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Teaching Mathematics: Foundations to Middle Years

Teaching Mathematics

Written by a group of highly respected and experienced teacher educators, the advice is based on current research and theories and, although written for pre-service teachers, there is much that will appeal to practising teachers.

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Zeno’s paradox

The finish line is 2 metres away. The runner must first reach half the distance to the finish line (1m) but when there must cover half of the remaining distance (1/2 m). Having done that the runner must cover half of the new remainder (1/4m), then 1/8m, then 1/16m etc.

Will the runner ever reach the finish line?

The run consists of an infinite number of finite distances, which Zeno concluded must take an infinite time, which is to say that it is never completed!

However, we know in reality that the runner will reach the finish line. How can we show that an infinite process ends?

Let D be the distance covered.

D               = 1 + ½ + ¼ + 1/8 + 1/16 + 1/32 …

½ D           = ½ + ¼ + 1/8 + 1/16 + 1/32 …

D – ½ D     = 1 + (½ + ¼ + 1/8 + 1/16 + 1/32 …) – (½ + ¼ + 1/8 + 1/16 + 1/32 …)

D – ½ D     = 1

½ D           = 1

D               = 2

 

This is only one of Zeno's paradoxes.