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| − | + | ==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 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: | ||
| − | <center><math>Z=a+bi</math></ | + | <center><math>\Z=a+bi</math></center> |
where <center><math>i^2 = -1</math></center> | where <center><math>i^2 = -1</math></center> | ||
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In electrical engineering because the letter i is reserved to denote current, the letter j replaces i such that, | In electrical engineering because the letter i is reserved to denote current, the letter j replaces i such that, | ||
<center><math>j^2=-1</math></center>. | <center><math>j^2=-1</math></center>. | ||
| + | |||
| + | |||
| + | ==Representations== | ||
| + | Complex numbers can be represented in cartesian or rectangular form as shown above or in polar form as shown below. | ||
| + | <center><math>\z = A\,\mathrm{e}^{i \varphi}\,</math></center> | ||
| + | As according to eulers identity the expression above can be expanded to Cartesian form by | ||
| + | <center><math>\Z=Acos(\varphi)+isin(\varphi)</math></center> | ||
| + | |||
| + | The conjugate of a complex number is defined as: | ||
| + | |||
| + | |||
| + | The can be represented graphically in the complex plane: | ||
| + | ==Properties== | ||
| + | |||
| + | :* Addition: <math>\,(a + bi) + (c + di) = (a + c) + (b + d)i</math> | ||
| + | :* Subtraction: <math>\,(a + bi) - (c + di) = (a - c) + (b - d)i</math> | ||
| + | :* Multiplication: <math>\,(a + bi) (c + di) = ac + bci + adi + bd i^2 = (ac - bd) + (bc + ad)i</math> | ||
| + | :* Division: <math>\,\frac{(a + bi)}{(c + di)} = \left({ac + bd \over c^2 + d^2}\right) + \left( {bc - ad \over c^2 + d^2} \right)i\,,</math> | ||
| + | where ''c'' and ''d'' are not both zero. | ||
Revision as of 15:20, 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.
As according to eulers identity the expression above can be expanded to Cartesian form by
The conjugate of a complex number is defined as:
The can be represented graphically in the complex plane:
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.
