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title TARGET DECK FILE TAGS tags
Complex Numbers Obsidian::STEM algebra::complex
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).

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%%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).

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%%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).

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%%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).

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%%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).

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%%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).

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%%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).

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%%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).

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%%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).

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%%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).

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%%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.

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%%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.

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%%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).

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%%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).

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%%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).

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%%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).

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%%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).

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%%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).

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%%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).

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%%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).

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%%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).

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%%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).

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%%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).

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%%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).

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%%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).

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%%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).

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%%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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Bibliography