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In mathematics, the round off method refers to replacing a number with an approximate value that consists of a shorter, simpler, or clearer representation. For example, replacing ＄36.4476 with ＄36.45, the fraction 312/937 with 1/3, or the expression √3 with 1.732. rounding and approximation in maths are often done to get a value that is easier to report and communicate than the original. Rounding can also play a significant role in avoiding misleadingly precise reporting of a computed number, measurement or approximation.

A quantity which is calculated as 146,478 but is known to be accurate only to within a few hundred units is generally better stated as "about 146,500".

892.3682 becomes:

1,000 when rounding up to the nearest 1,000

900 when rounding up to the nearest 100

890 when rounding up to the nearest 10

892 when rounding up to the nearest one (1)

892.4 When rounding up to the nearest 10th

892.37 When rounding up to the nearest 100th

892.368 When rounding up to the nearest (1,000th)

Rounding of exact numbers introduces some round off errors in numerical methods in the reported result. Rounding is almost inescapable when reporting many calculations –particularly when

Dividing two numbers in integer or fixed-point arithmetic;

When calculating mathematical functions such as sines, square roots, logarithms

When using a floating-point depiction using a fixed number of significant digits.

In an array of calculations, these rounding errors usually gather, and in certain ill-conditioned cases, they may make the outcome meaningless.

Wondering about how to round scientific figures? When rounding significant figures, we apply the standard rules of rounding numbers, besides that non-significant digits to the left of the decimal are significantly replaced with 0’s (zeros). In scientific rounding, the digit rounds up if the next digit is greater than or equal to 5 whereas it rounds down if the following digit is less than 5.

If the 1st non-significant digit is less than 5, then the least significant digit will remain the same (unchanged).

If the 1st non-significant digit is greater than 5, the least significant digit is increased by 1.

If the 1st non-significant digit is 5, the least significant digit can either be increased or left unchanged

Most standards recommended for science and technology, led by ASTM E29, make use of the “five-even” rule. By this rule, for example, the value 76.50, rounded to the closest unit, becomes 76, to make the last significant digit of the outcome even. In the same manner, 89.50, rounded to the closest unit, becomes 90.

To round to the 1st (or leading) digit of the numerical expression, we would require following these rules: Consider the digit following the leading one (i.e, the 2nd digit). If the 2nd digit is 5 or larger than 5, just add 1 to the leading digit and replace the rest of the digits with 0’s. Remember that the symbol ≈ is read as 'is approximately equal to'.

Example:

Round the given 4639.75 to the closest whole number

Solution:

In such a situation, we would require to round the number off the last digit before the decimal point.

Since the next digit on its right is 7 that is greater than 5, thus we increase 9 by 1. Since

We cannot directly replace 9 by 10, so in this case, we replace 9 by 0 and make the increment in the next digit on its left by 1. Hence, the rounded value will be 4640 (and can be left that way, or as 4.640 × 103).

Note: Usually, if we consist of a number ending in a string of 9’s, then we have to repeat the last step above as many times as required.

When rounding whole numbers, if the digit is 0, 1, 2, 3, or 4, do not create any changes in the rounding digit. All digits that are on the right of the requested rounding digit become 0.

If the digit is 5, 6, 7, 8, or 9, the rounding digit rounds by one number. All digits that are on the right of the requested rounding digit becomes 0.

FAQ (Frequently Asked Questions)

Q1. What are the Rules for Rounding Whole Numbers?

Answer: When performing rounding numbers, you should understand the term "rounding digit." When working with whole numbers and rounding to the nearest 10, the rounding digit will be the 2^{nd} number from the right—or the 10's place. When rounding to the closest 100, the 3^{rd} place from the right is the rounding digit—or the 100's place.

First, identify what your rounding digit is and then look to the digit on the right side of the expression.

Q2. What are the Rules for Decimal Numbers?

Answer: First identify what your rounding digit is and then look to the right side of it.

If that digit is 4, 3, 2, or 1, just drop all digits to the right-hand side.

If that digit is 5, 6, 7, 8, or 9, simply add one to the rounding digit and drop all digits to the right side of it.

Q3. What is Meant by Bankers Rounding Algorithm?

Answer: Some teachers prefer an alternative method to round decimal numbers, sometimes known as the "Banker's Rule," that offers more accuracy. When the 1^{st} digit dropped is 5 and there are no digits subsequently or the digits next are 0’s (zeros), make the preceding digit even (i.e., round off to the closest even digit). Using this rule, 4.315 and 4.325 both round to 4.32— rather than 4.325 roundings up to 4.33—when rounded off to the closest 100^{th}. The rationale behind the 3^{rd} rule is that almost half of the time the number will be rounded up and for the other half it will be rounded down.