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 :- ... 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/
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