Lining Up Decimal PointsDate: 03/03/2003 at 18:16:11 From: Rashonda Subject: Decimals Why do addition and subtraction have to be lined up by the decimals but division and multiplication don't have to be? Date: 03/04/2003 at 14:48:48 From: Doctor Ian Subject: Re: Decimals Hi Rashonda, That's a really good question! It's easy to just take it for granted, without ever stopping to think about it. Suppose I have 3 cherries, 2 apples, 3 pears and you have 4 apples, 1 pear, 3 watermelons and we want to know what we have together. Could we do this? 3 cherries 2 apples 3 pears 4 apples 1 pear 3 watermelons ---------- -------- ------------- 7 3 6 That doesn't make any sense at all, does it? We'd have to make sure that we're adding the same kinds of things: 3 cherries 2 apples 3 pears 4 apples 1 pear 3 watermelons ---------- -------- ------- ------------- 3 cherries 6 apples 4 pears 3 watermelons This is also what's going on when we add decimals. When we write a number like '123.4', that's really just a very compact way of writing the sum 1 * 100 + 2 * 10 + 3 * 1 + 4 * 1/10 And when we write a number like 5.432, we're just writing the sum 5 * 1 + 4 * 1/10 + 3 * 1/100 + 2 * 1/1000 Now, following the example of the fruit, can we do this? 1 * 100 2 * 10 3 * 1 4 * 1/10 5 * 1 4 * 1/10 3 * 1/100 2 * 1/1000 ------- -------- --------- -------- 6 6 6 6 Does that make any more sense than adding cherries to apples? No! So we need to line up terms where the digits have the same meaning: 1 * 100 2 * 10 3 * 1 4 * 1/10 5 * 1 4 * 1/10 3 * 1/100 2 * 1/1000 ------- ------ ----- -------- --------- ---------- 1 * 100 2 * 10 8 * 1 8 * 1/10 3 * 1/100 2 * 1/100 But how do we do this? By lining up the decimal points! So lining up the decimal points is just an easy way of making sure that we're not adding apples to pears, so to speak. Does that make sense? Okay, so much for addition and subtraction. What about multiplication and division? Let's consider the product 3.21 * 23.4 ------- Now, what we'd _really_ like is to get rid of those decimal points entirely, or at least temporarily. We can use a trick to do that. Note that 3.21 = 32.1 * 1/10 = 321 * 1/100 and 23.4 = 234 * 1/10 So 3.21 * 23.4 = (321 * 1/100) * 23.4 'Save' two decimal places = (321 * 1/100) * (234 * 1/10) 'Save' one deimal place = (321 * 234) * (1/100 * 1/10) = (321 * 234) * 1/1000 = 75114 * 1/1000 = 75.114 'Restore' three decimal places So we can just forget about the decimal points while we're doing the multiplication, so long as we know where to put the decimal point when we're done. In practice, we do this: 3.21 <-- 'save' two decimal places * 23.4 <-- 'save' one decimal place ------- 321 * 234 ------- 75114 * 1/1000 <-- 'restore' three decimal places to get 75.114 So in a sense, the answer to the question "Why don't we have to line up the decimal points when we multiply?" is that we remove them until we're done. You can't line up what isn't there. What about division? _____ 3.21 ) 23.4 We do a slightly different trick here, which is essentially the same thing we do when we find equivalent fractions: 23.4 23.4 100 ---- = ---- * --- 3.21 3.21 100 2340 = ---- 321 So now we have ________ 321 ) 2340.0 Once again, a little extra work up front lets us get rid of the decimal point in the divisor. So once again, there's nothing to line up. I hope this helps! Write back if you'd like to talk more about this, or anything else. - Doctor Ian, The Math Forum http://mathforum.org/dr.math/ |
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