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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> | ||
| Line 10: | Line 10: | ||
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> | ||
| + | where A is the magnitude or modulus. | ||
| + | As according to eulers identity the expression above can be expanded to Cartesian form by | ||
| + | <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== | ||
| + | |||
| + | :* 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. | ||
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.
