(New page: the sum of a real number and an imaginary number. An imaginary number is a multiple of i, where i is the square root of -1. Complex numbers can be expressed in the form a + bi, where a and...)
 
 
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the sum of a real number and an imaginary number. An imaginary number is a multiple of i, where i is the square root of -1. Complex numbers can be expressed in the form a + bi, where a and b are real numbers. They have the algebraic structure of a field in mathematics. In engineering and physics, complex numbers are used extensively to describe electric circuits and electromagnetic waves.  
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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 <math>\sqrt-1</math>.
  
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for example of complex number.
  
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calculate this.
  
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For a complex number a + bi, a is called the real part and b is called the imaginary part. Thus, the complex number -2 + 3i has the real part -2 and the imaginary part 3. Addition of complex numbers is performed by adding the real and imaginary parts separately. To add 1 + 4i and 2 - 2i, for example, add the real parts 1 and 2 and then the imaginary parts 4 and -2 to obtain the complex number 3 + 2i. The general rule for addition is
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<math>(4+3\sqrt2i)-(2-\sqrt2i)</math>
  
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(a+ bi) + (c+di) = (a+ c) + (b+d)i  
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answer:
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        <math>= 4+3\sqrt2i - 2+\sqrt2i</math>
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        <math>= (4-2) + (3\sqrt2 + \sqrt2)i</math>
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        <math>=2 + 4\sqrt2i</math>
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another example,
  
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Multiplication of complex numbers is based on the premise that i×i=-1 and the assumption that multiplication distributes over addition. This gives the rule
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change to <math>a+bi</math> form.
  
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(a+bi) × (c+di) = ( ac-bd) + (ad+bc)i
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<math>((1+i)/(1-i))^4</math>
  
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for example,
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answer:
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(1 + 4i) × (2 - 2i) = 10 + 6i
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          <math>=\left(\frac{\left(1+i\right)^2}{\left(1-i\right)\times\left(1+i\right)}\right)^4</math>
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If z=a+bi is any complex number, then, by definition, the complex conjugate of z is
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and the absolute value, or modulus, of z is
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          <math>=\left(\frac{1+2i+\left(i\right)^2}{1^2-\left(i\right)^2}\right)^4</math>
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For example, the complex conjugate of 1 + 4i is 1 - 4i, and the modulus of 1 + 4i is
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          <math>=\left(\frac{1+2i-1}{1-\left(-1\right)}\right)^4</math>
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          <math>=\left(\frac{2i}{2}\right)^4</math>
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          <math>=i^4</math>
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          <math>=((i)^2)^2</math>
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          <math>=(-1)^2=1</math>
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          <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 $
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