Difference between revisions of "Triangle Inequality"
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| + | The triangle inequality is a very important geometric and algebraic property that we will use frequently in the future. | ||
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| + | <blockquote style="background: white; border: 1px solid black; padding: 0.5em;"> | ||
| + | '''Theorem 1 (Triangle Inequality):''' Let <math>a</math> and <math>b</math> be real numbers. Then <math>\mid a + b \mid \leq \mid a \mid + \mid b \mid</math>. | ||
| + | </blockquote> | ||
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| + | :*'''Proof of Theorem:''' For <math>a</math> and <math>b</math> as real numbers we have that <math>-\mid a \mid \leq a \leq \mid a \mid</math> and <math>-\mid b \mid \leq b \leq \mid b \mid</math>. If we add these inequalities together we get that <math>-\mid a \mid - \mid b \mid \leq a + b \leq \mid a \mid + \mid b \mid</math> or rather <math>-\left ( \mid a \mid + \mid b \mid \right ) \leq a + b \leq \left ( \mid a \mid + \mid b \mid \right )</math> which is equivalent to saying that <math>\mid a + b \mid \leq \mid a \mid + \mid b \mid</math>. <math>\blacksquare</math> | ||
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| + | There are also some other important results similar to the triangle inequality that are important to mention. | ||
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| + | <blockquote style="background: white; border: 1px solid black; padding: 0.5em;"> | ||
| + | '''Corollary 1:''' If <math>a</math> and <math>b</math> are real numbers then <math>\mid \mid a \mid - \mid b \mid \mid \leq \mid a - b \mid</math>. | ||
| + | </blockquote> | ||
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| + | :*'''Proof of Corollary 1:''' We first write <math>a = a - b + b</math> and therefore applying the triangle inequality we get that <math>\mid a \mid = \mid (a - b) + b \mid \leq \mid a - b \mid + \mid b \mid</math> and therefore <math>\mid a \mid \leq \mid a - b \mid + \mid b \mid</math>. Subtracting <math>\mid b \mid</math> from both sides we get that <math>\mid a \mid - \mid b \mid \leq \mid a - b \mid</math>. | ||
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| + | :*Now we write <math>b = b - a + a</math> and therefore applying the triangle inequality we get that <math>\mid b \mid = \mid (b - a) + a \mid \leq \mid b - a \mid + \mid a \mid</math> and therefore <math>\mid b \mid \leq \mid b - a \mid + \mid a \mid</math> and subtracting <math>\mid a \mid</math> from both sides we get that <math>\mid b \mid - \mid a \mid \leq \mid b - a \mid</math> which is equivalent to <math>\mid a \mid - \mid b \mid \geq - \mid b - a \mid</math>. | ||
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| + | :*Therefore <math>\mid \mid a \mid - \mid b \mid \mid \leq \mid a + b \mid</math>. <math>\blacksquare</math> | ||
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| + | <blockquote style="background: white; border: 1px solid black; padding: 0.5em;"> | ||
| + | '''Corollary 2:''' If <math>a</math> and <math>b</math> are real numbers then <math>\mid a - b \mid \leq \mid a \mid + \mid b \mid</math>. | ||
| + | </blockquote> | ||
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| + | :*'''Proof of Corollary 2:''' By the triangle inequality we get that <math>\mid a + b \mid \leq \mid a \mid + \mid b \mid</math> and so then <math>\mid a + (-b) \mid \leq \mid a \mid + \mid -b \mid = \mid a \mid + \mid b \mid</math>. Therefore <math>\mid a - b \mid \leq \mid a \mid + \mid b \mid</math>. <math>\blacksquare</math> | ||
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| + | <blockquote style="background: white; border: 1px solid black; padding: 0.5em;"> | ||
| + | '''Corollary 3:''' If <math>a_1, a_2, ..., a_n \in \mathbb{R}</math> then <math>\mid a_1 + a_2 + ... + a_n \mid \leq \mid a_1 \mid + \mid a_2 \mid + ... + \mid a_n \mid</math>.</blockquote> | ||
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| + | :*'''Proof of Corollary 3:''' We note that <math>\mid a_1 + a_2 + ... + a_n \mid = \mid a_1 + (a_2 + ... + a_n) \mid \leq \mid a_1 \mid + \mid a_2 + ... + a_{n} \mid</math> by the triangle inequality. Applying the triangle inequality multiple times we eventually get that <math>\mid a_1 + a_2 + ... + a_n \mid \leq \mid a_1 \mid + \mid a_2 \mid + ... + \mid a_n \mid</math>. <math>\blacksquare</math> | ||
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| + | ''A more formal proof of Corollary 3 can be carried out by Mathematical Induction.'' | ||
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== Licensing == | == Licensing == | ||
Content obtained and/or adapted from: | Content obtained and/or adapted from: | ||
* [http://mathonline.wikidot.com/the-triangle-inequality The Triangle Inequality, mathonline.wikidot.com] under a CC BY-SA license | * [http://mathonline.wikidot.com/the-triangle-inequality The Triangle Inequality, mathonline.wikidot.com] under a CC BY-SA license | ||
Revision as of 15:02, 27 November 2021
The triangle inequality is a very important geometric and algebraic property that we will use frequently in the future.
Theorem 1 (Triangle Inequality): Let and be real numbers. Then .
- Proof of Theorem: For and as real numbers we have that and . If we add these inequalities together we get that or rather which is equivalent to saying that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mid a + b \mid \leq \mid a \mid + \mid b \mid} . Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \blacksquare}
There are also some other important results similar to the triangle inequality that are important to mention.
Corollary 1: If Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b} are real numbers then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mid \mid a \mid - \mid b \mid \mid \leq \mid a - b \mid} .
- Proof of Corollary 1: We first write Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a = a - b + b} and therefore applying the triangle inequality we get that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mid a \mid = \mid (a - b) + b \mid \leq \mid a - b \mid + \mid b \mid} and therefore Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mid a \mid \leq \mid a - b \mid + \mid b \mid} . Subtracting Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mid b \mid} from both sides we get that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mid a \mid - \mid b \mid \leq \mid a - b \mid} .
- Now we write Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b = b - a + a} and therefore applying the triangle inequality we get that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mid b \mid = \mid (b - a) + a \mid \leq \mid b - a \mid + \mid a \mid} and therefore Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mid b \mid \leq \mid b - a \mid + \mid a \mid} and subtracting Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mid a \mid} from both sides we get that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mid b \mid - \mid a \mid \leq \mid b - a \mid} which is equivalent to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mid a \mid - \mid b \mid \geq - \mid b - a \mid} .
- Therefore Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mid \mid a \mid - \mid b \mid \mid \leq \mid a + b \mid} . Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \blacksquare}
Corollary 2: If Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b} are real numbers then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mid a - b \mid \leq \mid a \mid + \mid b \mid} .
- Proof of Corollary 2: By the triangle inequality we get that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mid a + b \mid \leq \mid a \mid + \mid b \mid} and so then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mid a + (-b) \mid \leq \mid a \mid + \mid -b \mid = \mid a \mid + \mid b \mid} . Therefore Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mid a - b \mid \leq \mid a \mid + \mid b \mid} . Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \blacksquare}
Corollary 3: If Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a_1, a_2, ..., a_n \in \mathbb{R}} then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mid a_1 + a_2 + ... + a_n \mid \leq \mid a_1 \mid + \mid a_2 \mid + ... + \mid a_n \mid} .
- Proof of Corollary 3: We note that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mid a_1 + a_2 + ... + a_n \mid = \mid a_1 + (a_2 + ... + a_n) \mid \leq \mid a_1 \mid + \mid a_2 + ... + a_{n} \mid} by the triangle inequality. Applying the triangle inequality multiple times we eventually get that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mid a_1 + a_2 + ... + a_n \mid \leq \mid a_1 \mid + \mid a_2 \mid + ... + \mid a_n \mid} . Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \blacksquare}
A more formal proof of Corollary 3 can be carried out by Mathematical Induction.
Licensing
Content obtained and/or adapted from:
- The Triangle Inequality, mathonline.wikidot.com under a CC BY-SA license
