiB
Markdown
246 lines
8.7 KiB
Markdown
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---
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title: Complex Numbers
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TARGET DECK: Obsidian::STEM
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FILE TAGS: algebra::complex
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tags:
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- algebra
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- complex
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---
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## Overview
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The set $\mathbb{C}$ of **complex numbers** is defined by $$\mathbb{C} = \{a + bi \mid a, b \in \mathbb{R}\},$$
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where $i$ is the **imaginary number** defined as $i^2 = -1$.
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%%ANKI
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Basic
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How is set the complex numbers denoted?
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Back: As $\mathbb{C}$.
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Reference: John B. Fraleigh, _A First Course in Abstract Algebra_, Seventh edition, Pearson new international edition (Harlow: Pearson, 2014).
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<!--ID: 1735870487309-->
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END%%
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%%ANKI
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Basic
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How is set $\mathbb{C}$ defined in set-builder notation?
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Back: $\mathbb{C} = \{a + bi \mid a, b \in \mathbb{R}\}$
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Reference: John B. Fraleigh, _A First Course in Abstract Algebra_, Seventh edition, Pearson new international edition (Harlow: Pearson, 2014).
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<!--ID: 1735870487353-->
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END%%
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%%ANKI
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Basic
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Which of $\mathbb{R}$ or $\mathbb{C}$ is a subset of the other?
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Back: $\mathbb{R} \subseteq \mathbb{C}$
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Reference: John B. Fraleigh, _A First Course in Abstract Algebra_, Seventh edition, Pearson new international edition (Harlow: Pearson, 2014).
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<!--ID: 1735870487367-->
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END%%
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%%ANKI
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Basic
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What is $i$ called?
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Back: The imaginary number.
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Reference: John B. Fraleigh, _A First Course in Abstract Algebra_, Seventh edition, Pearson new international edition (Harlow: Pearson, 2014).
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<!--ID: 1735870487383-->
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END%%
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%%ANKI
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Basic
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How is the imaginary number typically denoted?
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Back: As $i$.
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Reference: John B. Fraleigh, _A First Course in Abstract Algebra_, Seventh edition, Pearson new international edition (Harlow: Pearson, 2014).
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<!--ID: 1735870487411-->
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END%%
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%%ANKI
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Basic
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$i$ was invented to provide a solution to what equation?
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Back: $x^2 = -1$
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Reference: John B. Fraleigh, _A First Course in Abstract Algebra_, Seventh edition, Pearson new international edition (Harlow: Pearson, 2014).
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<!--ID: 1735870487426-->
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END%%
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%%ANKI
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Basic
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What is the solution of $x^2 = -1$?
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Back: $i$
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Reference: John B. Fraleigh, _A First Course in Abstract Algebra_, Seventh edition, Pearson new international edition (Harlow: Pearson, 2014).
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<!--ID: 1735870487437-->
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END%%
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%%ANKI
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Cloze
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Real number {$r$} is identified with complex number {$r + 0i$}.
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Reference: John B. Fraleigh, _A First Course in Abstract Algebra_, Seventh edition, Pearson new international edition (Harlow: Pearson, 2014).
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<!--ID: 1735870487452-->
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END%%
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%%ANKI
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Cloze
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What real number is identified with $-\pi + 0i$?
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Back: $-\pi$
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Reference: John B. Fraleigh, _A First Course in Abstract Algebra_, Seventh edition, Pearson new international edition (Harlow: Pearson, 2014).
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<!--ID: 1735870487459-->
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END%%
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%%ANKI
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Cloze
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What real number is identified with $\pi + 2i$?
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Back: N/A.
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Reference: John B. Fraleigh, _A First Course in Abstract Algebra_, Seventh edition, Pearson new international edition (Harlow: Pearson, 2014).
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<!--ID: 1735870487463-->
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END%%
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%%ANKI
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Basic
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What is the horizontal axis of the complex plane typically called?
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Back: The real axis.
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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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<!--ID: 1735870487466-->
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END%%
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%%ANKI
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Basic
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What is the vertical axis of the complex plane typically called?
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Back: The imaginary axis.
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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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<!--ID: 1735870487469-->
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END%%
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%%ANKI
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Cloze
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The complex plane is formed from the {$x$}-axis and {$yi$}-axis.
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Reference: John B. Fraleigh, _A First Course in Abstract Algebra_, Seventh edition, Pearson new international edition (Harlow: Pearson, 2014).
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<!--ID: 1735870487472-->
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END%%
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%%ANKI
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Basic
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Which number is plotted on the complex plane below?
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![[complex-plane-point.png]]
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Back: $2 + i$
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Reference: John B. Fraleigh, _A First Course in Abstract Algebra_, Seventh edition, Pearson new international edition (Harlow: Pearson, 2014).
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<!--ID: 1735870487475-->
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END%%
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%%ANKI
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Cloze
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Real numbers are plotted on a {line} whereas complex numbers are plotted on a {plane}.
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Reference: John B. Fraleigh, _A First Course in Abstract Algebra_, Seventh edition, Pearson new international edition (Harlow: Pearson, 2014).
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<!--ID: 1735870487478-->
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END%%
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## Operations
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Addition and multiplication of complex numbers are done in the natural way. Given complex numbers $a + bi$ and $c + di$, we have that $$\begin{align*} (a + bi) + (c + di) & = (a + c) + (b + d)i \\ (a + bi) \cdot (c + di) & = (ac -bd) + (ad + bc)i \end{align*}$$
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The **absolute value** of $a + bi$, denoted $\lvert a + bi \rvert$, is defined as $$\lvert a + bi \rvert = \sqrt{a^2 + b^2}.$$
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%%ANKI
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Basic
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Let $a + bi$ and $c + di$ be complex numbers. What is their sum?
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Back: $(a + c) + (b + d)i$
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Reference: John B. Fraleigh, _A First Course in Abstract Algebra_, Seventh edition, Pearson new international edition (Harlow: Pearson, 2014).
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<!--ID: 1735870487481-->
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END%%
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%%ANKI
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Basic
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Let $a + bi$ and $c + di$ be complex numbers. What is their product?
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Back: $(ac - bd) + (ad + bc)i$
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Reference: John B. Fraleigh, _A First Course in Abstract Algebra_, Seventh edition, Pearson new international edition (Harlow: Pearson, 2014).
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<!--ID: 1735870487484-->
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END%%
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%%ANKI
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Basic
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Is addition of complex numbers commutative?
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Back: Yes.
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Reference: John B. Fraleigh, _A First Course in Abstract Algebra_, Seventh edition, Pearson new international edition (Harlow: Pearson, 2014).
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<!--ID: 1735870487487-->
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END%%
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%%ANKI
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Basic
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Is addition of complex numbers associative?
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Back: Yes.
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Reference: John B. Fraleigh, _A First Course in Abstract Algebra_, Seventh edition, Pearson new international edition (Harlow: Pearson, 2014).
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<!--ID: 1735870487491-->
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END%%
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%%ANKI
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Basic
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What does it mean for addition of complex numbers to be commutative?
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Back: For $z_1, z_2 \in \mathbb{C}$, it follows that $z_1 + z_2 = z_2 + z_1$.
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Reference: John B. Fraleigh, _A First Course in Abstract Algebra_, Seventh edition, Pearson new international edition (Harlow: Pearson, 2014).
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<!--ID: 1735870487494-->
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END%%
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%%ANKI
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Basic
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Is multiplication of complex numbers commutative?
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Back: Yes.
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Reference: John B. Fraleigh, _A First Course in Abstract Algebra_, Seventh edition, Pearson new international edition (Harlow: Pearson, 2014).
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<!--ID: 1735870487498-->
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END%%
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%%ANKI
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Basic
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Is multiplication of complex numbers associative?
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Back: Yes.
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Reference: John B. Fraleigh, _A First Course in Abstract Algebra_, Seventh edition, Pearson new international edition (Harlow: Pearson, 2014).
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<!--ID: 1735870487502-->
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END%%
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%%ANKI
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Basic
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What does it mean for multiplication of complex numbers to be associative?
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Back: For $z_1, z_2, z_3 \in \mathbb{C}$, it follows that $z_1(z_2z_3) = (z_1z_2)z_3$.
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Reference: John B. Fraleigh, _A First Course in Abstract Algebra_, Seventh edition, Pearson new international edition (Harlow: Pearson, 2014).
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<!--ID: 1735870487506-->
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END%%
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%%ANKI
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Basic
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How is the absolute value of complex number $z \in \mathbb{C}$ denoted?
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Back: As $\lvert z \rvert$.
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Reference: John B. Fraleigh, _A First Course in Abstract Algebra_, Seventh edition, Pearson new international edition (Harlow: Pearson, 2014).
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<!--ID: 1735870829668-->
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END%%
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%%ANKI
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Basic
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Let $z \in \mathbb{C}$. How is $\lvert z \rvert$ defined?
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Back: Assuming $z = a + bi$, as $\lvert z \rvert = \sqrt{a^2 + b^2}$.
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Reference: John B. Fraleigh, _A First Course in Abstract Algebra_, Seventh edition, Pearson new international edition (Harlow: Pearson, 2014).
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<!--ID: 1735870829671-->
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END%%
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%%ANKI
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Basic
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Geometrically speaking, what does the absolute value of $z \in \mathbb{C}$ correspond to?
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Back: $z$'s distance from the complex plane's origin.
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Reference: John B. Fraleigh, _A First Course in Abstract Algebra_, Seventh edition, Pearson new international edition (Harlow: Pearson, 2014).
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<!--ID: 1735870829675-->
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END%%
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%%ANKI
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Basic
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Let $a + bi$ be a complex number. How is $\sqrt{a^2 + b^2}$ more compactly written?
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Back: As $\lvert a + bi \rvert$.
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Reference: John B. Fraleigh, _A First Course in Abstract Algebra_, Seventh edition, Pearson new international edition (Harlow: Pearson, 2014).
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<!--ID: 1735870829679-->
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END%%
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%%ANKI
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Basic
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What geometric theorem motivates the definition of complex numbers' absolute values?
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Back: The Pythagorean theorem.
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Reference: John B. Fraleigh, _A First Course in Abstract Algebra_, Seventh edition, Pearson new international edition (Harlow: Pearson, 2014).
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<!--ID: 1735870829684-->
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END%%
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## Bibliography
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* “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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* John B. Fraleigh, _A First Course in Abstract Algebra_, Seventh edition, Pearson new international edition (Harlow: Pearson, 2014).
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