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Multi-Digit Decimal Numbers


Date: 09/07/2001 at 15:30:05
From: Tristan
Subject: Reading  multi digit negative decimal numbers

I'm writing an essay on electrons and I have wound up with a number 
representing the lifetime of an electron charge. The problem I am 
having is actually reading the number because it starts with zeros 
and the remainder of the number is so long. If the number were only a 
few digits (like .007 or .0076, etc.) then it would seem simple, but 
the actual number in question is much longer and so even though I 
know it's still simple, I can't determine exactly how I should say it 
aloud. The number is: 0.0072973525220505560582620625237164

Without the first two zeros I would say the number is 7.2 nonillion, 
but since the 7 is in the thousandth's place, then the number I am 
getting is 7.2 decillion. The number itself is correct, so all I need 
to know is what to call it. I can't say it's seven thousand 
nonillionth's because that's the same as seven decillion, right? The 
more I look at the number the more impossible it looks. Please help! 
Thank you!

Tris


Date: 09/07/2001 at 23:03:14
From: Doctor Peterson
Subject: Re: reading  multi digit negative decimal numbers

Hi, Tris. I hope you don't mind my having fun with this question, 
because I think it illustrates some very useful points.

My first reaction to this was, why would you want to read a number 
like this aloud? And in fact, that is the right question to start 
with. The purpose of your reading determines how to do it.

Actually, there's another question to ask first: Are you sure the 
number is really as accurate as you have said? You have a lot of 
significant figures, so you'd better be able to justify them! I 
strongly suspect that the number ought to be something more like 
0.007297.

But let's assume it's a valid number. Why are you saying it aloud? If 
you're simply reading to yourself, you don't bother with it at all; 
you say "about seven thousandths" or "point zero zero seven dot dot 
dot" or even "(some number)." And if you're presenting your paper to 
an audience, you could probably do the same thing. (Well, the first 
two at least.) After all, why do you need all those digits? If your 
hearers are typing it into their calculators as you say it, they need 
the digits (and you'll want to dictate it digit by digit, "point zero 
zero seven two ...," so they can do so). But ordinarily, what's 
important is not the digits, but the size of the number; and with 
decimals, it's just the first few digits that count. Either of my 
first two suggestions will accomplish that. If your purpose is rather 
to impress them with the precision (and, you hope, the accuracy) of 
your number, you can add "and so on for 32 digits," or just show it on 
a screen and wait for applause.

Now, you asked how to pronounce it as a fraction. I'll tell you, but 
with a caveat: of all the ways to say it, this communicates the least 
to your audience. It thoroughly hides the size of the number, and 
overwhelms people with details instead. It's impossible to copy down, 
and takes forever to say. Did you believe your teachers when they said 
to pronounce decimals this way? I think your example clearly shows why 
they were wrong (at least for long numbers like this). That's why I'm 
interested in your question.

Okay. Let's write the number as a fraction:

    0.0072973525220505560582620625237164
    ------------------------------------
    1.0000000000000000000000000000000000

becomes

        72 973 525 220 505 560 582 620 625 237 164
    ----------------------------------------------
    10 000 000 000 000 000 000 000 000 000 000 000

The numerator (in the American system) is about 72 nonillion; the 
denominator is 10 decillion. So the fraction is

    72 nonillion, 973 octillion, 525 septillion, 220 sextillion,
    505 quintillion, 560 quadrillion, 582 trillion, 620 billion,
    625 million, 237 thousand, 164 ten-decillionths.

Now, in a scientific context, you wouldn't have written the number 
this way in the first place; you'd use (surprise!) scientific 
notation:

    7.2973525220505560582620625237164 * 10^-3

This, of course, deals with all the issues I've raised; that's what 
it's for. Saying it this way, we can say as many digits as we like 
without obscuring the size of the number. Still, I wouldn't say the 
fractional part in fractional terms, because there's no need.

I'll make one final comment: in writing such a precise number, it's 
good to give the reader a way to keep track of the digits, and the 
standard way is to insert spaces:

    0.007 297 352 522 050 556 058 262 062 523 716 4

Notice that the groups of three start at the decimal; that's a clue 
that we don't bother to read such a number as a fraction, since this 
spacing doesn't help in pronouncing the numerator. Interesting, isn't 
it?

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/   
    
Associated Topics:
Elementary Large Numbers

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