History of Exponent Notation

Date: 11/03/2003 at 11:23:22
From: Lauren
Subject: history of the exponent

Why is the symbol for exponents a smaller number above the base?  How
did it become this?  What is the history of it?

Date: 11/03/2003 at 12:36:21
From: Doctor Peterson
Subject: Re: history of the exponent

Hi, Lauren.

There's a good source for this kind of information listed in our FAQ:

  Earliest uses of mathematical symbols
    http://jeff560.tripod.com/mathsym.html 

Look under "symbols of operation". Among other things, it says:

  Nicole Oresme (c. 1323-1382) used numbers to indicate powering in
  the fourteenth century, although he did not use raised numbers.
 
  Nicolas Chuquet (1445?-1500?) used raised numbers in Le Triparty
  en la Science des Nombres in 1484. However, in Chuquet's notation,
    3                   3
  12  actually meant 12x  (Cajori vol. 1, page 102). 

  In 1634, Pierre Hérigone (or Herigonus) (1580-1643) wrote a, a2,
  a3, etc., in Cursus Mathematicus, which was published in several
  volumes from 1634 to 1637; the numerals were not raised, however
  (Cajori vol. 1, page 202, and Ball). 

  In 1636 James Hume used Roman numerals as exponents in L'Algèbre
  de Viète d'vne methode novelle, claire, et Facile. Cajori writes
  (vol. 1, pages 345-346): 

      In 1636 James Hume brought out an edition of the algebra of
      Vieta, in which he introduced a superior notation, writing
      down the base and elevating the exponent to a position above
      the regular line and a little to the right. The exponent was
                                                   iii      3
      expressed in Roman numerals. Thus, he wrote A    for A .
      Except for the use of Roman numerals, one has here our
      modern notation. Thus, this Scotsman, residing in Paris, had
      almost hit upon the exponential symbolism which has become
      universal through the writings of Descartes. 

  In 1637 exponents in the modern notation (although with positive
  integers only) were used by Rene Descartes (1596-1650) in
  Geometrie. Descartes tended not to use 2 as an exponent, however,
                                  2
  usually writing aa rather than a , perhaps because aa occupies no
                   2
  less space than a .

As you see, lots of other ideas were floating around while algebraic 
notation was first being developed, and this one survived because it 
had some very strong advantages.  One that I like is the asymmetry of
the notation; if a notation like "a^b" were used, as we do in our
e-mail and in some programming languages, then it would make people
expect the operation to be commutative, like a+b = b+a, whereas it is
NOT true that a^b = b^a. So putting the exponent off the line makes it
clear that it is to be treated differently from other numbers in an
expression.

A disadvantage of the notation, of course, is that it is hard to type,
and does not fit in an ordinary line of type in a book. (That's why
exponents are commonly printed in a smaller size, though: to attempt
to fit them in!) Fortunately, today word-processors can handle
complicated notations fairly easily.

If you have any further questions, feel free to write back.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/