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==Representations== | ==Representations== | ||
| − | Complex numbers can be represented in | + | Complex numbers can be represented in Cartesian or rectangular form as shown above or in polar form as shown below. |
| − | <center><math>\ | + | <center><math>\Z = A\mathrm{e}^{i\varphi}\,</math></center> |
| + | where A is the magnitude or modulus. | ||
As according to eulers identity the expression above can be expanded to Cartesian form by | As according to eulers identity the expression above can be expanded to Cartesian form by | ||
<center><math>\Z=Acos(\varphi)+isin(\varphi)</math></center> | <center><math>\Z=Acos(\varphi)+isin(\varphi)</math></center> | ||
| − | + | To form the conjugate of a complex number in rectangular form, one just reverses the sign of the imaginary part: | |
| − | + | <center><math>\bar(Z)=a-bi</math></center> | |
| − | + | In polar form it is formed by making the argument negative: | |
| − | + | <center><math>\bar(Z) = A\mathrm{e}^{-i\varphi}\,</math></center> | |
==Properties== | ==Properties== | ||
Latest revision as of 15:49, 4 September 2008
Introduction and Definition
In mathematics the complex number system describes the set of all numbers that have both a real and an imaginary component. A complex number Z can be represented in rectangular form as:
In electrical engineering because the letter i is reserved to denote current, the letter j replaces i such that,
Representations
Complex numbers can be represented in Cartesian or rectangular form as shown above or in polar form as shown below.
where A is the magnitude or modulus. As according to eulers identity the expression above can be expanded to Cartesian form by
To form the conjugate of a complex number in rectangular form, one just reverses the sign of the imaginary part:
In polar form it is formed by making the argument negative:
Properties
- Addition: $ \,(a + bi) + (c + di) = (a + c) + (b + d)i $
- Subtraction: $ \,(a + bi) - (c + di) = (a - c) + (b - d)i $
- Multiplication: $ \,(a + bi) (c + di) = ac + bci + adi + bd i^2 = (ac - bd) + (bc + ad)i $
- Division: $ \,\frac{(a + bi)}{(c + di)} = \left({ac + bd \over c^2 + d^2}\right) + \left( {bc - ad \over c^2 + d^2} \right)i\,, $
where c and d are not both zero.
