Approximations
Sometimes it is useful to use an approximation of a number--that is, a
number which is close to the actual number but easier to think about and to use
in computations. For example, when shopping for a new car, it may not be
important that one car costs exactly $32,945.05 while another costs exactly
$41,234.78; we are only concerned with the fact that one costs approximately
$33,000 while the other costs approximately $41,000. An approximation has
fewer significant figures than the
actual number--you will learn more about significant figures.
For now, just think of an approximation as a number that is close to the given number.
Rounding
When an approximation, rather than an exact number, is required, we use a
technique called rounding. We can round numbers to the nearest hundred,
ten, one, tenth, hundredth, etc. Rounding gives the closest number to the
original whose last non-zero digit is in the specified place. To round a number
to a certain place--the hundreds place, for example--look at the numeral
directly to the right of that place. If the numeral is 0, 1, 2, 3, or 4,
then leave the numeral in the hundreds place (or in whichever place we are
rounding) as it is and change all the numerals to the right of it to zeros. If
the numeral is 5, 6, 7, 8, or 9, then add 1 to the numeral in the hundreds place
and change all the numerals to the right of it to zeros. For example, 1,923
rounds to 1,900 and 679 rounds to 700. 1,900 is the closest number to 1,923
which contains 0's in the tens and ones places, and 700 is the closest number
to 679 which contains 0's in the tens and ones places.
If the number is a decimal, drop all the zeros after the decimal point after
rounding. For example, 63.789 rounds in the tenths place to 63.8.
Rounding decimals is just like rounding whole numbers, with one key difference:
when rounding to a place to the left of the decimal point, change all the
numerals to the right of the rounding place but to the left of the decimal point
to zeros, and drop all the numerals to the right of the decimal point. When
rounding to a place to the right of the decimal point, drop all the numerals to
the right of the rounding place instead of changing them to zeros. Thus,
567.235 rounded to the nearest ten is 570, and 567.235 rounded to the nearest
tenth is 567.2.
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Example 1. Round 7,803 to the nearest hundred.
Since there is a "0" in the tens place, we leave the "8" as is. The answer is
7,800.
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Example 2. Round 4,019,576 to the nearest thousand.
Since there is a "5" in the hundreds place, we add 1 to the "9" in the thousands
place. The answer is 4, 019, 000 + 1, 000 = 4, 020, 000.
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Example 3. Round 63.52 to the nearest one.
Since there is a "5" in the tenths place, we add 1 to the "3" in the ones place.
The answer is 63 + 1 = 64.
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Example 4. Round 7,802.45123 to the nearest thousandth.
Since there is a "2" in the ten thousandths place, we leave the "1" as is. The
answer is 7,802.451.
