What Is Algebra?

But when we come to the end of our arithmetic we do not content ourselves with guesses; we proceed to algebra -- that is to say, to dealing logically with the fact of our own ignorance. ...

Instead of guessing whether we are to call it nine, or seven, or a hundred and twenty, or a thousand and fifty, let us agree to call it x, and let us always remember that x stands for the Unknown. ... This method of solving problems by honest confession of one's ignorance is called Algebra.

The Arabs had some cousins who lived not far off from Arabia and who called themselves Hebrews. A taste for Algebra seems to have run in the family. Three Algebras grew up among the Hebrews. I should think they are the grandest and most useful that ever were heard of or dreamed of on earth.

Mary Everest Boole
Philosophy And Fun Of Algebra,
London: C. W. Daniel, LTD, 1909

Algebra is a branch of mathematics that deals with properties of operations and the structures these operations are defined on. Elementary Algebra that follows the study of arithmetic is mostly preoccupied with operations on sets of whole and rational numbers and solving first and second order equations. What puts elementary algebra a step ahead of elementary arithmetic is a systematic use of letters to denote generic numbers.

Mastering of elementary algebra which is often hailed as a necessary preparatory step for the study of Calculus, is as often an insurmountable block in many a career. However, the symbolism that is first introduced in elementary algebra permeates all of mathematics. This symbolism is the alphabet of the mathematical language.

The word "algebra" is a shortened misspelled transliteration of an Arabic title al-jebr w'al-muqabalah (circa 825) by the Persian mathematician known as al-Khowarismi [Words, p. 21]. The al-jebr part means "reunion of broken parts", the second part al-muqabalah translates as "to place in front of, to balance, to oppose, to set equal." Together they describe symbol manipulations common in algebra: combining like terms, moving a term to the other side of an equation, etc.

In its English usage in the 14th century, algeber meant "bone-setting," close to its original meaning. By the 16th century, the form algebra appeared in its mathematical meaning. Robert Recorde (c. 1510-1558), the inventor of the symbol "=" of equality, was the first to use the term in this sense. He, however, still spelled it as algeber. The misspellers proved to be more numerous, and the current spelling algebra took roots.

Thus the original meaning of algebra refers to what we today call elementary algebra which is mostly occupied with solving simple equations. More generally, the term algebra encompasses nowadays many other fields of mathematics: geometric algebra, abstract algebra, boolean algebra, s-algebra, to name a few.


  1. S. Schwartzman, The Words of Mathematics, MAA, 1994

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