Another consideration in rounding is when you are required to round to "an appropriate number of significant digits". What are significant digits? Well, they're sort of the "interesting" or "important" digits. (They're sometimes also called "significant figures".) They are the digits which give us useful information about the accuracy of a measurement. (We'll get to the topic of "appropriateness" later.)
Here are some examples of how to count the number of significant digits:
3.14159 has six significant digits. That is to say, all the numerals ("digits") give us useful information.
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1000 has one significant digit: only the 1 is interesting (only it tells us anything specific); we don't know anything for sure about the hundreds, tens, or units places; the zeroes may just be placeholders; they may have rounded something off to get this value.
1000.0 has five significant digits: the ".0" tells us something interesting about the presumed accuracy of the measurement being made; namly, that the measurement is accurate to the tenths place, but that there happen to be zero tenths.
0.00035 has two significant digits: only the 3 and 5 tell us something; the other zeroes are placeholders, only providing information about relative size.
0.000350 has three significant digits: the last zero tells us that the measurement was made accurate to that last digit, which just happened to have a value of zero.
1006 has four significant digits: the 1 and 6 are interesting, and we have to count the zeroes, because they're between the two interesting numbers.
560 has two significant digits: the last zero is just a placeholder.
560. : notice that "point" after the zero! This has three significant digits, because the decimal point tells us that the measurement was made to the nearest unit, so the zero is not just a placeholder.
560.0 has four significant digits: the zero in the tenths place means that the measurement was made accurate to the tenths place, and that there just happen to be zero tenths; the 5 and 6 give useful information, and the other zero is between significant digits, and must therefore also be counted.
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If you need to express your answer as being "accurate to" a certain place, here's how the language works with the above examples:
3.14159 is accurate to the hundred-thousandths place
1000 is accurate to the thousands place
1000.0 is accurate to the tenths place
0.00035 is accurate to the hundred-thousandths place
0.000350 is accurate to the millionths place (note the extra zero)
1006 is accurate to the units place
560 is accurate to the tens place
560. is accurate to the units place (note the decimal point)
560.0 is accurate to the tenths place
Here are the basic rules for significant digits:
1) All nonzero digits are significant.
2) All zeroes between significant digits are significant.
3) All zeroes which are both to the right of the decimal point and to the right of all non-zero significant digits are themselves significant.
Here are some rounding examples; each number is rounded to four, three, and two significant digits.
To do my rounding, I have to start with the first significant digit, which is the 7. Then I count to the right from there. The first four significant digits of 742,396 are the 7, the 4, the 2, and the 3. Just to the right of the 3 is a 9. Because this value is "5 or greater", I have to round the 3 up to 4. I replace the remaining digits (the 9 and the 6) with zeroes. Then:
742,400 (four significant digits)
To round 742,396 to three places, I start again with the 7 and include the next two digits, being the 4 and the 2. Since the next digit is a 3, which is "less than 5", I leave the 2 alone; I don't round up. I replace the three digits after the comma with zeroes. Then:
742,000 (three significant digits)
To round 742,396 to two places, I use only the first two digits, which are followed by a 2, so I don't round up. Instead, I just replace the final four digits with zeroes, to get:
740,000 (two significant digits)
To round 0.07284 to four significant digits, I start with the first significant digit, which is the 7. (The zero between the decimal point and the 7 is not significant, as it serves only to "place" the 7 into the hundreds place. It provides no information about the accuracy of the following digits.) There are only three more digits, so all of them will be included in my answer. Since no digit follows the 4, there is no information about rounding, so I'll just leave the 4 as it is.
0.07284 (four sig-digs)
When rounding 0.07284 to three sig-digs, the final sig-dig is the 8, which is followed by the 4. Since 4 is less than 5, I won't round up. Because these sig-digs are after the decimal point, I will not replace the 4 with a zero. If I tacked a zero after the 8, I would be adding an improper digit, as it would be "significant" (according to the rules) but wrong (according to the original number they gave me). Instead, I drop that last digit entirely. Then:
0.0728 (three sig-digs)
To round 0.07284 to two sig-digs, I use the 7 and the 2. Since the 2 is followed by an 8, I'll round the 2 up to 3; I'll drop everything that follows.
0.073 (two sig-digs)
The first significant digit in 231.45 is the 2. The next three digits are 3, 1, and 4. Since the 4 is followed by a 5, I'll round the 4 up to 5. Because the original 5 came after the decimal point, I'll drop that digit (and place) from my answer.
231.5 (four sig-digs)
To round 231.45 to two significant digits, I'll only use the first three digits; the 2, the 3, and the 1. Because the 1 is followed by a 4, I won't round the 4 up. I'll drop everything after the decimal point. I'll also drop the decimal point itself, since my final sig-dig isn't a zero. (Had the value in the ones place been a zero, I'd have needed the decimal point to make clear that the zero was significant.) So I get:
231 (three sig-digs)
To round 231.45 to two sig-digs, I'll use only the first two digits; the 2 and the 3. Because the 3 is followed by a 1, I won't be rounding up. I will need to replace the 1 with a zero, to keep the number from collapsing to just "23", which obviously would be wrong.
230 (two sig-digs)