In significant figures whle rounding up if right most digit to be removed is 5 if even then no chenge why increase by 1 when its odd
Answers
Step-by-step explanation:
Now that "everyone" has a calculator that will give a result to six or eight (or more) figures, it is important that we know how to round the answer off correctly. The typical rule taught is that you round up with five or more and round down with four or less.
THIS RULE IS WRONG!
However, please do not rush off to your elementary school teacher and read 'em the riot act!
The problem lies in rounding "up" (increasing) the number that is followed by a 5. For example, numbers like 3.65 or 3.75, where you are to round off to the nearest tenth.
OK, let's see if I can explain this. When you round off, you change the value of the number, except if you round off a zero. Following the old rules, you can round a number down in value four times (rounding with one, two, three, four) compared to rounding it upwards five times (five, six, seven, eight, nine). Remember that "rounding off" a zero does not change the value of the number being rounded off.
Suppose you had a very large sample of numbers to round off. On average you would be changing values in the sample downwards 4/9ths of the time, compared to changing values in the sample upward 5/9ths of the time.
This means the average of the values AFTER rounding off would be greater than the average of the values BEFORE rounding.
This is not acceptable.
We can correct for this problem by changing the rule for rounding 5 rounding "off" (keeping the number the same) in fifty percent of the roundings-even numbers followed by a 5. Then, on average, the roundings "off" will cancel out the roundings "up."
The following rules dictate the manner in which numbers are to be rounded to the number of figures indicated. The first two rules are more-or-less the old ones. Rule three is the change in the old way.
When rounding, you examine the digit following (i.e., to the right of) the digit that is to be the last digit in the rounded off number. The digit you are examining is the first digit to be dropped.
If that first digit to be dropped is less than 5 (that is, 1, 2, 3 or 4), drop it and all the digits to the right of it.
If that first digit to be dropped is more than 5 (that is, 6, 7, 8 or 9), increase by 1 the number to be rounded, that is, the preceeding figure (to the digit being dropped).
If that first digit to be dropped is 5, round the digit that is to rounded off so that it will be even. Keep in mind that zero is considered to be even when rounding off.
The rules above are a bit technical, so here are some examples.
Example #1 - Suppose you wish to round 62.5347 to four significant figures. Look at the fifth digit. It is a 4, a number less than 5. Therefore, you will simply drop every digit after the fourth, and the original number rounds off to 62.53. (rule #1 above)
Example #2 - Round 3.78721 to three significant figures. Look at the fourth digit. It is 7, a number greater than 5, so you round the original number up to 3.79. (rule #2 above)
Example #3 - Round 726.835 to five significant figures. To do this, you must look at the sixth digit. It is a 5, so now you must look at the fifth digit also. That is a 3, which is an odd number, so you round the original number up to 726.84. (rule #3 above)
Example #4 - Round 24.8514 to three significant figures. Look at the fourth digit. It is a 5, so now you must also look at the third digit. It is 8, an even number, so you simply drop the 5 and the figures that follow it. The original number becomes 24.8. (rule #3 above)
Here are some more examples for rule #3.
Example #5 - Round 23.55 to the 0.1 place. To do this, you must look at the hundreths place (remember, we are going to keep the tenths place in our answer). It is a five, so now we look at the next digit inward (the tenth place) and see it is a five, an odd number. Since we are rounding off a 5 (in the hundreths place), we must round to an even number. The answer is 23.6.
Example #6 - Round 23.65 to the 0.1 place. To do this, you must look at the hundreths place (remember, we are going to keep the tenths place in our answer). It is a five, so now we look at the next digit inward (the tenth place) and see it is a six, an even number. Since we are rounding off a 5 (in the hundreths place), we must round to an even number.