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I was taught that there are two different methods for obtaining results for multiplication/division or addition/subtraction with decimals. For multiplication/division the result will have the least amount of significant figures of the multiplicands or the dividends. For addition/subtraction, the addend or the number being subtracted with the least amount of decimal places will be the amount of decimal places for the result. But what if both multiplication AND addition is being used? I guess I haven't gotten up to that yet. If anyone can help I'd appreciate it, thanks!

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Please don't use abbreviations like that, and especially not in the title. Why put all those people who will be reading your question to all the trouble of figuring out what that means when you can spell it out with only a handful of keystrokes?! –  joriki Sep 25 '11 at 12:53
    
title fixed ... –  GEdgar Sep 25 '11 at 13:07
    
Well I thought of all people, the ones on this stackexchange network would know! I mean look at all the other titles! I have no freaking idea what they mean at all!! –  David Sep 25 '11 at 13:16
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@David A good title uses standard mathematical vocabulary appropriate for the level of the question, so even if you don't know what a 'vertex' and a 'polytope' are (to pick a random example from the front page) an expert in mathematics will. But even an expert in mathematics may not know what 'sig figs' are if they're not a native English speaker (and perhaps even if they are!) If you're unsure, don't abbreviate. No damage done though! –  Chris Taylor Sep 25 '11 at 13:42
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1 Answer

up vote 2 down vote accepted

I'll answer your question by example. We'll work with three numbers

$$a = 52.4$$ $$b = 0.96$$ $$c = 2.193$$

Here $a$ has three significant figures and one decimal place, $b$ has two significant figures and two decimal places, and $c$ has four significant figures and three decimal places.

If we multiply, the number of significant figures of the result is the same as the multiplicand with the least number of significant figures, so although we have $a \times c = 114.9132$ if we keep all the digits of the result, one of the multiplicands only has three significant figures, so you would write

$$a\times c = 115$$

If you are adding, the number of decimal places of the result is the same as the addend with the least number of decimal places, so although we have $a+c = 54.593$ if we keep all the digits, one of the addends only has one decimal place, so you would write

$$a + c = 54.6$$

If you are adding and subtracting in the same calculation you apply the rules sequentially. So to compute $b + a\times c$, the result of $a\times c$ has three significant figures, and no decimal places. Then when adding $b$, one of the numbers in the sum has two decimal places and the other has none, so the result will have zero decimal places of accuracy. You perform the calculation keeping all the digits and round at the end. Keeping all the digits we have $b + a\times c = 115.8732$. Now rounding that result to zero decimal places gives

$$b + a\times c = 116$$

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This is brilliant! And one question? Multiplication has higher precedence, so that is why you did it before the addition, right? –  David Sep 25 '11 at 14:00
    
Yes, that's the convention - multiplication before addition. –  Chris Taylor Sep 25 '11 at 14:02
    
I figured it was that. I thought it was just in programming but i guess i was wrong. –  David Sep 25 '11 at 14:03
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Mathematics is different to coding in that in a given programming language, the expression a * b + c has a definite value, and normally the multiplication has higher precedence (though not in all languages! For example, q has right-to-left semantics, so that expression would actually return $a\times(b+c)$) whereas in mathematics it's just a convention that you do the multiplication first, and if there's any chance of ambiguity at all you should use parentheses. –  Chris Taylor Sep 25 '11 at 14:09
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(cont.d) A slash for division can be used in mathematics too, but its precedence is different from those of programming language -- it binds stronger than addition and subtraction but weaker than the invisible multiplication sign. So $a+b/cd$ means $a+\frac{b}{c\times d}$. And in mathematics the slash is non-associative, so "$a/b/cd$" is not meaningful in math. –  Henning Makholm Sep 25 '11 at 14:48
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