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I tried Google, and I couldn't find the answer, so I have to have ask this question here. Is this correct?

one millionth of a billionth of a trillionth of a second.

I can write one 'undecillionth' of a second, but I don't feel that many people would imagine the weight of the said term like they can with millionth to trillionth?

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  • What you have reads okay to me, assuming the information is correct.
    – ralph.m
    Commented Dec 29, 2018 at 7:42
  • I guess only Carl Sagan said things like "a billionth of a trilliionth" in a vain attempt to seem more intelligible.
    – GEdgar
    Commented Dec 29, 2018 at 15:39
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    When speaking to people who might not be familiar with exponentiation, it would be far more considerate of your audience to say something like One part in ten followed by 35 zeroes](google.com/…) Commented Dec 29, 2018 at 19:01
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    ...where of course you could save yourself a bit of maths with One part in one followed by 36 zeroes, but there aren't so many written instances of that version, simply because it's less "accessible" to the non-mathematically-minded. Commented Dec 29, 2018 at 19:03
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    @GEdgar Oh, was an attempt to seem more intelligible the reason for such turns of phrase? I thought that the reason was to create a vague impression that the number is awesomely small (or large, as the case may be) rather than just stating the number in a matter-of-fact way.
    – Rosie F
    Commented Dec 30, 2018 at 7:57

2 Answers 2

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Is this correct?

one millionth of a billionth of a trillionth of a second.

I can write one 'undecillionth' of a second, […]

That is not correct.

In the "short scale" (now used almost universally in the English-speaking world), "undecillionth" has the value you want (10−36), but "one millionth of a billionth of a trillionth" does not: one millionth is 10−6, one billionth is 10−9, and one trillionth is 10−12, so one millionth of one billionth of one trillionth is 10−6−9−12 = 10−27. So you should instead write "a trillionth of a trillionth of a trillionth".

In the "long scale" (formerly used in the UK, and still used in many non-English-speaking countries), "one millionth of a billionth of a trillionth" has the correct value (10−6−12−18 = 10−36), but "undecillionth" does not (it's 10−66); rather, you would use "sextillionth". Furthermore, even though "one millionth of a billionth of a trillionth" is correct, it's unnecessarily long, because in the "long scale", a millionth of a billionth is a trillionth; so it would be more sensible to say "a trillionth of a trillionth" (10−18−18 = 10−36). (That said, "a millionth of a billionth of a trillionth" may do a better job evoking the connotation of "really super super tiny", if you want more than just a neutral/factual/objective description.)


Zebrafish's answer correctly describes how you would read 10−36 aloud; but I feel that this is a misleading answer to your question, because we would not normally write out "ten to the power of minus thirty-six" in prose. Rather, we would use either the mathematical notation, 10−36, or a prose phrasing along the lines of your suggestion (e.g. "a trillionth of a trillionth of a trillionth").

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  • Zebrafish's answer uses "negative 36", not "minus 36", so it does not "correctly describe how you would read it aloud".
    – Andrew Leach
    Commented Dec 30, 2018 at 8:41
  • @AndrewLeach: I think "negative thirty-six" is perfectly correct, albeit more formal and less common. (Zebrafish's answer mentions both.)
    – ruakh
    Commented Dec 30, 2018 at 17:45
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Note, "negative" as used in this answer is interchangeable with "minus". Thanks to a number of commenters pointing this out, NGram Viewer and Google search both show "minus" as being more frequent than "negative". "Negative" may be more common in formal use.

Exponentiation such as you've shown is usually said, in your example either:

Ten raised to the power of negative 36

or

Ten to the power of negative 36

or

Negative 36th power of 10 (less common)

or

Ten to the negative 36th power

or more casually as:

Ten to the negative 36th

Notice that in some cases we usually use the cardinal number (36) whereas in other cases we use the ordinal number (36th) when referring to the exponent. Although what I've listed are the more common ways, the cardinal and ordinal forms aren't absolutely universal. For example, you may hear:

10 to the negative 36

However, in my opinion this is much less common.

_

The exponent is usually shown as a superscript to the right of the base. In that case, b^n is called "b raised to the n-th power", "b raised to the power of n", or "the n-th power of b".
Wikipedia: Exponentiation

Notice how it's said in this Khan Academy video, it's said in the shorter way I described, ie., he says ten to the twenty-third.

Specifically in your case of seconds, it would be said:

Ten to the negative 36th seconds
or
Ten raised to the power of negative 36 seconds
or
Ten to the power of negative 36 seconds
or
36th power of 10 seconds (somewhat less common in my opinion, I don't recommend this)

Finally, about the word "undecillion" or "undecillionth", I would advise against using that unless comprehension is not important. Firstly, it's an uncommon word. Secondly, as shown by the dictionary definition below, it can be two different numbers, I assume as a result of the differences between the short and long scales which (historically) separated Europe and the US.

undecillion
a cardinal number represented in the U.S. by 1 followed by 36 zeros, and in Great Britain by 1 followed by 66 zeros.
Random House Unabridged Dictionary

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    We often say "ten to the minus thirty-six" for that amount where I come from (UK). Commented Dec 29, 2018 at 9:27
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    @MichaelHarvey Ten to the minus thirty-six sounds a lot more natural to me too. (I'm Canadian, but I don't know if this has anything to do with regionalism.) Using the words power and negative seems unnecessarily long and formal. However, I suppose this could be a case where writing it and speaking it are distinct things. As well as the level of formality you want to achieve. Commented Dec 29, 2018 at 17:44
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    There are almost twice as many written instances of Ten to the power of minus (X) as there are of Ten to the power of negative (N). Commented Dec 29, 2018 at 18:53
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    @Zebrafish: No - what we're saying is it's more natural to use minus, not negative. Commented Dec 29, 2018 at 18:55
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    To the minus. ;) Commented Dec 29, 2018 at 18:55

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