Is American Numbering System Illogical

Cecil's Columns/Staff Reports
1

In Cecil’s answer to “What is a gazillion?” he posted a quote that included the following comment (from Doctor Math) on “what is a billion”:

“In the United States, it means 10^9; in Great Britain, it means 10^12. The Brits add 6 zeroes per step up, and we add 3. So a British “trillion” is 10^18. In a sense, the British system makes more sense–billion, trillion, quadrillion, etc., indicate 2, 3, and 4 from the roots of the names. If you think of them as meaning 2, 3, and 4 groups of 6 zeroes, everything makes good sense–and it makes no sense in the U.S. system.”

Actually, this last comment, that the US system makes no sense is not true. Doctor Math simply doesn’t understand it. The British system names these numbers according to the following system:

10^(61) = 10^6 = million
10^(62) = 10^12 = billion
10^(63) = 10^18 = trillion
10^(64) = 10^24 = quadrillion
10^(6*5) = 10^30 = quintillion
and so on

Which reflects the mathematical system explained in the quote. The US system, however, is also based on mathematical logic. In the American system, start at one thousand, instead of one million.

10^3+(30) = 10^3 = thousand
10^3+(31) = 10^6 = million
10^3+(32) = 10^9 = billion
10^3+(33) = 10^12 = trillion
10^3+(34) = 10^15 = quadrillion
10^3+(35) = 10^18 = quintillion
and so on

So contrary to the claims of Doctor Math above, there is a completely reasonable, and easily remembered logic behind the American system as well.

It sure bugs me when people assume there is no reason for something simply because THEY can’t figure it out.

2

Welcome to the Straight Dope Message Board, Rhomphaia. It’s customary to post a link to the column you’re discussing in the first post. Here it is:
How much is a gazillion?

The column is written by Dex, of the Straight Dope staff, not by Cecil. So, this thread will soon be moved over to the Comments on Mailbag Items forum, where such staff writings are discussed.

Don’t worry about it, most new people make a couple of mistakes. You sound like someone with strong opinions, so I hope you’ll stick around and keep posting.

3

Now to comment on your point.

It’s true, there is logic to the fact that the American system has a named value every third decimal place. What Dr. Math is complaining about is that the names are based on Greek numerical prefixes, and they don’t seem to line up to anything.

In the Brittish system, a quadrillion has four groups of six zeroes. In the US system, a quadrillion has 15 zeroes, which doesn’t have four of anything.

Regardless, I’ve read many comments from Dopers from overseas that say that even in the UK, they are using the American system. The names may not make sense to a Greek scholar, but they are becoming the standard.

4

The greater logic seen in the British system is obviously that an exponent that is given a new name, based on sequential Latin units prefixes, always only upon each increment of 6, starting with zero, is following a less arbitrary pattern than is an exponent that doesn’t have equal named increments until the second step, after only taking an increment of 3 in its first step up from zero.

Rather than saying one thousand million the British can alternatively say one milliard. However, a billiard is a shot in the game of billiards, and I think the Brits just say ‘one thousand billion’, ‘one thousand trillion’, etc. for the intermediate 10^3-factored steps between their -illions.

Ray (is still trying to get beyond a googoltriplex.)

5

Thanks, Saltire, for helping me on the netiquette of this forum. I’ll try to do better next time.

In your reply you said:

“In the Brittish system, a quadrillion has four groups of six zeroes. In the US system, a quadrillion has 15 zeroes, which doesn’t have four of anything.”

If you look back up at my example, you find that the US system DOES use sets, it simply adds 3 to those sets (since it starts at a thousand).

So a quadrillion is FOUR sets of three zeros, plus three zeros, yielding 15 zeros; a quintillion is FIVE sets of three zeros, plus three zeros, yielding 18 zeros. This pattern is uniform, and works every time.

It might not be as “neat” as the British system, where a quadrillion is simply four groups of six zeros, with nothing else added, but it DOES use groups based on the prefix of the number (2 for bi, 3 for tri, 4 for quad, 5 for quint, etc). Adding three zeros at the end does NOT negate the grouping pattern that exists.

So long as you know what the prefix means, remember that it refers to groups of THREE zeros, and remember to ADD three more zeros at the end, you can logically figure out any giant number quickly and efficiently. For example: duodecillion = twelve groups of three zeros (36), plus three zeros, yielding 1 followed by 39 zeros. Quick, logical, easy, and yes, it follows a grouping pattern based on the Greek root of the prefix.

So as I said before, Doctor Math really is wrong about that.

6

I meant to end the first paragraph in my earlier post here with “after taking an different increment, one of 6, in its first step up from zero.” Sorry.

Ray

7

I’d just like to point out what an incredible coincidence it was that Doctor Math’s name coincides so closely to his chosen profession.

8

Hey Ray. It is very nice to discuss things with logical, intelligent people. A real pleasure.

You said, in part:

“The greater logic seen in the British system is obviously that an exponent that is given a new name, based on sequential Latin units prefixes, always only upon each increment of 6, starting with zero, is following a less arbitrary pattern than is an exponent that doesn’t have equal named increments until the second step, after only taking an increment of 6 in its first step up from zero.” (corrected according to your later post)

I suppose one could say there is greater logic in that. I’m not convinced. I don’t find greater logic in one formula over another. They are all math:

Greek Prefix = P

British formula for calculating the number of zeros:

(P*6)

American formula for calculating the number of zeros:

(p*3)+3

Both formulas work for every “illion” with a Greek prefix (million, billion, trillion, quadrillion, etc.). The British version is easier, I suppose(you only have to multiply, where in the devious US system, you have to multiply, and then ADD), but that does not necessarily translate to “greater logic.”

One could easily argue that the American system uses greater logic as it names EVERY step with a designation having a unique greek prefix, where the British system doesn’t (making the groups between quadrillion and quintillion, for example, a little more difficult to calculate from their names in the British system…and leading to names, such as milliard in England, and milliard and billiarde in Germany, that DO NOT follow a Greek prefix formula), which makes the US system “more logical.”

Beginning to sound like a “six one way, half a dozen another” situation.

Anyway, my real point was that Doctor Math said the US system “makes no sense,” and he is wrong. It makes perfect sense, and it DOES use the Greek prefixes. I was not really claiming it makes MORE sense, or even that it is easier than the British system, simply that the good Doctor’s claim that it makes “no sense” is wrong.

And I do believe I have proven that point.

9

>I suppose one could say there is greater logic in that. I’m >not convinced. I don’t find greater logic in one formula >over another.

Elegance = logic = elegance (only '6’s, no '3’s).

>They are all math:

The British was math, the American aftermath. :wink:

>Greek Prefix = P

I tell ya: Those are Latin prefixes, not Greek (except for ‘m-’ if it stands for ‘mono-’, and ‘tr-’ goes for either). But I guess it’s all Greek to you. No logic; just the way it is.

>The British version is easier, . . . . . .that does not
>necessarily translate to “greater logic.”

Easier = more elegant = more logical = more elegant = easier. :wink:

>One could easily argue that the American system uses >greater logic as it names EVERY step with a designation >having a unique greek prefix, where the British system >doesn’t. . .which makes the US system “more logical.”

Who says a “step” must be 3 unit increments of power? You’re not naming every step in your logic, Man (in Greek, Latin or English). Hey, if steps of 3 are “logicaler” than steps of 6, we English-speaking logical people should use “steps” of 1 (1-step at a time) – ten = 10, tentwo = 100, tenthree = 1000, tenfour = 10,000, etc. – keeping things in English. Or we could just do everything in a binary base, maybe in multiple dimensions, just to step things up a little.

>Beginning to sound like a “six one way, half a dozen >another” situation.

Naw, it’s ‘six one way and a quarter dozen the other’. Dozen that sound right? :wink:

>Anyway, my real point was that Doctor Math said the US >system “makes no sense,” and he is wrong. It makes perfect >sense, . . .

No, the US system doesn’t make “no sense”, but it only makes imperfect sense.

>I was not really claiming it makes MORE sense, or even that >it is easier than the British system, . . .

Well, it’s true that the British system doesn’t make any cents. . .only pence (for what that’s worth).

>. . .simply that the good Doctor’s claim that it makes “no >sense” is wrong.

In which case, he wouldn’t be a “good” doctor, now would he? He’d be a bad doctor, of course.

>And I do believe I have proven that point.

When you’ve proven a point you know it; you don’t just believe it. But these were all integers we were talking about; there weren’t any points in them. And you never informed us whether a zillion is the same in the UK as in the US or different. Somehow I think we’ve lost the point. It musta been sharper than us.

Ray (I lost count.)

10

LOL. That was a good post, Nanobyte. I laughed through the whole thing. You’ve a good, sharp wit. Well done.

Just have one response.

===========

>Greek Prefix = P

I tell ya: Those are Latin prefixes, not Greek (except for ‘m-’ if it stands for ‘mono-’, and ‘tr-’ goes for either). But I guess it’s all Greek to you. No logic; just the way it is.

==========

I am embarrassed at making this mistake. It is even worse because I actually read Koine Greek every day, and I studied Attic Greek in college.

Sigh. Egg on my face.

I deal with the Greek numbers so little I have to really stop and think about it to even remember what they are. I was concentrating on Doctor Math’s comment that the US system made no sense, and I never actually thought about it. Someone somewhere said Greek, and I just started using it without one second’s thought.

No excuse. Shame on me.

Actually, the “mi” could come from the Greek (feminine cardinal “one” is mia). The others include three overlaps from 1-10: tri at three, oct at eight, and dec at ten. But overlaps do not a pattern make, and these are clearly “primarily” Latin prefixes.

My humble apologies.

Other than that, my logic was flawless.

Hehehe.

You have a good day.

11

Welcome to the SDMB, and thank you for posting your comment.
Please include a link to the relevant article if it’s on the straight dope web site.
To include a link, it can be as simple as including the web page location in your post (make sure there is a space before and after the text of the URL).

The SD staff report can be found on-line at the link kindly provided by Saltire.

Since the article is an SD staff report, not a Straight Dope column, this thread is leaving the «Comments on Cecil’s Columns» forum and going to visit my colleague CKDextHavn in the «Comments on Mailbag Answers» forum.

moderator, «Comments on Cecil’s Columns»

12

The Greek thing was my fault. In the immortal words of Agent 86, “Sorry 'bout that, Chief.”

13

O sure, everybody understands that, but following that logic, a bicycle would have only one wheel. I think that’s all Dr. Math’s point.

14

Of course. The relatively lesser degree of illogicalness of the American system rests simply on the fact that the first jump, from 0 to 3, is only 1/2 the size of the following “sets”. Vulcan logic would have a straight ramp. KISS. But I guess a tempered first step makes it easier for American couch potatoes to get up the ramp. I think it’s time for the sun to “set” on this thread.

Ray

15

Nanobyte said:

“Of course. The relatively lesser degree of illogicalness of the American system rests simply on the fact that the first jump, from 0 to 3, is only 1/2 the size of the following “sets”. Vulcan logic would have a straight ramp. KISS. But I guess a tempered first step makes it easier for American couch potatoes to get up the ramp.”

I don’t quite follow you on this one. The American system does follow a straight line. There is a jump of three at every step. I don’t understand how the first jump is half the size of each successive jump. If you plot it on a graph, starting with a power of zero, you see that both are straight lines, with the American progression shifted to the left (when P=0, the US system = 3, the German system = 0).

When you draw a straight line on a graph, there is nothing more or less logical about one line over another, nor is one formula more or less logical than any other.

Maybe you can help me understand how the formula (p3)+3 is less logical than the formula (p6), and explain why. If you’ve taken any geometry, you see that these are simply formulas for straight lines. Why is a line that crosses the “y” axis at three less logical than one that crosses at the origin?

I must be slow today, because RM Mentock’s comment that following the logic of (p*3)+3, a bicycle would have only one wheel completely escaped me, since the US system multiplies by 2 where the prefix is “bi.”

:slight_smile:

Please help me with this, guys. Thanks.

16

Oh, and BTW. I realize it was just a typo, but I can’t resist. Hey, I can resist anything…except temptation.

:slight_smile:

Ray said

“The relatively lesser degree of illogicalness of the American system rests simply on the fact that the first jump, from 0 to 3, is only 1/2 the size of the following “sets”.”

Just thought I’d point out that “a lesser degree of illogicalness” is the same thing as “a greater degree of logic,” and I don’t think Ray intended to say the US system was more logical.

I’m pretty sure he actually meant “a greater degree of illogicalness,” or “a lesser degree of logicalness.” And I do not have a problem with spontaneously coined new words, so long as I can decipher them.

Hehe.

17

OK, three wheels.

18

3 wheels? Inventing words? Oh, you mean a tripeddle velocimpeder.

OK, I blew it twice in my last post. I meant ‘less logicalness’, which I’ll allow to = ‘more logicalness’. But where is the invented word? ‘Logicalness’ and ‘illogicalness’ are in my moderate-sized dictionary. I was going to use ‘illogicality’, but that isn’t in it.

The other mistake I made is the same one I made before, switching ‘6’ and ‘3’: In the American system the first “set” of the ramp, from 0 to 1,000,000 is 6 powers of 10 and all the following ones are 3. The first set thus looks to me like an ADA design. I simply say this is not KISS and stick the tag ‘illogical’ on it because nobody can dig up any general need to have the first “set” different from the others; this scenario would seem to be a quirk of somewhat localized history. But since we’re all billionaires (by the US convention) in the US now :wink: , this shouldn’t bother us, right? Which does bring up another worry for me though: In Europe, would a billionaire (in any given currency) be called something like a ‘milliardaire’, or would such a person have to hang in there another American “set” before (s)he won the match?

Ray (omnilogical)

19

Ray said,

“OK, I blew it twice in my last post. I meant ‘less logicalness’, which I’ll allow to = ‘more logicalness’. But where is the invented word? ‘Logicalness’ and ‘illogicalness’ are in my moderate-sized dictionary. I was going to use ‘illogicality’, but that isn’t in it.”

Well, maybe you need a better dictionary. Mine does have illogicalness (you were correct there), but it also has illogicality. I have never heard the word illogicalness before. I learned something. Thanks.

:slight_smile:

As for the rest of your comments, I give up. I would love to be there when you take an algebra course and tell the instructor that the formula for the line Y=(x3)+3 is less logical than the formula for the line Y=(x6).

Just don’t say it out loud in front of the whole class. You might not live that one down for a while.

And in reference to Mentock’s post, how does multiplying by 2 when the prefix is “bi” (as in “bicycle”…with TWO wheels) equal three? The US system doesn’t multiply by 3 until the prefix is “tri.” I’m afraid your superior grasp of mathematics has left me in the dust. Could you explain your logic?

Please.

20

That last sentence doesn’t make any sense at all. I take “tri” to refer to ‘trillion’. The US system multiplies by 3 powers of 10 between a million and a billion and between a billion and a trillion, as well as thereafter (but by 6 between 1 and a million).

And as to your math-class ramblings, the logical issue is not as to the end result, but as to the process of arriving at it. It is not logical (under normal conditions) to fly from San Francisco to Oakland via New Delhi, though, if you care to go to such a place, your getting their by any means causes a logical result. Hell, if bicycles are where it’s at, and time is not of the essence, wait around until the bicycle lobby actually gets CalTrans to put bike lanes into the new bridge design and then hangout until the new infinite-earthquake-proof bridge gets build. . .just don’t hold your breath ('tain’t logical).

Ray