TOPICS # Absolute Square Min Max Min Max Re Im The absolute square of a complex number , also known as the squared norm, is defined as (1)

where denotes the complex conjugate of and is the complex modulus.

If the complex number is written , with and real, then the absolute square can be written (2)

If is a real number, then (1) simplifies to (3)

An absolute square can be computed in terms of and using the Wolfram Language command ComplexExpand[Abs[z]^2, TargetFunctions -> Conjugate ].

An important identity involving the absolute square is given by (4) (5) (6)

If , then (6) becomes (7) (8)

If , and , then (9)

Finally, (10) (11) (12)

Complex Argument, Complex Modulus, Complex Number, Imaginary Part, Real Part, Sign

## Explore with Wolfram|Alpha More things to try:

## Cite this as:

Weisstein, Eric W. "Absolute Square." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/AbsoluteSquare.html
{"ft_lang_label":"__label__en","ft_lang_prob":0.8310421,"math_prob":0.9680514,"size":615,"snap":"2023-40-2023-50","text_gpt3_token_len":171,"char_repetition_ratio":0.15711948,"word_repetition_ratio":0.0,"special_character_ratio":0.31544715,"punctuation_ratio":0.104347825,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9964441,"pos_list":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120],"im_url_duplicate_count":[null,null,null,null,null,null,null,1,null,null,null,null,null,null,null,null,null,null,null,1,null,null,null,null,null,null,null,null,null,null,null,null,null,1,null,1,null,1,null,1,null,1,null,1,null,1,null,1,null,1,null,1,null,1,null,1,null,1,null,1,null,1,null,1,null,1,null,1,null,1,null,1,null,1,null,1,null,1,null,1,null,1,null,1,null,1,null,1,null,1,null,1,null,1,null,1,null,1,null,1,null,1,null,1,null,1,null,1,null,1,null,1,null,1,null,1,null,1,null,null,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2023-12-04T07:37:38Z\",\"WARC-Record-ID\":\"<urn:uuid:165db068-b0d9-4374-b2f1-309c8c038ba5>\",\"Content-Length\":\"84005\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:9c8210f4-70ac-4f5f-9cd7-b4693c4cb9c5>\",\"WARC-Concurrent-To\":\"<urn:uuid:c08baf09-89db-4940-bb7b-43b3c293cb6a>\",\"WARC-IP-Address\":\"140.177.8.25\",\"WARC-Target-URI\":\"https://mathworld.wolfram.com/AbsoluteSquare.html\",\"WARC-Payload-Digest\":\"sha1:QXYK2H77S7YA5AT466HTYV7P6LA7IEUB\",\"WARC-Block-Digest\":\"sha1:F3XPWRT2KILCYJISNNVO2QVXM3BBNTAS\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2023/CC-MAIN-2023-50/CC-MAIN-2023-50_segments_1700679100525.55_warc_CC-MAIN-20231204052342-20231204082342-00021.warc.gz\"}"}