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| − | complex number combined sum of a real number and an imaginary number. the basic expression of complex number is a + bi.(a and b are real numbers) An imaginary number is a multiple of i, it mean i is | + | complex number combined sum of a real number and an imaginary number. the basic expression of complex number is a + bi.(a and b are real numbers) An imaginary number is a multiple of i, it mean i is <math>\sqrt-1</math>. |
| + | |||
| + | for example of complex number. | ||
| + | |||
| + | calculate this. | ||
| + | |||
| + | <math>(4+3\sqrt2i)-(2-\sqrt2i)</math> | ||
| + | |||
| + | answer: | ||
| + | <math>= 4+3\sqrt2i - 2+\sqrt2i</math> | ||
| + | <math>= (4-2) + (3\sqrt2 + \sqrt2)i</math> | ||
| + | <math>=2 + 4\sqrt2i</math> | ||
| + | another example, | ||
| + | |||
| + | change to <math>a+bi</math> form. | ||
| + | |||
| + | <math>((1+i)/(1-i))^4</math> | ||
| + | |||
| + | answer: | ||
| + | <math>=\left(\frac{\left(1+i\right)^2}{\left(1-i\right)\times\left(1+i\right)}\right)^4</math> | ||
| + | |||
| + | <math>=\left(\frac{1+2i+\left(i\right)^2}{1^2-\left(i\right)^2}\right)^4</math> | ||
| + | |||
| + | <math>=\left(\frac{1+2i-1}{1-\left(-1\right)}\right)^4</math> | ||
| + | |||
| + | <math>=\left(\frac{2i}{2}\right)^4</math> | ||
| + | |||
| + | <math>=i^4</math> | ||
| + | |||
| + | <math>=((i)^2)^2</math> | ||
| + | |||
| + | <math>=(-1)^2=1</math> | ||
| + | |||
| + | <math>=1 + 0i</math> | ||
Latest revision as of 10:46, 4 September 2008
complex number combined sum of a real number and an imaginary number. the basic expression of complex number is a + bi.(a and b are real numbers) An imaginary number is a multiple of i, it mean i is $ \sqrt-1 $.
for example of complex number.
calculate this.
$ (4+3\sqrt2i)-(2-\sqrt2i) $
answer:
$ = 4+3\sqrt2i - 2+\sqrt2i $ $ = (4-2) + (3\sqrt2 + \sqrt2)i $ $ =2 + 4\sqrt2i $
another example,
change to $ a+bi $ form.
$ ((1+i)/(1-i))^4 $
answer:
$ =\left(\frac{\left(1+i\right)^2}{\left(1-i\right)\times\left(1+i\right)}\right)^4 $ $ =\left(\frac{1+2i+\left(i\right)^2}{1^2-\left(i\right)^2}\right)^4 $ $ =\left(\frac{1+2i-1}{1-\left(-1\right)}\right)^4 $ $ =\left(\frac{2i}{2}\right)^4 $ $ =i^4 $ $ =((i)^2)^2 $ $ =(-1)^2=1 $ $ =1 + 0i $
