Complex Conjugate

Pronunciation: /kəmˈplɛks ˈkɒndʒəgɪt/ ?

Complex conjugates are two complex numbers of the form a+bi and a-bi where a and b are real numbers.[1] Complex conjugates are denoted using a bar over the variable: If z=a+bi then z bar = a-bi Complex conjugates have the property that, when added together, the result is a real number.

(a+bi)+(a-bi)
a+a+bi-bi
2a

Complex conjugates also have the property that, when multiplied together, the result is a real number.

(a+bi)*(a-bi)
a*a+a*bi-a*bi-bi*bi
a^2-bi^2

Complex conjugates are usually used when dividing complex numbers.

Properties of Complex Conjugates

conjugate of z+w equals the conjugate of z + the conjugate of w.
conjugate of z-w equals the conjugate of z - the conjugate of w.
conjugate of z*w equals the conjugate of z * the conjugate of w.
conjugate of z/w equals the conjugate of z / the conjugate of w.
The absolute value of z equals the absolute value of the conjugate of z.
The absolute value of z^2 equals z*conjugate of z equal conjugate of z*z
The conjugate of the conjugate of z equals z
e raised to the conjugate of z equals the conjugate of e raised to the z.
Table 1 - Properties of complex conjugates

References

  1. conjugate. merriam-webster.com. Encyclopedia Britannica. (Accessed: 2010-01-04). http://www.merriam-webster.com/dictionary/conjugate.
  2. E. G. Phillips. Functions of a Complex Variable, Eight edition, pp 4-5,14. Oliver and Boyd, 1961. (Accessed: 2010-01-17). http://www.archive.org/stream/functionsofacomp029605mbp#page/n17/mode/1up/search/conjugate.

Cite this article as:
Complex Conjugate. 2010-01-17. All Math Words Encyclopedia. Life is a Story Problem.org. http://www.allmathwords.org/en/c/complexconjugates.html.

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Revision History


2010-01-17: Added "References" (McAdams, David.)
2008-11-25: Changed equations to images (McAdams, David.)
2008-07-08: Changed equations from images to hot_Eqn (McAdams, David.)
2008-03-25: Revised More Information to match current standards (McAdams, David.)
2008-02-13: Added z-bar notation (McAdams, David.)
2007-07-12: Initial version (McAdams, David.)

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