160 lines
6.6 KiB
Markdown
160 lines
6.6 KiB
Markdown
---
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title: Absolute Value
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TARGET DECK: Obsidian::STEM
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FILE TAGS: algebra::abs
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tags:
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- algebra
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---
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## Overview
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Let $x \in \mathbb{R}$. The **absolute value** of $x$, denoted $\lvert x \rvert$, is defined as $$\lvert x \rvert = \begin{cases} x & \text{if } x \geq 0 \\ -x & \text{if } x \leq 0 \end{cases}$$
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%%ANKI
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Basic
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How is the absolute value of $x \in \mathbb{R}$ denoted?
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Back: $\lvert x \rvert$
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Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
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<!--ID: 1724609565708-->
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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 $x \in \mathbb{R}$ defined?
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Back: $\lvert x \rvert = \begin{cases} x & \text{if } x \geq 0 \\ -x & \text{if } x \leq 0 \end{cases}$
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Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
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<!--ID: 1724609565711-->
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END%%
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%%ANKI
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Basic
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The absolute value of $x \in \mathbb{R}$ considers what two cases?
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Back: Whether $x \geq 0$ or $x \leq 0$.
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Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
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<!--ID: 1724609565713-->
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END%%
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%%ANKI
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Basic
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Let $x \in \mathbb{R}$. When is $-\lvert x \rvert \leq x < \lvert x \rvert$?
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Back: When $x < 0$.
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Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
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<!--ID: 1724609565715-->
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END%%
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%%ANKI
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Basic
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Let $x \in \mathbb{R}$. When is $-\lvert x \rvert < x \leq \lvert x \rvert$?
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Back: When $x > 0$.
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Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
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<!--ID: 1724609565716-->
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END%%
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%%ANKI
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Basic
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Let $x \in \mathbb{R}$. When is $-\lvert x \rvert \leq x \leq \lvert x \rvert$?
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Back: Always.
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Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
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<!--ID: 1724609565717-->
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END%%
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%%ANKI
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Basic
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Let $x, a \in \mathbb{R}$ and $a \geq 0$. How is $\lvert x \rvert \leq a$ equivalently written as a chain of inequalities?
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Back: $-a \leq x \leq a$
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Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
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<!--ID: 1724609565718-->
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END%%
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%%ANKI
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Basic
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Let $x, a \in \mathbb{R}$ and $a \geq 0$. How is $\lvert x \rvert \leq a$ geometricaly depicted?
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Back:
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![[abs-value-geom.png]]
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Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
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<!--ID: 1724609565719-->
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END%%
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%%ANKI
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Basic
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Let $x, a \in \mathbb{R}$ and $a \geq 0$. How is $-a \leq x \leq a$ equivalently written using absolute value?
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Back: $\lvert x \rvert \leq a$
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Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
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<!--ID: 1724609565720-->
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END%%
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%%ANKI
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Basic
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Let $x, a \in \mathbb{R}$ and $a \geq 0$. How is $-a \leq x \leq a$ geometrically depicted?
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Back:
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![[abs-value-geom.png]]
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Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
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<!--ID: 1724609565721-->
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END%%
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## Triangle Inequality
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Let $x, y \in \mathbb{R}$. Then the **triangle inequality** of $\mathbb{R}$ states $$\lvert x + y \rvert \leq \lvert x \rvert + \lvert y \rvert$$
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%%ANKI
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Basic
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What does the triangle inequality of $\mathbb{R}$ state?
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Back: For $x, y \in \mathbb{R}$, $\lvert x + y \rvert \leq \lvert x \rvert + \lvert y \rvert$.
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Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
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<!--ID: 1724609565722-->
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END%%
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%%ANKI
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Basic
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Why is the triangle inequality named the way it is?
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Back: The length of a triangle side is $\leq$ the sum of the other two sides.
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Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
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<!--ID: 1724609565723-->
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END%%
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%%ANKI
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Basic
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What algebraic inequality is demonstrated in the following?
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![[triangle-inequality.png]]
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Back: The triangle inequality of $\mathbb{R}$.
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Reference: “Triangle Inequality.” In _Wikipedia_, July 1, 2024. [https://en.wikipedia.org/w/index.php?title=Triangle_inequality](https://en.wikipedia.org/w/index.php?title=Triangle_inequality&oldid=1232015318).
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<!--ID: 1724609565724-->
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END%%
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%%ANKI
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Basic
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What degenerate triangle justifies use of $\leq$ over $<$ in the triangle inequality of $\mathbb{R}$?
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Back:
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![[triangle-inequality-degenerate.png]]
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Reference: “Triangle Inequality.” In _Wikipedia_, July 1, 2024. [https://en.wikipedia.org/w/index.php?title=Triangle_inequality](https://en.wikipedia.org/w/index.php?title=Triangle_inequality&oldid=1232015318).
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<!--ID: 1724609565725-->
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END%%
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%%ANKI
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Basic
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What two chains of inequalities can be added together to prove the triangle inequality of $\mathbb{R}$?
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Back: $-\lvert x \rvert \leq x \leq \lvert x \rvert$ and $-\lvert y \rvert \leq y \leq \lvert y \rvert$.
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Reference: “Triangle Inequality.” In _Wikipedia_, July 1, 2024. [https://en.wikipedia.org/w/index.php?title=Triangle_inequality](https://en.wikipedia.org/w/index.php?title=Triangle_inequality&oldid=1232015318).
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<!--ID: 1724609565726-->
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END%%
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%%ANKI
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Basic
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What does the general triangle inequality of $\mathbb{R}$ state?
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Back: For real numbers $a_1, \ldots, a_n$, $$\left\lvert \sum_{k=1}^n a_k \right\rvert \leq \sum_{k=1}^n \lvert a_k \rvert$$
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Reference: “Triangle Inequality.” In _Wikipedia_, July 1, 2024. [https://en.wikipedia.org/w/index.php?title=Triangle_inequality](https://en.wikipedia.org/w/index.php?title=Triangle_inequality&oldid=1232015318).
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<!--ID: 1724611618744-->
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END%%
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%%ANKI
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Basic
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Let $a_1\, \ldots, a_n \in \mathbb{R}$. What is the following a generalization of? $$\left\lvert \sum_{k=1}^n a_k \right\rvert \leq \sum_{k=1}^n \lvert a_k \rvert$$
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Back: The triangle inequality of $\mathbb{R}$.
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Reference: “Triangle Inequality.” In _Wikipedia_, July 1, 2024. [https://en.wikipedia.org/w/index.php?title=Triangle_inequality](https://en.wikipedia.org/w/index.php?title=Triangle_inequality&oldid=1232015318).
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<!--ID: 1724611618749-->
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END%%
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## Bibliography
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* Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
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* “Triangle Inequality.” In _Wikipedia_, July 1, 2024. [https://en.wikipedia.org/w/index.php?title=Triangle_inequality](https://en.wikipedia.org/w/index.php?title=Triangle_inequality&oldid=1232015318). |