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 | + | The '''imaginary unit''', <math>i=\sqrt{-1}</math>, is the fundamental component of all [[complex numbers]]. In fact, it is a complex number itself. |
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Let's begin by computing powers of <math>i</math>. | Let's begin by computing powers of <math>i</math>. | ||
| − | <math> | + | <math>i^1=\sqrt{-1}</math> |
| − | <math> | + | <math>i^2=\sqrt{-1}\cdot\sqrt{-1}=-1</math> |
| − | <math> | + | <math>i^3=-1\cdot i=-i</math> |
| − | <math> | + | <math>i^4=-i\cdot i=-i^2=-(-1)=1</math> |
| − | <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 -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. | ||
Revision as of 16:09, 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.
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.
