Decimals in Word Form: Basics

We’ve been looking at the place value concept, and writing number in expanded form(s); but how about the word form of decimals? This can be confusing at several points. We’ll start with reading a number and writing its word form, and then do the reverse.

Reading decimals: the basics

We’ll start with this, from 1997:

Writing Numbers in Words

Write each number in words: 1) 73.78007 2) 2.900087 (3) 34.4939.

Doctor Sonya answered, starting with place value:

Dear Leah,

The key to writing numbers in words is knowing the name of the place where each digit is. For example, the number 461 has a 4 in the "hundreds" place, a 6 in the "tens" place, and a 1 in the "ones" place. 

Have you learned about place value in school yet? Place value is the key to this problem. Using our above example, 461 in words is:

   4 hundreds (400) plus
   6 tens (60) plus
   1 ones (1).

We change this around a little bit so that it is in standard English, and we get "four hundred sixty-one". (Remember that 6 tens is sixty.)

Here we are going from (short) expanded form, “400 + 60 + 1”, to “mixed form”, “4 hundreds, 6 tens, and 1 one”, to English word form where 6 tens becomes “sixty”, 1 one becomes “one”, and the last two are combined to make “sixty-one”. We’ll be trusting that you know that last part; the hard part is the decimal.

The question you are asking us is a little bit more complicated, because it has decimals in it and so you also have to worry about the names of the places after the decimal point. Let's take some decimals.  I choose 4.337829. Here are the names of the places. I'll put a star in each place I'm talking about:

    4.*37829    tenths place
    4.3*7829    hundredths place
    4.33*829    thousandths place
    4.337*29    ten thousandths place
    4.3378*9    hundred thousandths place
    4.33782*    millionths place 

Do you see a pattern here?

We saw this in the post about place value for decimals. Now, how do we fit those six places into a word form? Do we have to write, “4 and 3 tenths and 3 hundredths and 7 thousandths and …”? No, we make it a little easier:

Remember that a decimal is just another way of writing a fraction.  For example:

    0.3 is really 3/10

In words, 3/10 is "three tenths," so 0.3 is also called "three tenths," and the 3 is in the "tenths" place.

The same is true for 0.33 = 33/100.  Can you figure our what 33/100 is in words?

What we’ve done here is to combine all the decimal places into one, lumping it as a single fraction, “thirty-three hundredths”. If you don’t see how we can do that, we’ll be seeing it later.

Let's try to do one of the examples you sent us:

     73.78007

1) First, what's before the decimal point? You know what that is: it's seventy-three.  

2) Now for after the decimal point. The first thing you have to do is figure out how to say the number as if the decimal point weren't there.

   4.9832 --> 9832 --> nine thousand eight hundred thirty-two
   0.364  -->  364 --> three hundred sixty-four

In our example, 73.78007 --> 78007 --> seventy-eight thousand seven.

So far, we have the whole part, and the numerator of the fraction part.

3) The next step is to figure out what the name of the smallest place (the place farthest to the right) is.  In our number, it's the hundred thousandths place.

4) Finally, you say the whole number after the decimal, followed by the name of this place:  

     seventy-eight thousand seven hundred thousandths  

   This is the way to say everything after the decimal.

5) Now that we know how to say everything before the decimal and after the decimal, we just have to connect it with an "and":

   "seventy-three" and "seventy-eight thousand seven" hundred thousandths.

Now you can do your other examples the same way.

Of course, the quotation marks are not part of the answer; and as we’ll see later (next time), it’s a good idea to add in a hyphen:

73.78007 = seventy-three and seventy-eight thousand seven hundred-thousandths

Yes, that’s a little awkward, and we’ll be discussing that fact next time, too.

A shortcut

How about that issue of turning the whole decimal part into a single fraction, rather than lots of separate places? Here is a 2001 question:

Using Fractions to Read Decimals

I am having trouble reading decimals. Can you please tell me how to read in words, decimals such as:

  .02
  .20
  .020
  .002
  .025
  .0256
  .02456

Thank you very much for your help,

I answered:

Hi, Julius.

Here's one way to keep track of the meaning of a decimal.

Write the decimal, and below it put a 1 (under the one's place in the number) followed by a decimal point and a zero for every decimal place in the number:

    0.02
    ----
    1.00

Now take out both decimal points, and you have the fraction equivalent to the decimal:

    002
    ---
    100

So 0.02 means "2 hundredths." Similarly,

    0.02456   002456
    ------- = ------
    1.00000   100000

so 0.02456 means "2,456 hundred-thousandths."

Dividing a number by 1 (or putting 1 under it as a denominator) leaves the value unchanged, so the first step does nothing but turn the decimal into a fraction (though an ugly one, with decimals in it). The second step was actually to multiply the numerator and denominator by the same number, a power of ten that moved the decimal point all the way to the right in both numbers. This, again, leaves the value unchanged (because the decimal point was moved the same number of places in both parts); and it turned it into a proper fraction, with whole numbers in both parts. (This was the reason for using as many zeros after the decimal point as there are decimal places in the original number.)

You don't really have to write all this out; just count the decimal places in the number, and use that number of zeros to picture the denominator.

So, to write out 0.33, we put 33 on top, and a 1 followed by two zeros on the bottom: $$0.33 = \frac{33}{100} =\text{ thirty-three hundredths}$$

What we're actually doing is writing a "fraction" using decimals that is equal to your decimal number (since a denominator of 1 doesn't change the number); then multiplying the numerator and denominator of this number by ten repeatedly until we have two whole numbers and can say the fraction.

As I said before.

Really long decimals

Let’s take it up another notch, with this question from 2001:

Multi-Digit 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

We’ll be looking at numbers like “nonillion” eventually; one nonillion is 1,000,000,000,000,000,000,000,000,000,000! I answered, warning Tris that I would be saying more than he needed:

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.

Never decide to do something difficult without considering why you want 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.

Not knowing how the number was calculated, I can’t say for sure, but it is very likely that at some point Tris had multiplied or divided by a number that was only accurate to a few decimal places, and by the rules of significant digits, only that many digits should be shown. There is rarely (if ever) a real need for such a precise decimal number!

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.

These reflect what I have actually done in practice. In reading for myself, I only have to get the general size of a number in my mind; if I need the actual number for a calculation, I will go back and copy it. If I were showing a PowerPoint with a long number, I might just point out “… and we get this number …”.

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.

If I read the number at all, it would be as “zero point zero zero seven two nine …”.

But it’s time to actually answer the question, the same way I did above for a simpler one:

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.

There. Satisfied?

(The reference to “the American system” reflects the fact that what Americans call a billion is not what Europeans have traditionally called a billion. I’ll get to that eventually, too.)

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.

The exponent part makes the size of the number explicit; and you can say (or look at) as many decimal places as you choose, without losing any significant information. The number is about 7 thousandths.

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?

If we were intending to read this as a fraction, we’d write it as “0.0 072 973 525 220 505 560 582 620 625 237 164”, to make the 72 nonillion clear. But we don’t!

Reading words, writing decimals

Now let’s reverse the process. What if you are given a number in words, and have to write the decimal form? We’ll also think a little more deeply about some of the subtleties of the word form. Here is a question from 1998:

Writing Decimals from Words

Can you help me with my assignment? Here's a sample question:

Write the decimal for five hundred seventeen thousandths.                      

Please help,
Lisa

Doctor Rob answered:

Lisa,

The first thing to do is to figure out where the fraction bar should go. The numerator will be a simple number, and the denominator will be a number with -ths on the end (like "three tenths": the numerator is "three" and the denominator is 10 because of "tenths"). Since these are specified to be decimals in the statement of the problem, the denominator will be a power of 10: tenths, hundredths, thousandths, ten thousandths, hundred thousandths, millionths, and so on.

So before we start writing anything, we have to analyze the structure of the words, finding clues to group them into numerator and denominator.

Figure out starting from the back where that part starts. Put the fraction bar just before that, and drop the "-ths" off the end. Then write the number before the fraction bar as the numerator, and the number after the fraction bar as the denominator. In your example:

   "five hundred seventeen thousandths"

since "seventeen thousandths" is not a power of 10, it cannot be the denominator, so the fraction bar must go between these words. Then you have:

   "five hundred seventeen / thousand"

or:

   517/1000

Having pulled off the denominator at the end, what’s left is the numerator.

But observe: If we didn’t expect this to be a decimal, it could very well have been “five hundred seventeen-thousandths”, that is, \(\displaystyle\frac{500}{17,000}\)! Only the hyphen would tell us the difference. We’ll discuss this in detail next time. Words are ambiguous.

Next, we have to turn that fraction into a decimal, by reversing the process we saw earlier:

Now divide the denominator into the numerator by moving the decimal points of each to the left the same number of places as the number of zeroes in the denominator (in this case, three places left):

   .517/1.000 = .517

and that is your answer.

In other words, we have moved the decimal point in 517 from the right, to the left as many places as there are zeros in 1000.

WARNING: Sometimes there are two answers, and you can't tell which is correct. As an example, "One hundred ten thousandths" could mean 110/1000 = .110, or 100/10000 = .0100 (think about it). When the words are spoken, a slight hesitation indicates the position of the fraction bar, removing the ambiguity. The first would be pronounced as if it were written, "One hundred ten, thousandths," and the second as if it were written, "One hundred, ten thousandths."  Of course the comma is never written, just understood.

Good luck!

Again, a hyphen would help: “One hundred ten thousandths” is \(\displaystyle\frac{110}{1000} = 0.110\), whereas “One hundred ten-thousandths” is \(\displaystyle\frac{100}{10,000} = 0.0100\).

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