Triangle Inequality

The triangle inequality is a very important geometric and algebraic property that we will use frequently in the future.

Theorem 1 (Triangle Inequality): Let 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 $b$ be real numbers. Then $\mid a+b\mid \leq \mid a\mid +\mid b\mid$.

• Proof of Theorem: For $a$ and $b$ as real numbers we have that $-\mid a\mid \leq a\leq \mid a\mid$ and $-\mid b\mid \leq b\leq \mid b\mid$. If we add these inequalities together we get that $-\mid a\mid -\mid b\mid \leq a+b\leq \mid a\mid +\mid b\mid$ or rather $-\left(\mid a\mid +\mid b\mid \right)\leq a+b\leq \left(\mid a\mid +\mid b\mid \right)$ which is equivalent to saying that $\mid a+b\mid \leq \mid a\mid +\mid b\mid$. $\blacksquare$

There are also some other important results similar to the triangle inequality that are important to mention.

Corollary 1: If $a$ and $b$ are real numbers then $\mid \mid a\mid -\mid b\mid \mid \leq \mid a-b\mid$.

• Proof of Corollary 1: We first write $a=a-b+b$ and therefore applying the triangle inequality we get that $\mid a\mid =\mid (a-b)+b\mid \leq \mid a-b\mid +\mid b\mid$ and therefore $\mid a\mid \leq \mid a-b\mid +\mid b\mid$. Subtracting $\mid b\mid$ from both sides we get that $\mid a\mid -\mid b\mid \leq \mid a-b\mid$.

• Now we write $b=b-a+a$ and therefore applying the triangle inequality we get that $\mid b\mid =\mid (b-a)+a\mid \leq \mid b-a\mid +\mid a\mid$ and therefore $\mid b\mid \leq \mid b-a\mid +\mid a\mid$ 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