Accuracy of Approximate Numbers

A52

Mathematicians, especially those pursuing pure mathematics, prefer to work with exact numbers such as π , √2 , 5/7 .

In the practical settings of measurement and computation, there is a need to approximate the actual value of a number, especially when a long or infinite decimal is involved. In order for such approximations to be of service, conventions are needed for the expression of inexact numbers.

We use the word “rounding” to describe the process of converting an exact number to an approximate form with a specified degree of accuracy. There are three main ways to round a given number:

  1. to the nearest power of 10
  2. to a certain number of significant figures
  3. to a certain number of decimal places

Rounding to the Nearest Power of 10

This kind of approximation is most often applied to integers, such as 56,742. The given power of 10 specifies the key digit, k, in the number to be rounded. We look at the digit immediately right of k and, if it is 5 or greater, we increase k by 1. All digits right of k are then reduced to zero.

Example:  Re-write 3471 to the nearest 100.

Solution:  3471 ≈ 3500 .  The original 100’s digit, 4 , is                                      rounded “up” because the original 10’s digit (7) is                           greater than 5.

Example 2:  Re-write 3471 to the nearest 10.

Solution:  3471 ≈ 3470 .  The original 10’s digit, 7 , is not changed because                  the 1’s digit is less than 5.


Significant Figure Approximation

Example:


Decimal Place Approximation