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What term describes the relationship between tenth, hundredth, thousandth, et cetera (1/10, 1/100, 1/1000, ...) and the number ten? (Despite what some may say, I don't accept that "decimal" is the answer.)

More specifically, in this context, I'm looking for the opposite of the term "multiple". Ten, hundred, thousand, et cetera are multiples of ten. Tenth, hundredth and thousandth are what of ten? The closest thing I can think of is "multiplicative inverses of multiples of ten". Is there no better choice of word or words?

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  • $\begingroup$ Powers are still multiples. You're correct in that I can call them "negative powers of ten", but, again, is there no better choice? $\endgroup$
    – Chris Shiherlis
    Apr 23 at 17:46
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    $\begingroup$ Calling powers multiples is as sensible as calling multiples sums, by which we can transitively infer that calling powers sums is equally sensible. Something in all this is unsound and counterintuitive to standard English usage of these terms. I recommend that you not do so. $\endgroup$
    – tchrist
    Apr 23 at 18:37
  • $\begingroup$ But as clarification, do you wish an answer that captures the fractional nature, as well as the mathematical value, of the series? $\endgroup$
    – David
    Apr 23 at 22:45
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    $\begingroup$ As you've asked the question, the answer would probably be "negative powers of 10". However, you might be looking for the word reciprocal. Wikipedia seems to disfavor the term "reciprocal" in favor of "multiplicative inverse", which would seem clearer. $\endgroup$
    – Nat
    Apr 24 at 18:38
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    $\begingroup$ @Chris Maybe "decimal submultiples", similar to the terminology used for units of measurement in physics and engineering e.g. physics.nist.gov/cuu/Units/prefixes.html "decimal multiples and submultiples of SI units". You'd have to clearly define it in the context, to avoid confusion with math "multiples". $\endgroup$
    – dxiv
    Apr 25 at 4:42

5 Answers 5

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They are negative powers of ten.

Consider that
ten to the positive power of two is ten squared (100)
ten to the positive power of three is a thousand (1000)

and so forth.

ten to a negative power is exemplified by
ten to the minus one = 0.1
ten to the minus two = 0.01

and so forth.

Britannica

power of 10, in mathematics, any of the whole-valued (integer) exponents of the number 10. A power of 10 is as many number 10s as indicated by the exponent multiplied together. Thus, shown in long form, a power of 10 is the number 1 followed by n zeros, where n is the exponent and is greater than 0; for example, 106 is written 1,000,000. When n is less than 0, the power of 10 is the number 1 n places after the decimal point; for example, 10−2 is written 0.01.

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  • $\begingroup$ Or Powers of 10 with negative exponents! ixl.com/math/lessons/powers-of-10 $\endgroup$
    – user 66974
    Apr 23 at 18:07
  • $\begingroup$ They may be equivalent to negative powers of ten, but this does not capture the verbal formulation in outdated mathematical language, which I think is what the poster wants. That is to say something to describe 1/10, 1/100 1/1000 etc. in a way that differentiates it from 0.1, 0.01, 0.001 etc. I suppose one could ask him (and why he wants to to use archaic maths representations). $\endgroup$
    – David
    Apr 23 at 18:52
  • $\begingroup$ While I accept that this is correct, a negative power may be too much for laymen or children. $\endgroup$
    – Chris Shiherlis
    Apr 23 at 19:09
  • $\begingroup$ @ChrisShiherlis Agreed, but the whole concept is not for the mathematically uneducated, is it? Similar constraints apply to many aspects of linguistic usage on this site. The site is not aimed at the simplest issues; and in any case not everything can be expressed in terms that the average primary school child can comprehend. We can only answer the questions; it is in the nature of things that we cannot always make answers accessible to all. $\endgroup$
    – Anton
    Apr 23 at 20:36
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You wrote:

More specifically, in this context, I'm looking for the opposite of the term "multiple".

The opposite of a "multiple" is a "factor". For example, 15 is a multiple of 3, and 3 is a factor of 15. The term "divisor" is sometimes used as a synonym for "factor" (although they are not really the same concept).

However, that does not seem to be what you are actually looking for. You then wrote:

Ten, hundred, thousand, et cetera are multiples of ten. Tenth, hundredth and thousandth are what of ten?

It is true that 10, 100, 1000, etc. are multiples of ten, but so are 20, 570, etc. You seem to be focussing on the fact that they are powers of 10. In that case, we can say that 10, 100, 1000, etc. are positive powers of 10:

101 = 10
102 = 100
103 = 1000
etc.

and that 1/10, 1/100, 1/1000, etc. are negative powers of 10 (as others have pointed out):

10−1 = .1 = 1/10
10−2 = .01 = 1/100
10−3 = .001 = 1/1000
etc.

By the way, I noticed elsewhere on this page that you didn't like the term "negative power". Because .001-1/3 = 10, .001 = the negative one-third root of 10. However, we don't usually use negative fractional roots.

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    $\begingroup$ This post contains some interesting commentary, but is it an answer? @Anton seems to have provided a better example, IMO. $\endgroup$
    – EJ Mak
    Apr 24 at 17:59
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They are:

fractions of one in which the denominator is an integral positive power of ten

if one really wants to describe their fractional nature. It doesn’t fit the pattern in the question, but I think that’s a deficiency in the question, and it’s in the same spirit.

PS

Of course a simpler way this is described is as a “decimal fraction with a numerator of one”.

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On a slide rule, we have the standard C and D scales. Slide rule operators are facile with the CI and DI scales. Here the "I" is for "inverse." So etymologically speaking, "inverses" or "inverse powers" might be the best answer!

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More specifically, in this context, I'm looking for the opposite of the term "multiple".

I think the words "divisions" and "subdivisions" may be the terms you seek.

In the Imperial system, quantities are commonly subdivided by powers of two. We typically divide an inch into half-inches, quarter-inches, eighths, sixteenths, etc...

In the Metric system, of course, it is subdivided by powers of ten... divisions into ten equal parts.

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