--- title: Complex Numbers TARGET DECK: Obsidian::STEM FILE TAGS: algebra::complex tags: - algebra - complex --- ## Overview The set $\mathbb{C}$ of **complex numbers** is defined by $$\mathbb{C} = \{a + bi \mid a, b \in \mathbb{R}\},$$ where $i$ is the **imaginary number** defined as $i^2 = -1$. %%ANKI Basic How is set the complex numbers denoted? Back: As $\mathbb{C}$. Reference: John B. Fraleigh, _A First Course in Abstract Algebra_, Seventh edition, Pearson new international edition (Harlow: Pearson, 2014). END%% %%ANKI Basic How is set $\mathbb{C}$ defined in set-builder notation? Back: $\mathbb{C} = \{a + bi \mid a, b \in \mathbb{R}\}$ Reference: John B. Fraleigh, _A First Course in Abstract Algebra_, Seventh edition, Pearson new international edition (Harlow: Pearson, 2014). END%% %%ANKI Basic Which of $\mathbb{R}$ or $\mathbb{C}$ is a subset of the other? Back: $\mathbb{R} \subseteq \mathbb{C}$ Reference: John B. Fraleigh, _A First Course in Abstract Algebra_, Seventh edition, Pearson new international edition (Harlow: Pearson, 2014). END%% %%ANKI Basic What is $i$ called? Back: The imaginary number. Reference: John B. Fraleigh, _A First Course in Abstract Algebra_, Seventh edition, Pearson new international edition (Harlow: Pearson, 2014). END%% %%ANKI Basic How is the imaginary number typically denoted? Back: As $i$. Reference: John B. Fraleigh, _A First Course in Abstract Algebra_, Seventh edition, Pearson new international edition (Harlow: Pearson, 2014). END%% %%ANKI Basic $i$ was invented to provide a solution to what equation? Back: $x^2 = -1$ Reference: John B. Fraleigh, _A First Course in Abstract Algebra_, Seventh edition, Pearson new international edition (Harlow: Pearson, 2014). END%% %%ANKI Basic What is the solution of $x^2 = -1$? Back: $i$ Reference: John B. Fraleigh, _A First Course in Abstract Algebra_, Seventh edition, Pearson new international edition (Harlow: Pearson, 2014). END%% %%ANKI Cloze Real number {$r$} is identified with complex number {$r + 0i$}. Reference: John B. Fraleigh, _A First Course in Abstract Algebra_, Seventh edition, Pearson new international edition (Harlow: Pearson, 2014). END%% %%ANKI Cloze What real number is identified with $-\pi + 0i$? Back: $-\pi$ Reference: John B. Fraleigh, _A First Course in Abstract Algebra_, Seventh edition, Pearson new international edition (Harlow: Pearson, 2014). END%% %%ANKI Cloze What real number is identified with $\pi + 2i$? Back: N/A. Reference: John B. Fraleigh, _A First Course in Abstract Algebra_, Seventh edition, Pearson new international edition (Harlow: Pearson, 2014). 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). 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). 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). 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). 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). END%% ## 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). 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). 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). 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). 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). 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). 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). 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). 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). 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). 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). 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). 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). 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).