有效數字(如何決定答案中的有效數字?)

Question

我在美國拿化學,有以下問題;
suppose four ostrich feathers weigh 1.00lb. Assuming that each feather is equal in mass, how many milligrams does a single feather weigh?

我所解出的答案為: 0.25lb*453.6g*1000=113,400

textbook上面的答案為 1.13*10的五次方,所以是三個有效數字.

為什麼答案為三個有效數字呢?如何決定答案應該有幾個有效數字呢?

那再請問,以這題為例,怎麼判定要寫1.13 x 10^5(three sig fig)還是要寫113x 10^3(also three sig figs)?

Answer 1

有效數字的算法是從左開始往右算到第一位不準確的位數為止.往後的計算裡, 誰和不準的數字運算也是不準.一個有效數字最多只能有一位不準數.
題目中的1.00磅有3位有效數字.4,453.6和1000都是常數.計算結果:
1.00*453.68*1000/4=113420,其中3420是不準的,只能保留一位.又由於113,000容易誤解有效數字的位數,故採用scientific form 1.13x105.
A number in scientific form can't have more than 1 digit before decimal point.

Answer 2

怎麼判定要寫1.13 x 10^5(three sig fig)還是要寫113x 10^3(also three sig figs)?

>>>就是科學記號呀，普化第一章就有敎了...

Answer 3

your problem lies how you set up the equation~
the questions give 1.00lb (three sig figs), therefore, you can't just divide by 4 to give you 0.25
(or at least, you have to write 0.250 with three sig figs)
so, in order to get the correct amount of sig figs.....
1. you need to show 1.00lb / 4 = 0.250lb
"4" in this case have infinite amount of sig figs (ie. 4.000000000000000000000000000.....) because you are implying you want to divide by "exactly" 4, not 4.1 4.2 4.3 4.001...etc
2. all coversion factors (ex. grams <--> lb) should have more sig figs than the number you're plugging in
in this case, 453.6 is good because it has one more sig fig than the number plugged in (0.250)
also, unit prefixes like milli, centi, micro.. etc also have infinite number of sig figs
3. how many sig figs should you report? depends on how many steps may be involved
if you're adding or subtracting, there's a high probability you might gain or lose sig figs
if you're multiplying or dividing, you will never have more sig figs than what the question gave
in this case, only one step is required to solve, therefore you apply the multiplying rule which will give you three sig figs at the end
*it's always a good idea to check if your answer with the numbers in the question.. if there's more sig figs in your answer and you didn't do any addition.... there's a good chance you didn't get your sig figs right*
4. write in scientific notation whenever you can, esp with results like 113,400 where there are trailing zeros
(1.00 lb /4)(453.6 g/lb)(1000 mg/g) = 1.13 x 10^5
btw, it is also NOT a good idea to write 113,000
why? because the trailing zeros are ambiguous!! this answer could have 3 to 6 sig figs
and if you intended to only have 3, and your paper accidentally give a dot at the end of the last zero, your answer now has 6 sig figs
which is also why it's best to write in scientific notation whenever you can

2007-01-23 13:57:26 補充：
why 1.13 x 10^5?

very simple~ coz that's the way to write scientific notations

always " 'number' point something" where the number is any from 1 - 9

for example:
120. in scientific notation with 2 sig figs would be 1.20 x 10^2

0.13 would be 1.3 x 10^-1

2007-01-23 13:57:33 補充：
anyway.. a good way to remeber is that the number before the 10^power is always greater than 1 and less than 10
so... 0.53 x 10^2 would not be correct -----should be---> 5.3 x 10^1
and 11.2 x 10^-5 would not be correct -----should be---> 1.12 x 10^-6