Significant digits/figures question?
I’m doing a ‘lab’ for physics, and I’ve got to express my answers three different times, one with the raw answer the calculator outputs, then in scientific notation, then scientific notation with the proper number of significant digits.
This one’s the age of the solar system in seconds:
4.6 x 10^9 x 365.2425 x 24 x 3600 = 145161979200000000
Now, how many significant digits is it supposed to have.. ? Seven because of the 365.2425, or is it just two because of the 4.6 billion? I was told that the 24 and 3600 are irrelevant because they’re exact and are whole numbers.
Here’s another example that I think is right:
1 x 365.2425 x 24 x 3600 = 31556952
3.1556952 x 10^7
3.1556950 x 10^7
(7 sig figs, because of the 365.2425).
Can anyone shed some light on this for me? Thank you!
I think you have analyzed the question correctly.
The 3 answers would be:
145161979200000000
1.451619792 x 10^17
1.5 x 10^17
Actually, this final answer has 2 significant digits, but the concept of significant digits is a little “squishy.” With the 2 significant digits here (1.5), we are saying that we know the value is between 1.45 and 1.55. That is, we know it with an accuracy of plus or minus 0.05 out of 1.5, i.e., the possible error is 0.05 / 1.5 = 3.3%.
However, the possible percentage error in our original data point (4.6) was only 0.05 / 4.6 = 1.1%. So we are understating the degree of accuracy by using only 2 significant digits in our answer. In other words, 2 significant digits of accuracy are considerably more accurate when the first digit is 4 than when it is 1 (and is greatest when the first digit is 9).
Bottom line: The number of significant digits is a measure of the potential percentage error in a value, but it is only a ROUGH measure of the percentage error.
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