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Syllables from One to One Million

Date: 03/12/2003 at 11:20:01
From: Kathleen
Subject: Counting from one to one million

How many syllables are there when counting from one to one million?

My daughter is using this information for a math project. She has 
identified most numbers from one to seven syllables. She is doing 
this without using the word "and." She has identified the number from 
one to one million to have the most syllables in it to be 777,777. 
After seven syllables the math gets very involved for a 4th grade 
student.

We estimated that there are about 14 million syllables from one to 
one million (excluding the word "and").


Date: 03/12/2003 at 16:40:10
From: Doctor Peterson
Subject: Re: Counting from one to one million

Hi, Kathleen.

This seems like an awfully big job for a 10-year-old, though the 
complexity is mostly in finding a way to organize the work; I wouldn't 
be surprised if the estimate of 14 million (which is exactly what I 
estimated before starting the big job) is all that is expected. Or is 
this a self-imposed assignment?

But it's an interesting challenge, so I spent some time looking not 
only for a solution, but for a way to organize my work so that I could 
explain it clearly. That's not easy! And once I'd done it, I found a 
number of errors in my work, which I would not have been able to catch 
if I hadn't written it all out carefully.

First, for an estimate, which will let us check that our answer makes 
sense: a "typical" number would be something like 555,555, or five 
hundred fifty-five thousand, five hundred fifty-five, which has 14 
syllables; there are 1,000,000 numbers to count, so we have about 
14,000,000 syllables in all. The actual number will be less than that, 
since many numbers are shorter.

Let's start the real counting. First look at the numbers from 1 to 9:

  one, two, three, four, five, six, seven, eight, nine
   1    2     3     4     5     6    7 8     9     10

  1
  ...
  9
  --
  total syllables: 10

Now the teens are special, too:

  ten, eleven, twelve, thirteen, fourteen, fifteen, sixteen,
   1   2 3 4     5       6   7     8   9   10  11   12  13
  seventeen, eighteen, nineteen
  14 15 16    17  18    19  20

  10
  11
  ...
  19
  --
  total syllables: 20

From here on, numbers are built up by combining numbers we've already 
looked at, so we can multiply rather than count each syllable. The 
next group is 20 - 99, which are pronounced as "Xty-Y". We'll count 
the "Xty"s by row, and the "Y"s by column:

  20  30  ...  90   total of "Xty"s = 17; repeated for 10 rows
  21  31       91
  ...
  29  39       99
  --
  total of "Y"s for each column = 10; repeated for 8 columns

So the total number of syllables is 17*10 + 10*8 = 250. We calculate 
this by counting the syllables in "twenty, thirty, ..." as 17 (two 
syllables each in eight numbers, plus an extra syllable in seventy), 
and seeing that each of those is repeated ten times (21, 22, ...), 
making a total of 170 syllables for the tens; and then counting "one, 
two, ..." to get 10, each of which is repeated 8 times (21, 31, 
41, ...), making a total of 80 syllables for the ones.

Now we can add up all the syllables from 1 to 99 and we get 10+20+250 
= 280 syllables.

Once we pass 99, we read numbers as "X hundred, Y-ty Z" from 100 
through 999. Again, we can count the hundreds by rows and the tens 
and ones by columns:

  100  200  ...  900  total of "X hundred"s = 10 + 9*2 = 28;
  101  201       901    repeated for 100 rows
  ...
  199  299       999
  --
  total of "Y-ty Z"s for each column = 280; repeated for 9 columns

"Hundred" is repeated 9 times in each row, adding 18 syllables to the 
count for each row in addition to the 10 for the numbers themselves. 
The number of syllables contributed by the tens and ones in each 
column is the count of 280 we already have. So the total number of 
syllables here is 28*100 + 280*9 = 5320 syllables; and from 1 to 999 
we have 280+5320 = 5600 syllables.

Finally, from here to 999,999, we read numbers as "X thousand, Y" 
where X and Y can be anything from 1 to 999:

  1,000 2,000 ... 999,000  total of "X thousand"s = 5600+999*2=7598;
  1,001 2,001     999,001    repeated for 1000 rows
  ...
  1,999 2,999     999,999
  -----
  total of "Y"s = 5600; repeated for 999 columns

The total here is therefore 7598*1000 + 5600*999 = 13,192,400, and 
the cumulative total from 1 to 999,999 is 5600 + 13,192,400 = 
13,198,000 syllables.

Finally, add on the 3 syllables for "one million" and you have a grand 
total of 13,198,003 syllables to count to a million. That's quite 
close to our estimate.

Organizing is the key to accomplishing this task; and that's a good 
lesson to learn. I don't know whether you'll be able to use this, but 
it was fun going beyond an estimate to an exact number. Here's the 
best we'd previously done in this area:

   Really Counting to One Billion
   http://mathforum.org/library/drmath/view/59179.html 

One more comment. It sounds as if your daughter is organizing this in 
the opposite direction, trying to group numbers by the number of 
syllables rather than finding the total numbers of syllables in each 
natural group. That would be an immensely difficult task; I don't 
think I'd want to do it without a computer! But the data would be 
very interesting to see; our result shows that the vast majority of 
numbers are close to 14 syllables, and it would be interesting to see 
just how many. That's not necessary, however, in order to find the 
total number of syllables.

If you have any further questions, feel free to write back.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/ 
Associated Topics:
High School Projects
Middle School Number Sense/About Numbers
Middle School Word Problems

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