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# Truncation and Rounding

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Truncation and RoundingTruncation:Truncation is the process in which number of digits after the decimal point is reduced to the desired level by removing the digits with least significance.This is required while converting fractions into decimals.

Some examples are provided below:Examples of truncation:Convert the following fractions into decimals and truncate these to three decimal places:1) 3/7              2) 5/9               3) 4/11            4 5/121) 3/7 = 0.42857…………Truncation up to three decimal places makes                       3/7 = 0.428As the digits from third place onwards are dropped.Similarly,2) 5/9 = 0.

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55555………. = 0.556Here the third digit has been rounded as well, which is a must to do while truncating.

Rounding off will be discussed in the subsequent section.

3) 4/11 = 0.363636…….. = 0.

3644) 5/12 = 0.41666666….. = 0.

417Rounding:Rounding is the process to account for truncating digits from a certain place onwards. There are certain rules for rounding off. These rules are:1) If a digit being dropped during truncation is more than 5 (i.

e.

6 to 9); then the last digit remaining after truncation is to be increased by 1.2) If a digit being dropped during truncation is less than 5 (i.e. 0 to 4); then the last digit remaining after truncation is kept unchanged.

3) If a digit being dropped during truncation is 5; then the last digit remaining after truncation is to be increased by 1 if it is n odd integer.Some examples are provided below:Truncate the following decimals to three decimal places by incorporating suitable rounding.1)      0.5657215….

. = 0.5662)      0.5654895….

. = 0.5653)      0.5655219….

= 0.5664)      0.5645892….  = 0.

564Applications:Truncation helps in making calculations easy. It is obvious as it is easier to carry a number with fewer digits after decimal than those having more digits. However, this causes truncation error.It is essential to truncate a decimal to the maximum number of allowed significant digits in case of calculations involving physical quantities.

Rounding is essential to account for the digits being dropped in the process of truncation. This helps in minimizing truncation error in the calculations involving large number of multiplication, division etc and each of which requires truncation. Thus applications of rounding is in making computations easy and reducing the truncation error.Truncation and rounding in real life applications (for grade 5-9)The important issue is how to determine as to truncation should be done up to how many decimal places.

The rule is that for measured quantities the measurement can be done only for certain number of significant digits and this number is guided by the least count on the measuring instrument. Like in case of length measurement by a measuring scale having millimeter markings one can do measurements in mm up to one decimal place and the last value will be most dubious or least significant. There is no meaning is putting any more digits right of it.Now let us consider an experiment in which you are given a wooden piece and asked to measure its length using a measuring scale having mm markings.

Your value comes to be 35.3 mm. You can be sure about 5 but not about 3 as there was no marking suggesting 0.3 on this scale.

So to be sure and accurate you gave the exercise to 4 more friends of your and asked them to report the length of the piece to you. They reported the following values:35.2 mm, 35.3 mm, 35.

4 mm and 35.2 mm and you have one more value measure by yourself i.e. 35.

3 mm.Naturally you will like to go fro the average length as an accurate measure of the length. The average length comes out to be:l = (35.2 + 35.

3 + 35.4 + 35.2 + 35.3)/5 = 35.

28 mmBut 5.28 mm cannot be accepted as a measuring scale cannot measure any better than one decimal place. Therefore, the resulting value must be truncated to one decimal place by suitable rounding and the value turns out to be 35.3 mm.

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