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What is rounding?
Rounding is a method of changing a given number by dropping one or more digits. The method gives a less accurate number, but one that might be easier to calculate with or use in estimation. Rounding produces an approximation.
For example, multiplying 5.3179 and 7.9824 would be difficult to do without a calculator. By rounding 5.3179 to 5 and rounding 7.9824 to 8, the approximate answer, 5 X 8 = 40 is a close guess for the actual answer: 42.44960496.
Rounding starts by selecting a multiple of ten to round to. The number, 465.254 could be rounded to the nearest,
Each of these uses the method of keeping the decimal place we selected by dropping all the digits less than that multiple of ten. The digits left are either increased or kept, depending on the dropped digits.
Common rounding drops 4, 3, 2, 1 and 0 without changing the digit we selected to round to, and increasing the digit we are rounding to by one if the digit dropped is a 5,6,7,8 or 9. With the common method, 64 would round to 60, and 65 would round to 70.
How is rounding used in the real world?
Rounding is inherent in most measurements. A tape measure may be marked at every 1/16th of an inch, but when a measurement is taken, it is unlikely that whatever is being measured will fall exactly on one of the marked lines. By estimating the closest mark, we are actually rounding to the nearest sixteenth inch.
This is true for measurement of weight, volume and prices as well. Whenever you purchase something by weight or volume, there is a hidden rounding that occurs so that the price you pay ends up as an even number of pennies, instead of some fraction of a cent.
A basic problem using rounding.
Fencing is available in 100 foot lengths (100ft per roll). How many rolls would be needed to fence a square area that is 62.154 feet on a side? (Round first to the nearest ones, then round your answer to the nearest 100.)
Rounding 62.154 to the nearest one gives 62. 62 X 4 (four sides to a square) = 248. Since you can only buy 100 foot rolls, you need 3. Note that this last rounding, if done with the common method would have given 2 rolls, but this would have left 48 more feet of fencing still needed. This method is called 'rounding up' or 'rounding to a ceiling'. It is used in real world situations when having more than enough is better than not having enough.
Who invented rounding?
Rounding as a method of estimation has been used since pre-history. However, it was only in the age of computers that the subject received serious attention. Because the number of digits a computer handles is limited (by available memory) and the number of calculations can be huge, rounding methods in computer science are an important subject.
In 1940, a standard for rounding numbers where 5 was the digit to be dropped was published. The problem was that a number like 62.5 was exactly between 62 and 63. With the common method, any number ending with a 5 would be rounded upwards and this made the estimates slightly higher - significant when there was a lot of data being used.
The 1940 rule, called 'unbiased rounding' says when this happens, round to the nearest even number, either up or down. So, 62.5 would then round to 62, but 63.5 would round to 64.
This new method gives a balancing effect and eliminates the bias.
An interesting fact about rounding.
In business, rounding can have a real impact. Suppose you purchase a hundred widgets at 1.225 cents a widget, your final cost is rounded up to $1.23 . However, if you make ten purchases of ten widgets each (total bought is still 100), you only pay, because of rounding, $0.12 for each set of ten, for a total of $1.20.
This might not seem like much (three cents saved) but over time and a large number of purchases, this adds up.