Counting Decimal PlacesDate: 05/22/2001 at 23:46:49 From: Craig Nadeau Subject: Decimals When multiplying decimals why does placing the decimal in the answer by counting the number of decimal places in the problem and counting from the right the same number of places work? I have searched the Internet and even looked in my math book, but I still can't find anything. I asked one of the teachers at my school and he didn't know. My own math teacher won't help because she says I need to learn it on my own whether I get it or not. I would really like to show her up and prove I'm not a complete idiot. All the answers I've tried are apparently not good enough for her. Date: 05/23/2001 at 09:16:22 From: Doctor Peterson Subject: Re: Decimals Hi, Craig. I tried searching our archives for this, hoping I could help you find the answer for yourself, but I didn't find a good complete answer. Here's how it works: Suppose we are multiplying 1.23 by 4.5. First, we can convert each number to a decimal by using all its digits as the numerator (with no decimal point), and using for a denominator a 1 followed by as many zeros as there were decimal places, so that 1.23 is 123/100. You may want to think of it this way, where I repeatedly move the decimal point to the right to multiply by 10, until everything is a whole number: 1.23 12.3 123. 1.23 = ---- = ---- = ---- 1.00 10.0 100. (Remember, if you multiply the numerator and denominator of a fraction by the same number, you haven't changed the value.) Writing both decimals as fractions, we have: 123 45 123*45 5535 1.23 * 4.5 = --- * -- = ------ = ---- = 5.535 100 10 100*10 1000 Now, when I multiplied the fractions, the denominator was the product of two powers of ten, and turned out to be a 1 followed by the total number of zeros in the two numbers, which is the total number of decimal places in the two decimals! When I convert back to a decimal at the end, that's how many decimal places I need. - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/ |
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