# 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.

(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.