Converting Fractions to DecimalsDate: 01/08/97 at 21:36:32 From: Susan Fredrickson Subject: Question Dear Dr. Math, Today in math class we were working on rational and irrational numbers and I asked the teacher what the square root of 2.25 is. Well, if you punch it into your calculater, it says 1.5. That's not a rational number because it's not an integer. You could do 1.5 over one and that wouldn't be a rational number, but if you doubled both the top and the bottom, it would be 3 over 2 and that would be a rational number. Doubling them is kind of like the opposite of reducing and I was wondering whether that had a name. Also, the equation 3/3 = 1 is correct but I don't see how that can be because 1/3 =.3333 and if you multiply 3.3 by three times, it won't be one. It will be 9.999999..... Thank you for your time. Date: 01/11/97 at 14:44:18 From: Doctor Gerald Subject: Re: Question Susan, Good questions - you are really thinking! First, remember that a rational number is one that CAN be written as one integer over another. So your argument that doubling 1.5 to get 3 over 2 shows that it really is rational. Both doubling and reducing have the same general name; it's called finding EQUIVALENT fractions for rational numbers. You can multiply or divide the top and bottom of any fraction by the same number and get an equivalent fraction. When you divide, you are reducing; when you multiply, you are doubling or tripling or whatever. Any fraction has countless equivalent fractions. For example: .15/.1 = .3/.2 = 1.5/1 = 3/2 = 6/4 = 12/8 = 60/40 = 300/200 = .... Can you write others that are equivalent to 1.5? Your second question gets into something that mathematicians call "limits." You are very insightful to recognize that: 3(1/3) = 3(.3333...) = .9999... = 1 .9999... is equal to 1 because no matter how small a difference between .9999... and 1 you ask for, I can write enough 9s to get within that difference. So suppose you want it within .00001 of 1. I can write .99999. This can go on and on. Here is another way of looking at it. Let's say n = .9999.... This means that 10 x n = 9.9999.... Now subtract 10n - n = 9.9999... - .9999... So 9n = 9.0 n = 1 This works because that 9s keep repeating and .9999... - .9999... = 0 You can do this with any repeating decimal to find the fraction that it is equivalent to. For example, what fraction is equivalent to 2.161616...? Let n = 2.161616... 100n = 216.161616... (Notice that this time you multiply by 100. Why?) 100n - n = 216.161616... - 2.161616... (Subtract) 99n = 214 n = 214/99 Plug 214/99 into your calculator and see what you get. Calculators can go only one way, from fractions to decimals. You have to figure out how to go from decimals to fractions. -Doctor Gerald, The Math Forum Check out our web site! http://mathforum.org/dr.math/ |
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