'algebraically' originates from 'algebra', specifically the word 'al-jabr', where 'al-' meant 'the' and 'jabr' meant 'reunion of broken parts'.

'al-jabr' transformed into the Medieval Latin word 'algebra', and eventually became the modern English word 'algebra'.

Initially, it meant 'reunion of broken parts', but over time it evolved into its current meaning of 'a branch of mathematics dealing with symbols and the rules for manipulating those symbols'.