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Why is the “powers of ten” line asymmetrical? and the grownup name of “googol”

May 1, 2012

Another kind of number line was the powers of 10, with 1 in the zeroes place as 100, and then each positive power of 10 corresponding to the same number of zeroes after the “1”. However, on the negative side, it seemed off. While “ten” (10¹) is “10.”, “tenth” (10-1) is not “.01”, but rather “.1”. Likewise, “hundred” (10²) is “100.”; while “hundredth” (-2) is not “.001”, but “.01”. “Thousand” (10³) is “1,000.” and “thousandth”-3 is not “.000,1”, but “.001”)

Here is a larger sample:

Million 10 6 1,000,000
hundred thousand 10 5 100,000
ten-thousand 10 4 10,000
thousand 10³ 1,000
hundred 10² 100
ten 10¹ 10
one 100 1
tenth 10-1 .1
hundredth 10-2 .01
thousandth 10-3 .001
ten thousandth 10-4 .000,1
hundred thousandth 10-5 .000,01
millionth 10-6 .000,001
ten millionth 10-7 .000,000,1

This appears to be a “reflectional symmetry” that is broken.
I found it so interesting to see another kind of “mirror image” of the numbers, even down to the grouping of zeroes in threes by the commas. But it was so off! It made me wonder for a while, should we have started off with “.1” as “oneth“, and then make “tenth” the next negative power? But that would throw a lot else off!

But the reason why it appears broken, is that we are looking a the axis of this symmetry, as the decimal point. But actually; it’s —the ones place, itself! “One” is what is defined as 100, and zero is the center of the number line, or the axis or fulcrum in which the symmetry hinges. The reason it appears off, is because the decimal point is noted next to (to the right of) the ones place; but it is really supposed to be marking the ones place itself; not the space between the ones and tens place. If we had developed a notation system using an accent above or below the ones place, then it would look more symmetrical. Then, “01”, with the “0” so noted as the ones place, would be “tenth”, instead of “.1”. So what mathematicians have often done, is to fill in the ones place with the “0” to the left of the decimal point, so you end up with a matching numer of zeroes even with the decimal point:

tenth 10-1 0.1
hundredth 10-2 0.01
thousandth 10-3 0.001
ten thousandth 10-4 0.000,1
hundred thousandth 10-5 0.000,01
millionth 10-6 0.000,001

This still throws the comma groupings off, however.

True “grownup” name of “googol”

Also, other technical number information: the proper name of number they call “googol” (10100) is ten duotrigintillion. Others have pointed this out (including Wikipedia, for starters), but I figured it myself before the Internet age by realizing that after the one, it will consist of 33 groups of three zeroes, with one left over; so it will be “ten-something-illion”. Then, notice the powers of 10 divisible by 3 have Latin Prefixes attached to the “illion”: billion, trillion, quadrllion, etc. The Latin number goes up with each set of three zeroes added. But it is offset. Billion is not two sets of zeroes, (106) but rather three (109), trillion is four sets (1012), etc. This is because “million” started out as 106, but the sets of zeroes were six. So billion was two sets of 6 zeroes, or 1012, and so on. This was the old British system, but the American system started with the same “million” and began increasing by sets of three. So 33 sets of zeroes would have the Latin term for “32”, rather than 33, and this is duotrigint.

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