To err is defined as to make a mistake or to do something wrong.(verb)
When you add 2+2 and get 5, this is an example of a situation where you err.
See err in Webster's New World College Dictionary
Origin: ME erren < OFr errer < L errare, to wander, go astray, err < IE *eras- > race, Ger irren, to err
See err in American Heritage Dictionary 4
intransitive verb erred, err·ing, errs
Origin: Middle English erren
Origin: , from Old French errer
Origin: , from Latin errāre, to wander; see ers- in Indo-European roots. Usage Note: The pronunciation (ûr) for the word err is traditional, but the pronunciation (ĕr) has gained ground in recent years, perhaps owing to influence from errant and error, and must now be regarded as an acceptable variant. The Usage Panel was split on the matter: 56 percent preferred (ûr), 34 percent preferred (ĕr), and 10 percent accepted both pronunciations.
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