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  "body": "\n## Summary\n\nTwo elements $a$ and $b$ commute for a [[binary operation|0M418FE7BB]] $\\ast$ when $a \\ast b = b \\ast a$.\nAn operation is [[commutative|099E72KTF8]] exactly when every pair of elements commutes, but in a noncommutative structure some pairs commute and some do not.\nIn a [[group|G5S1M9]], $a$ and $b$ commute if and only if their [[commutator|0PXW0BN7DW]] is the identity.\n[[Commuting matrices|8G3X8MPG7E]] are an important special case.\n\n## Explanations\n\n### L1 human\n\nTwo things commute when doing them in either order gives the same result.\nTurning a square by $90$ degrees and turning it by $180$ degrees commute: both orders give a turn by $270$ degrees.\nA turn by $90$ degrees and a flip across a diagonal do not commute: the two orders give different pictures.\nEvery element commutes with itself and with the identity element.\n",
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