Complex Conjugate

Pronunciation: /kəmˈplɛks ˈkɒndʒəgɪt/ Explain

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 complex number. merriam-webster.com. Encyclopedia Britannica. Merriam-Webster. Last Accessed 8/6/2018. https://www.merriam-webster.com/dictionary/conjugate%20complex%20number.
  2. Phillips, E. G.. Functions of a Complex Variable. 8th edition. pp 4-5,14. www.archive.org. Oliver and Boyd. 1961. Last Accessed 8/6/2018. http://www.archive.org/stream/functionsofacomp029605mbp#page/n17/mode/1up/search/conjugate. Buy the book

Cite this article as:

McAdams, David E. Complex Conjugate. 6/25/2018. All Math Words Encyclopedia. Life is a Story Problem LLC. http://www.allmathwords.org/en/c/complexconjugates.html.

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Revision History

6/25/2018: Removed broken links, updated license, implemented new markup, updated GeoGebra apps. (McAdams, David E.)
1/17/2010: Added "References". (McAdams, David E.)
11/25/2008: Changed equations to images. (McAdams, David E.)
7/8/2008: Changed equations from images to hot_Eqn. (McAdams, David E.)
3/25/2008: Revised More Information to match current standards. (McAdams, David E.)
2/13/2008: Added z-bar notation. (McAdams, David E.)
7/12/2007: Initial version. (McAdams, David E.)

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