Difference between revisions of "Imaginary unit"

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<math>i^1=\sqrt{-1}</math>
 
<math>i^1=\sqrt{-1}</math>
  
<math>i^2=\sqrt{-1}*\sqrt{-1}=-1</math>
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<math>i^2=\sqrt{-1}\cdot\sqrt{-1}=-1</math>
  
<math>i^3=-1*i=-i</math>
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<math>i^3=-1\cdoti=-i</math>
  
<math>i^4=-i*i=-i^2=-(-1)=1</math>
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<math>i^4=-i\cdoti=-i^2=-(-1)=1</math>
  
<math>i^5=1*i=i</math>
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<math>i^5=1\cdoti=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 2006=4*501+2, 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 16:08, 22 June 2006

The imaginary unit, $i=\sqrt{-1}$, 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. The most common type of problem involving it are sums, i.e. problems such as "Find the sum of $i^1+i^2+\ldots+i^{2006}$." Let's begin by computing powers of $i$.

$i^1=\sqrt{-1}$

$i^2=\sqrt{-1}\cdot\sqrt{-1}=-1$

$i^3=-1\cdoti=-i$ (Error compiling LaTeX. Unknown error_msg)

$i^4=-i\cdoti=-i^2=-(-1)=1$ (Error compiling LaTeX. Unknown error_msg)

$i^5=1\cdoti=i$ (Error compiling LaTeX. Unknown error_msg)

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 $i^1+i^2+\ldots+i^{4k}$ have a sum of zero (k is a natural number). Since $2006=4\cdot501+2$, the original series sums to the first two terms of the powers of i which equals -1+i.