Syllables from One to One MillionDate: 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/ |
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