history of mathematics - National STEM Centre
history of mathematics - National STEM Centre
history of mathematics - National STEM Centre
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Consider the following different cases.<br />
a / intersects AB.<br />
b / intersects an extension <strong>of</strong> AB.<br />
c [Harder] / is parallel to AB.<br />
12 Summaries and exercises<br />
2 Given a line segment AB and a point P on AB, use Euclid's rules to construct on<br />
AB a point Q such that AQ = BP.<br />
3 Explain how to construct a square equal in area to a given triangle.<br />
4 More Greek <strong>mathematics</strong><br />
Chapter summary<br />
• the major contributions and the background to the lives <strong>of</strong><br />
• Euclid (Chapter 3 and Activities 4.1 to 4.3)<br />
• Archimedes (Activities 4.6 and 4.7)<br />
• Apollonius (Activities 4.8 and 4.9)<br />
• Hypatia (text at the end <strong>of</strong> Chapter 4)<br />
• how to interpret Euclidean theorems <strong>of</strong> geometric algebra in modern algebraic<br />
notation (Activity 4.1)<br />
• how to use the Euclidean algorithm to find the highest common factor (Activity<br />
4.2)<br />
• how the ancient Greeks proved the infinity <strong>of</strong> primes (Activity 4.3)<br />
• the meaning <strong>of</strong> 'incommensurability' and the significance <strong>of</strong> its discovery to the<br />
ancient Greeks (Activity 4.4)<br />
• an example <strong>of</strong> how the Greeks continued to take an interest in numerical<br />
methods: Heron's algorithm for finding square roots (Activity 4.5)<br />
• how to use and interpret Archimedes's method for estimating n (Activity 4.6)<br />
• how Archimedes found the area under a parabolic arc (Activity 4.7)<br />
• how Apollonius defined and named the conic sections (Activities 4.8 and 4.9).<br />
Practice exercises<br />
1 Here is a proposition from Euclid.<br />
If there are two straight lines, and one <strong>of</strong> them can be cut into any number<br />
<strong>of</strong> segments, the rectangle contained by the two straight lines is equal to<br />
the rectangles contained by the uncut straight line and each <strong>of</strong> the<br />
segments.<br />
a Construct a diagram to represent the proposition.<br />
b Make an algebraic statement equivalent to the proposition.<br />
2 Use the Euclidean algorithm to find the greatest common divisor <strong>of</strong> 2689 and<br />
4001.<br />
3 Prove proposition 14 from Book VIII <strong>of</strong> Euclid's Elements that, if a 2 measures<br />
b 2 , then a measures b, and, conversely, that if a measures b, then a 2 measures b 2 .<br />
4 Write brief notes on Archimedes's contribution to <strong>mathematics</strong>.<br />
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