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Complex numbers are of the form x+iy, where x and y are real numbers. The complex number z=x+iy can be represented by a point in the Cartesian coordinate plane with abscissa x and ordinate y .Then the x axis is called real axis and the y axis is called the imaginary axis. | Complex numbers are of the form x+iy, where x and y are real numbers. The complex number z=x+iy can be represented by a point in the Cartesian coordinate plane with abscissa x and ordinate y .Then the x axis is called real axis and the y axis is called the imaginary axis. | ||
| − | Every complex number x+iy can be expressed in the form r(cos t +i sin t).This is called the polar form of the complex | + | Every complex number x+iy can be expressed in the form r(cos t +i sin t).This is called the polar form of the complex n |
| − | + | Some basic operations: | |
Addition | Addition | ||
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To add complex numbers in rectangular form, add the real components in order to get the real part of the result and add the imaginary components to get the imaginary part. | To add complex numbers in rectangular form, add the real components in order to get the real part of the result and add the imaginary components to get the imaginary part. | ||
| + | (a+ib) + (x+iy)= (a+x) +i(b+y) | ||
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| + | Subtraction | ||
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| + | To subtract complex numbers in rectangular form, subtract the real components in order to get the real part of the result and subtract the imaginary components to get the imaginary part. | ||
| + | (a+ib) - (x+iy)= (a-x) +i(b-y) | ||
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Multiplication | Multiplication | ||
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| + | In rectangular form : | ||
| + | (a+ib) * (x+iy)= (ax-by) +i(bx+ay) | ||
| + | |||
| + | For polar form : | ||
To multiply the numbers in the polar forms in the polar form multiply the magnitudes and add the angles. | To multiply the numbers in the polar forms in the polar form multiply the magnitudes and add the angles. | ||
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| + | Conjugate | ||
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| + | The conjugate (or complex conjugate) of the complex number a + bi is a - bi | ||
| + | Conjugates are important because of the fact that a complex number times its conjugate is real. | ||
Latest revision as of 14:43, 5 September 2008
Complex numbers are of the form x+iy, where x and y are real numbers. The complex number z=x+iy can be represented by a point in the Cartesian coordinate plane with abscissa x and ordinate y .Then the x axis is called real axis and the y axis is called the imaginary axis. Every complex number x+iy can be expressed in the form r(cos t +i sin t).This is called the polar form of the complex n
Some basic operations:
Addition
To add complex numbers in rectangular form, add the real components in order to get the real part of the result and add the imaginary components to get the imaginary part. (a+ib) + (x+iy)= (a+x) +i(b+y)
Subtraction
To subtract complex numbers in rectangular form, subtract the real components in order to get the real part of the result and subtract the imaginary components to get the imaginary part. (a+ib) - (x+iy)= (a-x) +i(b-y)
Multiplication
In rectangular form : (a+ib) * (x+iy)= (ax-by) +i(bx+ay)
For polar form : To multiply the numbers in the polar forms in the polar form multiply the magnitudes and add the angles.
Conjugate
The conjugate (or complex conjugate) of the complex number a + bi is a - bi Conjugates are important because of the fact that a complex number times its conjugate is real.
