Numbers and Numerals
Date: 11/20/98 at 11:14:53 From: Will Crall Subject: Numbers and numerals Dear Dr. Math, What is the difference between numbers and numerals? Will (6th Grade Student)
Date: 11/20/98 at 16:55:35 From: Doctor Peterson Subject: Re: Numbers and numerals Hi, Will. The difference is sort of like the difference between a person and his name. You are a person, and there is just one of you. But different people probably call you by different names. Your name isn't you, but it represents you. In the same way, a number is a thing that we talk about in math, such as "three," which is hard to define exactly. I might say it's the abstract property or "threeness" that is shared by any set of three things. A numeral, on the other hand, is any name or symbol for that number, such as "3" or "III" or "11 (binary)" or whatever animal face is used to represent a three in Mayan carvings. You can't see a three, but you can see things that can be described by the number 3, and you can see the numeral "3" that is used to represent it. We have to be careful sometimes not to confuse the numeral with the number. For example, a numeral "12" may have two digits, but the number 12 could be represented by numerals with other numbers of digits, or by numerals where the concept of digit is meaningless. We often get questions like "are there any numbers after a trillion?" when people really mean, "are there any names for numbers bigger than a trillion?" Similarly, the fact that no one can ever write out the numeral for the number we call "pi" completely doesn't mean the number doesn't exist! On the other hand, we do often use the word "number" to mean "numeral" when it isn't important to make that distinction. I don't get upset if someone says he's writing a number, when it's really a numeral he wrote. - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/
Date: 01/29/2002 at 04:48:01 From: Am Subject: Numerals and numbers What is the exact difference between numerals and numbers?
Date: 01/29/2002 at 09:07:49 From: Doctor Peterson Subject: Re: Numerals and numbers Hi! I did a little extra research to check up on my understanding here, and was a little surprised! A number is an abstract concept; a numeral is a way to express a number, usually in writing. For example, the number 5 can be thought of as the concept of "fiveness" which all sets of five objects have in common; it can be expressed using numerals such as 5, V, |||||, five, 101 (base 2), and so on. Since dictionary writers are often able to express meaning better than I can, I looked up "number" and "numeral" at m-w.com (Merriam- Webster), and found number: 1 a (1) : a sum of units : TOTAL (2) : COMPLEMENT 1b (3) : an indefinite usually large total <a number of members were absent> <the number of elderly is rising> (4) plural : a numerous group : MANY (5) : a numerical preponderance b (1) : the characteristic of an individual by which it is treated as a unit or of a collection by which it is treated in terms of units (2) : an ascertainable total <bugs beyond number> c (1) : a unit belonging to an abstract mathematical system and subject to specified laws of succession, addition, and multiplication; especially : NATURAL NUMBER (2) : an element (as Y') of any of many mathematical systems obtained by extension of or analogy with the natural number system ... 4 a : a word, symbol, letter, or combination of symbols representing a number b : a numeral or combination of numerals or other symbols used to identify or designate <dialed the wrong number> numeral: a conventional symbol that represents a number I was not surprised at how hard it is to define number, or by the fact that one definition tells us that "number" is often used to mean "numeral"; I was surprised that "numeral" is defined so briefly and without examples to clarify, for example, whether a numeral is a single digit ("1") or the whole "number" ("123"). In my understanding, it is the latter. Here are American Heritage's definitions (from www.bartleby.com): number: 1. Mathematics a. A member of the set of positive integers; one of a series of symbols of unique meaning in a fixed order that can be derived by counting. b. A member of any of the further sets of mathematical objects, such as negative integers and real numbers. 2. numbers Arithmetic. 3a. A symbol or word used to represent a number. b. A numeral or a series of numerals used for reference or identification: his telephone number; the apartment number. numeral: 1. A symbol or mark used to represent a number. I like this definition of number better; but "numeral" still gets an inadequate definition. My old paper copy is a little better: numeral: 1. A symbol, such as a letter, figure, or word used alone or in a group to denote a number. See Arabic numeral, Roman numeral. Arabic numerals. The numerical symbols 1, 2, 3, 4, 5, 6, 7, 8, 9, and 0. Roman numeral. One of the letters employed in the ancient Roman system of numeration... So they are clear that a numeral is a single symbol (or word, such as "one"), not the whole "number." But I think it is common in teaching to use the word "numeral" to refer to any representation of a number (as I suggested above) rather than just to the individual symbols of which it is made, particularly since the symbols can be combined in different ways (as in base 10 and base 2, which use some of the same set of symbols). In the cuneiform system, all numbers are made up of "wedges" | and > ; are those the numerals, or combinations like >|| that were made in specific patterns? I think the "single symbol" definition is hard to apply in general, and for our purposes is not sufficient. I don't always trust lexicographers to catch the nuances of mathematical usages, so you can take the dictionary references with a grain of salt. One more dictionary reference partially justifies my answer; here's what the Academic Press Dictionary of Science and Technology (listed in our FAQ) says for "numeral": a written representation of a fixed numerical quantity; in particular, the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. So the term is used _most specifically_ of digits, but _more generally_ of the whole representation of any number. I like that view. Here are two previous answers from our archives, which are clearer than what I've said, being untainted by dictionary nonsense: Numbers and Numerals http://mathforum.org/dr.math/problems/moore9.6.97.html The History of Numbers and Numerals http://mathforum.org/dr.math/problems/jacobs12.8.98.html We all agree that a numeral is a representation of a number. - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/
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