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שורה 1: |
| <math:math xmlns:math="http://www.w3.org/1998/Math/MathML"><math:semantics><math:mtable><math:mtr><math:mrow>
| | '''Archimedes''' was a [[Greek]] [[mathematician]] who is best known for the myriad mathematical |
| <math:mi>sin</math:mi> | | [[notation]]s that he invented, most of which are still in use today. His earliest work included |
| <math:mrow>
| | devising simple [[inline equation]]s such as <math>\sin x = \cos^2(y+t)</math> and |
| <math:mi>x</math:mi>
| | <math>x^2 + y^2 = -e^{-\theta}</math>. He pioneered the use of greek symbols such as |
| <math:mo math:stretchy="false"></math:mo>
| | <math>\alpha</math> in English writing. While performing complicated calculations such as |
| <math:mi>sin</math:mi>
| | <math>\sum_{i=1}^3 i^2 = 47</math>, he noticed that despite the baseline of the equation |
| </math:mrow>
| | lining up nicely with the surrounding text, the so-called [[displayed equation]] |
| <math:mrow>
| | : <math>\displayed\sum_{i=1}^3 i = 46, \qquad \textrm{unless} 46 \not= 47</math> |
| <math:mi>y</math:mi>
| | was probably better value. A similar effect occurred for integrals such as |
| <math:mo math:stretchy="false">=</math:mo>
| | <math>\int_0^1 \sin^2 x \, dx</math>. He marked this up using the kludgy "\displayed" |
| <math:mn>2</math:mn>
| | command, although he suspected that later and greater thinkers would come up with something |
| </math:mrow>
| | better. When he couldn't make up his mind he would write |
| <math:mi>sin</math:mi>
| | : <math>\displayed F(x) = \begin{cases} \left\uparrow\frac{\partial^2 G}{\partial u \partial v}\right\} |
| <math:mrow>
| | & \textrm{if the sky was \bf blue}, \\ A_0 + \cdots + A_k & \textit{if Troy was on the attack.} |
| <math:mfrac>
| | \end{cases}</math> |
| <math:mrow>
| | He also invented the polynomial rings <math>\mathbf{R}[x]</math>, |
| <math:mi>x</math:mi>
| | <math>\mathcal{C}[y]</math> and <math>\boldsymbol{\mathcal{C}[z]}</math>, and being |
| <math:mo math:stretchy="false"></math:mo>
| | fluent in Chinese he was comfortable writing things like |
| <math:mi>y</math:mi>
| | :<math>\displayed 钱 = \sqrt{不好},</math> |
| </math:mrow>
| | although historians have debated whether his Chinese really was all that good. |
| <math:mn>2</math:mn>
| |
| </math:mfrac>
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| <math:mo math:stretchy="false">∗</math:mo>
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| <math:mi>cos</math:mi> | |
| </math:mrow>
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| <math:mfrac>
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| <math:mrow>
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| <math:mi>x</math:mi>
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| <math:mo math:stretchy="false">−</math:mo>
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| <math:mi>y</math:mi>
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| </math:mrow>
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| <math:mn>2</math:mn>
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| </math:mfrac>
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| </math:mrow>
| |
| </math:mtr>
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| <math:mtr>
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| <math:mrow>
| |
| <math:mi>sin</math:mi>
| |
| <math:mrow>
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| <math:mi>x</math:mi>
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| <math:mo math:stretchy="false">−</math:mo>
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| <math:mi>sin</math:mi>
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| </math:mrow>
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| <math:mrow>
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| <math:mi>y</math:mi>
| |
| <math:mo math:stretchy="false">=</math:mo>
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| <math:mn>2sin</math:mn>
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| </math:mrow>
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| <math:mrow>
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| <math:mfrac>
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| <math:mrow>
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| <math:mi>x</math:mi>
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| <math:mo math:stretchy="false">−</math:mo>
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| <math:mi>y</math:mi>
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| </math:mrow>
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| <math:mn>2</math:mn>
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| </math:mfrac>
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| <math:mo math:stretchy="false">∗</math:mo>
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| <math:mi>cos</math:mi>
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| </math:mrow>
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| <math:mfrac>
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| <math:mrow>
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| <math:mi>x</math:mi>
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| <math:mo math:stretchy="false"></math:mo>
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| <math:mi>y</math:mi>
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| </math:mrow>
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| <math:mn>2</math:mn>
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| </math:mfrac>
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| </math:mrow>
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| </math:mtr>
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| <math:mtr>
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| <math:mrow>
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| <math:mi>cos</math:mi>
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| <math:mrow>
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| <math:mi>x</math:mi>
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| <math:mo math:stretchy="false"></math:mo>
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| <math:mi>cos</math:mi>
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| </math:mrow>
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| <math:mrow>
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| <math:mi>y</math:mi>
| |
| <math:mo math:stretchy="false">=</math:mo>
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| <math:mn>2cos</math:mn>
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| </math:mrow>
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| <math:mrow>
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| <math:mfrac>
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| <math:mrow>
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| <math:mi>x</math:mi>
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| <math:mo math:stretchy="false"></math:mo>
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| <math:mi>y</math:mi>
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| </math:mrow>
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| <math:mn>2</math:mn>
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| </math:mfrac>
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| <math:mo math:stretchy="false">∗</math:mo>
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| <math:mi>cos</math:mi>
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| </math:mrow>
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| <math:mfrac>
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| <math:mrow>
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| <math:mi>x</math:mi>
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| <math:mo math:stretchy="false">−</math:mo>
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| <math:mi>y</math:mi>
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| </math:mrow>
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| <math:mn>2</math:mn>
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| </math:mfrac>
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| </math:mrow>
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| </math:mtr>
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| <math:mtr>
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| <math:mrow>
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| <math:mi>cos</math:mi>
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| <math:mrow>
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| <math:mi>x</math:mi>
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| <math:mo math:stretchy="false">−</math:mo>
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| <math:mi>cos</math:mi>
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| </math:mrow>
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| <math:mrow>
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| <math:mi>y</math:mi>
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| <math:mo math:stretchy="false">=</math:mo>
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| <math:mrow>
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| <math:mo math:stretchy="false">−</math:mo>
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| <math:mn>2sin</math:mn>
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| </math:mrow>
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| </math:mrow>
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| <math:mrow>
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| <math:mfrac>
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| <math:mrow>
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| <math:mi>x</math:mi>
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| <math:mo math:stretchy="false"></math:mo>
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| <math:mi>y</math:mi>
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| </math:mrow>
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| <math:mn>2</math:mn>
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| </math:mfrac>
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| <math:mo math:stretchy="false">∗</math:mo>
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| <math:mi>sin</math:mi>
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| </math:mrow>
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| <math:mfrac>
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| <math:mrow>
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| <math:mi>x</math:mi>
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| <math:mo math:stretchy="false">−</math:mo>
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| <math:mi>y</math:mi>
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| </math:mrow>
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| <math:mn>2</math:mn>
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| </math:mfrac>
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| </math:mrow>
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| </math:mtr>
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| </math:mtable>
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| <math:annotation math:encoding="StarMath 5.0">sin x + sin y = 2 sin{{x + y} over 2} * cos {{x-y} over 2} newline sin x - sin y = 2sin{x-y} over 2 * cos {x+y} over 2 newline cos x + cos y = 2cos {x+y} over 2* cos {x-y} over 2 newline cos x - cos y = -2sin {x+y} over 2 * sin {x-y} over 2</math:annotation>
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| </math:semantics>
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| </math:math>
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