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. It has a [[magnitude]] of 1, and can be written as <math>1 \mathrm{cis} \left(\frac{\pi}{2}\right)</math>. |
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==Problems== | ==Problems== | ||
| − | + | === Introductory === | |
| − | + | #Find the sum of <math>i^1+i^2+\ldots+i^{2006}</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. | + | ==Solutions == |
| + | === Introductory === | ||
| + | #Let's begin by computing powers of <math>i</math>. | ||
| + | #:<math>i^1=\sqrt{-1}</math> | ||
| + | #:<math>i^2=\sqrt{-1}\cdot\sqrt{-1}=-1</math> | ||
| + | #:<math>i^3=-1\cdot i=-i</math> | ||
| + | #:<math>i^4=-i\cdot i=-i^2=-(-1)=1</math> | ||
| + | #:<math>i^5=1\cdot i=i</math> | ||
| + | #: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 <math>-1+i</math>. | ||
== See also == | == See also == | ||
Revision as of 17:11, 23 September 2007
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The imaginary unit, 
, is the fundamental component of all complex numbers. In fact, it is a complex number itself. It has a magnitude of 1, and can be written as 
.
Problems
Introductory
- Find the sum of
Solutions
Introductory
- 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 
.
.
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 
.
