Difference between revisions of "Imaginary unit"
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The '''imaginary unit''', <math>i=\sqrt{-1}</math>, is the fundamental component of all [[complex numbers]]. In fact, it is a complex number, itself. | The '''imaginary unit''', <math>i=\sqrt{-1}</math>, is the fundamental component of all [[complex numbers]]. In fact, it is a complex number, itself. | ||
| − | The imaginary unit shows up frequently in contest problems. | + | |
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| + | ==Problems== | ||
| + | The imaginary unit shows up frequently in contest problems. One type of problem involving it is sums, i.e. problems such as "Find the sum of <math>i^1+i^2+\ldots+i^{2006}</math>." | ||
Let's begin by computing powers of <math>i</math>. | Let's begin by computing powers of <math>i</math>. | ||
Revision as of 11:41, 20 July 2006
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The imaginary unit, 
, is the fundamental component of all complex numbers. In fact, it is a complex number, itself.
Problems
The imaginary unit shows up frequently in contest problems. One type of problem involving it is sums, i.e. problems such as "Find the sum of 
."
Let's begin by computing powers of 
.
We can now stop because we have come back to our original term. This means that the sequence i, -1, -i, 1 repeats. Note that this sums to 0. That means that all sequences 
have a sum of zero (k is a natural number). Since 
, the original series sums to the first two terms of the powers of i, which equals -1+i.
