Difference between revisions of "Imaginary unit"
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| + | ==Problems== | ||
| + | One common problem involving the imaginary unit is along the lines of: | ||
| + | "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>. | ||
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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 <math>i^1+i^2+\ldots+i^{4k}</math> have a sum of zero (k is a natural number). Since <math>2006=4\cdot501+2</math>, the original series sums to the first two terms of the powers of i, which equals -1+i. | 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 <math>i^1+i^2+\ldots+i^{4k}</math> have a sum of zero (k is a natural number). Since <math>2006=4\cdot501+2</math>, the original series sums to the first two terms of the powers of i, which equals -1+i. | ||
| + | |||
| + | == See also == | ||
| + | * [[Algebra]] | ||
| + | * [[Complex numbers]] | ||
| + | * [[Geometry]] | ||
Revision as of 09:55, 29 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
One common problem involving the imaginary unit is along the lines of:
"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.
