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What is a Sign?

Date: 04/21/2003 at 13:27:59
From: Emma
Subject: Signs: operational and positive/negative

Can you do anything in math without signs?

I don't believe this is possible because all the math I have done 
concerned signs. Addition, subtraction, multipication, division, 
positive, and negative signs. Is a radical a sign?  Is an exponent a 
sign? When doing 4(3) aren't the parentheses signs?


Date: 04/21/2003 at 15:01:40
From: Doctor Peterson
Subject: Re: Signs: operational and positive/negative

Hi, Emma.

This is an interesting question, but we have to start by asking what 
you mean by "sign." My first question is, is even a _number_ a sign?

The Merriam-Webster dictionary defines "sign," in part, as

  1 a : a motion or gesture by which a thought is expressed or a
        command or wish made known
    b : SIGNAL 2a
    c : a fundamental linguistic unit that designates an object or
        relation or has a purely syntactic function
    d : one of a set of gestures used to represent language; also
      : SIGN LANGUAGE
  2 : a mark having a conventional meaning and used in place of
      words or to represent a complex notion
  3 : one of the 12 divisions of the zodiac
  4 a (1) : a character (as a flat or sharp) used in musical notation
      (2) : SEGNO
    b : a character (as ) indicating a mathematical operation; also
      : one of two characters + and - that form part of the symbol
        of a number and characterize it as positive or negative

So in some sense any communication other than words can be called a 
sign (def. 1), which might include everything we write in math, 
including numbers and even words; in a more specific sense (def. 4b), 
only operations would be called signs; more specifically yet, signs 
are only the positive and negative signs on a number. It isn't clear 
whether the more general mathematical definition should include 
parentheses, but I would assume so, since they are closely related to 
the operations. So you have to decide for yourself which definition 
you mean in your question.

But even if you include as signs every symbol we use in math except 
numbers and letters, I think the answer to your question is this: 
Yes, we do math all the time without writing, and therefore not using 
signs; and in fact for most of history there _were_ no signs of 
operations, and everything was written out either in words or in 
simple abbreviations. As an example, see this page, which shows some 
early notation, in which symbols have started to take over from 
words, but many words and abbreviations are still used:

   Earliest Uses of Grouping Symbols - Jeff Miller
   http://jeff560.tripod.com/grouping.html 
       _______________
  B in D quad + B in D is used to represent B(D2 + BD).

This expression from the 1600's, translated from Latin into English, 
would look something like this:

  B by (D sq + B by D)

where you can see that the words "by" (short for "multiplied by") 
and "sq" (short for "squared") are mixed with symbols for addition 
and grouping.

Not too long before this, symbols of any sort were very rare. Here is 
an example of algebra (which today is almost inconceivable without 
writing down equations) as explained by one of its inventors:

Abu Ja'far Muhammad ibn Musa Al-Khwarizmi
http://www-history.mcs.st-and.ac.uk/Mathematicians/Al-Khwarizmi.html 

  Al-Khwarizmi then shows how to solve the six standard types of
  equations. He uses both algebraic methods of solution and
  geometric methods. For example to solve the equation x^2 + 10x = 39
  he writes [11]:- 

  ... a square and 10 roots are equal to 39 units. The question
  therefore in this type of equation is about as follows: what is
  the square which combined with ten of its roots will give a sum
  total of 39? The manner of solving this type of equation is to
  take one-half of the roots just mentioned. Now the roots in the
  problem before us are 10. Therefore take 5, which multiplied by
  itself gives 25, an amount which you add to 39 giving 64. Having
  taken then the square root of this which is 8, subtract from it
  half the roots, 5 leaving 3. The number three therefore
  represents one root of this square, which itself, of course is 9.
  Nine therefore gives the square.

It is very awkward to do this level of math without having any way to 
write out what you are doing; that is why the invention of modern 
algebraic notation was so important in the development of math. But 
with just an abacus and Roman numerals or Egyptian hieroglyphics, 
people did ordinary calculations for centuries without even a symbol 
for '+'. So although important math is very hard to do without 
symbols, the world got along fine without them for a very long time!

But on the other hand, if you merely mean, "Is there anything you do 
in math that can't be considered a "sign" in the sense of being an 
operation that you _could_ write as a symbol?", then I think we have 
to say no. The operations we represent by symbols are the verbs of 
math, and without them, there is nothing to be done.

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

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/ 
Associated Topics:
Elementary Definitions
Elementary Math History/Biography
High School Definitions
High School History/Biography
Middle School Definitions
Middle School History/Biography

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