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| − | In any polynomial involving i, i.e. <math>c1*i^n+c2*i^{n-1}+...+c</math> we can express the even powers of i as either 1 or -1. Thus, any polynomial in i can be expressed as <math>c1+c2*i</math> where c1 and c2 are any real number constants. This also establishes the set {1, i} as a basis for C as a vector space over all real numbers. | + | In any polynomial involving i, i.e. |
| + | |||
| + | <math>c1*i^n+c2*i^{n-1}+...+c</math> | ||
| + | |||
| + | we can express the even powers of i as either 1 or -1. Thus, any polynomial in i can be expressed as | ||
| + | |||
| + | <math>c1+c2*i</math> | ||
| + | |||
| + | where c1 and c2 are any real number constants. This also establishes the set {1, i} as a basis for C as a vector space over all real numbers. | ||
More importantly for our class, euler's formula: | More importantly for our class, euler's formula: | ||
Latest revision as of 19:56, 4 September 2008
In any polynomial involving i, i.e.
$ c1*i^n+c2*i^{n-1}+...+c $
we can express the even powers of i as either 1 or -1. Thus, any polynomial in i can be expressed as
$ c1+c2*i $
where c1 and c2 are any real number constants. This also establishes the set {1, i} as a basis for C as a vector space over all real numbers.
More importantly for our class, euler's formula:
$ e^{i\pi}=\cos(\theta)+i*\sin(\theta) $
also
$ e^{-i\pi}=\cos(\theta)-i*\sin(\theta) $
