Academic journal article Australian Mathematics Teacher

I: The Ultimate Mystery

Academic journal article Australian Mathematics Teacher

I: The Ultimate Mystery

Article excerpt

For any real number x, the quantity e can be expressed as a convergent infinite series:

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(If you are unsure about this, take x = 1 for example. This makes the left hand side [e.sup.1] = e = 2.71828 ... Now on the right, evaluate successively the terms

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and observe how these terms approach more and more closely to the value of e.)

Now it can be shown that this series is still meaningful when x is replaced by a complex number. An interesting example is when we replace x by in:

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The mathematician Leonhard Euler (17071783) discovered that this series has a very simple sum: in fact it turns out that [e.sup.i[pi]] is a real number!

1. Table 1 gives the approximate values of some of the powers of [pi]. Use these values to complete Table 2 where we sum the first few terms of the series for [e.sup.i[pi]].

2. Each of the sums we have found is a complex number. Plot the corresponding points in the Argand diagram, and join the pairs of successive points to form a "spiral". …

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