history of mathematics - National STEM Centre
history of mathematics - National STEM Centre
history of mathematics - National STEM Centre
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156<br />
12 Summaries and exercises<br />
using the circle (x-8) 2 +(y + j) =8 2 +(f) and the parabola x2 =y. Explain why<br />
the intersection <strong>of</strong> this circle and the parabola provides the solution.<br />
3 Use the method <strong>of</strong> Descartes to solve the equation y 1 = -ay + b 2<br />
9 The beginnings <strong>of</strong> calculus<br />
Chapter summary<br />
• how Euclid defined a tangent to a circle (Activity 9.1)<br />
• how Archimedes constructed a tangent to a spiral (Activity 9.2)<br />
• how Fermat found maxima and minima, and how he used this in his general<br />
method to construct tangents to a curve (Activities 9.3 and 9.4)<br />
• how Descartes's definition <strong>of</strong> a tangent, and how his method for constructing<br />
tangents differed from Fermat's definition and method (Activity 9.5)<br />
• the development <strong>of</strong> integral notation (Activities 9.6 and 9.7)<br />
• Leibniz: gradient and area (Activity 9.8)<br />
• Newton and differentiation (Activity 9.9).<br />
Practice exercises<br />
1 Write brief notes on Leibniz's contribution to <strong>mathematics</strong>.<br />
2 What three long-standing classes <strong>of</strong> problems were solved by the introduction <strong>of</strong><br />
the calculus? Explain why calculus was important in the solution <strong>of</strong> these problems.<br />
10 'Two minuses make a plus'<br />
Chapter summary<br />
how to visualise addition and multiplication on a number line (Activity 10.1)<br />
subtracted numbers (Activity 10.2)<br />
an example <strong>of</strong> subtracted numbers arising in ancient China (Activity 10.3)<br />
Chinese rules for combining negative numbers (Activity 10.4)<br />
a geometric approach to multiplying negative numbers using Indian, Arab and<br />
Greek sources (Activities 10.5 to 10.7)<br />
thinking about why there is a gap in development (Activity 10.8)<br />
some particular difficulties with negative numbers (Activities 10.9 and 10.10)<br />
a debate based on source material (Activity 10.11).<br />
Practice exercises<br />
The practice exercises for this chapter are combined with those in Chapter 11.<br />
77 Towards a rigorous approach<br />
Chapter summary<br />
• thinking about the validity <strong>of</strong> pro<strong>of</strong> by extrapolation (Activity 11.1 and 11.4)<br />
• using analogy to justify the rules for combining negative numbers (Activity 11.2)