The Oneths PlaceDate: 09/29/98 at 21:30:33 From: Cag Subject: Transition Math Dr. Math, Why is there not a oneths place? My classmates and I were wondering. It is weird that decimals begin with tenths, hundredths, and so on. We appreciate your help. Thank you. Date: 09/30/98 at 17:02:15 From: Doctor Peterson Subject: Re: Transition Math Hi, Cag. This is a good question. It could be answered by just saying "There isn't one, so accept it!", but it deserves some more thought. Let's think about it a moment: What are "oneths"? A "oneth" would be 1/1, so there already is a "oneths" place. It's called the ones' place. You don't need another. As I'll explain in a moment, this is a lot like the reason there is no -0: negative zero is the same as positive zero, so we only need one of them in the number line. What you're noticing is that the place values don't seem symmetrical, or balanced. On the left we have 1, 10, 100, and so on. On the right we have 1/10, 1/100, and so on. Shouldn't everything on the right match up with something on the left, 1/10 with 10 and so on? Or to put it another way, when we convert, say, 1/1000 to decimals, we find that although 1000 has three zeroes, 0.001 has only two! Why can't it be simpler? Place values really are symmetrical. The problem is that the center is at the ones' place, not at the decimal point where we would like it to be. If you think of the ones as being in the middle, then as you move to the left, you multiply by 10 each time, and as you move to the right, you divide by 10 each time. The trouble is caused by the decimal point: it's off-center. It was natural to put it to the right of the ones' place, to separate the fraction part of a number from the integer part, but that takes away the symmetry. What you have to do is to think of the ones' digit as special, and count places to the left or right of that. For example, 1000 has two zeroes to the left of the ones', and 0.001 has two zeroes to the right of the ones'. The 1 in 1000 is three places to the left of the ones' place, and the 1 in 0.001 is three places to the right of the ones' place. If we wrote numbers by underlining the ones' place rather than putting a decimal point after it, there would be no confusion. Another way to look at this is to relate place values to the number line. Each place corresponds to a power of ten: 123.45 means this: 1 2 3 . 4 5 * * * * * 100 10 1 1/10 1/100 Each of these place values is 1 either multiplied or divided by some number of tens: 1*10*10 1*10 1 1/10 1/10/10 We can write this using an exponent (which looks like "^2" here, and like a small number raised above the other in a book). Positive exponents mean you multiply 1 by that many tens, and negative exponents mean you divide by that many tens: 10^2 10^1 10^0 10^-1 10^-2 Now look at the exponents: they form a number line, and the zero is not at the decimal point, but at the ones' place. <-----+-------+-------+-------+-------+-----> 2 1 0 -1 -2 In the number line we have not only positive and negative values, but also zero, which is neither positive nor negative but in between. It doesn't belong on either side, but in the middle. On the number line we can see it that way, but in a number the 10^0 place has to go on one side of the decimal point. There aren't both a positive and a negative zero, and there aren't both ones and oneths. The ones' place, just like 0 on the number line, belongs in the middle. I hope that helps. It's interesting how we tend to expect things to be symmetrical. Mathematicians and scientists often expect it too, and when things don't seem balanced, they try to find out why. So keep wondering about things like this. - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/ |
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