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Comparing Numbers

. . .

which have

Rounding Errors

Math - Comparing Numbers which have Rounding Errors

Comparing Numbers - Which have Rounding Errors . . .

Comparing "Numbers" without using a Number Line:

Direct Comparison , without using a number line, has many practical

advantages because it does not require a physical drawing.

Although a number line provides a visual model which is easy to understand, it is not feasible to use it in most situations. The following limitations make a number line cumbersome and time consuming to use:

  • There is not enough space to physically plot more than a few numbers on the same line graph (number line).
  • Plotting extremely small numbers, or extremely large numbers, or numbers with completely different orders of magnitude, or numbers which are almost equal may each require dramatically different labeling of scale marks. In addition, each situation may require completely different intervals marked on the number line.

Direct Comparison eliminates the problems listed above. Direct comparison does not require a physical drawing.

Real Numbers are sequential. By comparing place values, any two numbers can be compared directly - once the numbers are written in the same standard notation.

Math - Comparing Numbers which have Rounding Errors Rounded Numbers . . .
- Comparing numbers which have been rounded

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Note: Rounding a number makes it much more convenient to use the number.

However, rounding a number always introduces an error - called the rounding error .

The question is: Can you afford to ignore the error? Is the rounding error small, relative to the value of the rounded number?

Often the answer to that question is "yes". For example, how many people live in your country? The answer is usually rounded to millions. It makes no sense to list an exact number, down to the last individual person. The rounding error is small, relative to hundreds of millions of people.

Rounding errors accumulate . Do not use rounded numbers in calculations because the rounding errors compound and accumulate. Round the final result, not the numbers used in intermediate calculations.

(Have you ever had a bank, or a grocery store, round their intermediate calculations in such a way that you lose money?)

To guard against the inaccuracies introduced by accumulated rounding errors, guard digits should be used. This is the opposite of rounding. Guard digits are extra digits which are included in calculations to compensate for possible rounding errors. For example, if calculations are to be carried out to the hundredths place, use the thousands place or beyond in your intermediate calculations.

Example : Multiply the following three numbers together: 2.301, 1.046, and 3.111 . Round the final result to the nearest hundredth.

Round Numbers before multiplication :

(2.30)(1.05)(3.11) = 7.51

DO NOT Round Numbers before multiplication :

(2.301)(1.046)(3.111) = 7.49

Final Result : The calculated results are different due to the extra three rounding errors introduced when the numbers were rounded before being multiplied.