Question
Vishal estimated a number as 2000 when he rounded
the number to 1, 2 or 3 significant figures. Ethan was shocked by Vishal's
answer and he asked Vishal what is the smallest and largest possible number
such that the estimated number is 2000. Can you help Vishal answer Ethan?

Introduction
This is
taken from ShingLee’s Secondary 1 Mathematics (7th edition) textbook. It is a good question to test if students really understand the meaning
of significant figures! If you have not
done so, please read my article Significant Figures Made Easy.
To round
certain number to n significant figures:
Step 1: Identify
the first nonzero digit from the left.
Step 2: From this
digit, start highlighting n
significant figures.
Step 3: Look at the
digit just after the last highlighted digit.
Round up if
this is 5 and above, otherwise do nothing.
Always remember to include the placeholder ‘0’s.

Solution
This
question requires one to think backwards.
You are given the number
2000. If it is correct to 1
significant figures, then what are the possible numbers that round to
it? If it is correct to 2
significant figures, then what possible numbers round to it? If it is correct to 3
significant figures, what are the possible numbers round to that? The question is not specific about what sort
of “number” they meant, so I shall assume that they refer to real numbers.
Technical Remark
In case you are wondering, for
example, why in the answer for 2
significant figures i.e. the interval
1 950 < x < 2 050, the x cannot be made equal 2 050, this is because it would be rounded to 2 100. A number like 2 049.9999999 (seven ‘9’s after the decimal point) will be rounded down to 2 000. Even if there were one trillion ‘9’s after the decimal point, as long as the number of decimals terminates (i.e. is finite), it will still be rounded down to 2 000.
1 950 < x < 2 050, the x cannot be made equal 2 050, this is because it would be rounded to 2 100. A number like 2 049.9999999 (seven ‘9’s after the decimal point) will be rounded down to 2 000. Even if there were one trillion ‘9’s after the decimal point, as long as the number of decimals terminates (i.e. is finite), it will still be rounded down to 2 000.
Once a number touches 2 050
exactly, it will be rounded up to
2 100 for 2
significant figures. So
technically, there is no largest number, because you can keep on adding more
and more 9’s behind as long as it is a finite number. But you cannot add an infinite number of 9’s
because that would touch 2 050.
2 050 is not the largest number, as it is not among the possible numbers, but 1 950 is the lowest number, because it is among all the possible numbers. Because there is actually no largest number, I used the word “largest” in quotation marks. [ Optional: The interested reader may want to check up infimum and supremum ] These are some very fine details that the textbook authors and editors forgot. Even if the editor has a PhD, the PhD could be in “mathematics education” but not “mathematics”, and may have forgotten the rigours of the real numbers system. The answers given at the back of the book are not only wrong, one pair of answers was missing. This caused some confusion among parents and students. I suspect they employed a nonprofessional parttimer to do up the answers in order to save costs.
2 050 is not the largest number, as it is not among the possible numbers, but 1 950 is the lowest number, because it is among all the possible numbers. Because there is actually no largest number, I used the word “largest” in quotation marks. [ Optional: The interested reader may want to check up infimum and supremum ] These are some very fine details that the textbook authors and editors forgot. Even if the editor has a PhD, the PhD could be in “mathematics education” but not “mathematics”, and may have forgotten the rigours of the real numbers system. The answers given at the back of the book are not only wrong, one pair of answers was missing. This caused some confusion among parents and students. I suspect they employed a nonprofessional parttimer to do up the answers in order to save costs.
Suitable Levels
* Lower Secondary Mathematics
(usually secondary 1 ~ equivalent to about grade 7)
* other syllabuses that involve estimation
and approximation
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