[Maths Class Notes] on Rounding Methods Pdf for Exam

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.

Examples of Rounding Up Math

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”.

Examples of How to Round Numbers

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 and Adjusting Method of Exact Numbers

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.

Scientific Rounding            

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.

Rules for Rounding-Off

• 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

ASTM E29 Rounding

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.

Leading Digit Approximation

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’.

Solved Examples on Math Rounding        

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.

Fun Facts

• 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.