Quod Erat Demonstrandum

2008/09/04

F.2 Mathematics: significant figures

Filed under: Junior Form Mathematics — johnmayhk @ 10:30 下午

F.2 students, please fill in the following blanks.

(a) -0.0012000 has __________ significant figures.

(b) 12000 has __________ significant figures.

Answer

(a) -0.0012000 has 5 significant figures.

The leftmost zeros are not significant (important).
The rightmost zeros are significant.
It is because, the rightmost zeros tell us how accurate the number is.
If -0.0012000 is an approximation, it will be ‘more accurate’ than -0.00120 (say).
Why?
Let me discuss a bit further.
If -0.0012000 is an approximation, then what are possible values of the corresponding true value?
The true value should lie between -0.00120005 and -0.00119995, hence the maximum absolute error is 0.00000005.
For -0.00120, the corresponding true value should lie between -0.001205 and -0.001195, hence the maximum absolute error is 0.000005.
By comparing the maximum absolute error, we may draw the conclusion that -0.0012000 is more accurate than -0.00120 when it comes to expressing an approximate value of the corresponding true value.
That’s why I mentioned that ‘the rightmost zeros tell us how accurate the number is’ and hence the rightmost zeros are significant.

May be F.2 students did not know what I was writing above, doesn’t matter, I just give a poor but efficient teaching strategy: 記口訣

「左零唔數右零數。」

(b) Actually, we don’t know how many significant figures for 12000.

If 12000 is an exact number, then 12000 has 5 significant figures.
If 12000 is the result of correcting the corresponding true value to the nearest ten, then then 12000 has 4 significant figures.
If 12000 is the result of correcting the corresponding true value to the nearest hundred, then then 12000 has 3 significant figures.
If 12000 is the result of correcting the corresponding true value to the nearest thousand, then then 12000 has 2 significant figures.

Can 12000 be a number having 1 significant figure?

No! There are 2 non-zero digits!

Finally, we may write

12000 has at least 2 significant figures.

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