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Chapter 0: Problem 31

Approximate each number (a) rounded and (b) truncated to three decimal places. $$ 28.65319 $$

Short Answer

Expert verified
Rounded: 28.653, Truncated: 28.653

Step by step solution

01

- Understanding the number

Examine the given number: 28.65319. We will round and truncate it to three decimal places.
02

- Rounding to three decimal places

To round the number to three decimal places, look at the fourth decimal place (1 in this case). Since it is below 5, round down the third decimal place. The rounded number is 28.653.
03

- Truncating to three decimal places

To truncate the number to three decimal places, simply cut off the number after the third decimal place without rounding. The truncated number is 28.653.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

rounding numbers
Rounding numbers allows us to simplify complicated numbers while maintaining a value close to the original. When rounding numbers to a specific number of decimal places, we look at the digit immediately following the last desired decimal place.

Similarly, rounding to three decimal places involves examining the fourth decimal place. If this digit is 5 or higher, increase the third decimal place by one.

If it is below 5, keep the third decimal place as it is. In our example, 28.65319 becomes 28.653 because the fourth decimal place is 1, which is less than 5.
decimal places
Decimal places refer to the positions of digits to the right of the decimal point. Each decimal place represents a power of ten. The first decimal place is the tenths place, the second is the hundredths, and the third is the thousandths.

When we round or truncate a number, deciding how many decimal places to keep is crucial. For example, stating a number rounded to three decimal places means showing three digits after the decimal point. This decision influences the precision and accuracy of our results.

In our exercise, rounding or truncating 28.65319 to three decimal places requires maintaining only the digits within these first three positions after the decimal point.
truncation
Truncation is another method to simplify numbers by directly cutting off digits after a certain decimal place without rounding them. This method often leads to a slightly less accurate representation of the original number compared to rounding.

To truncate a number to three decimal places, simply remove all digits after the third decimal place. For instance, 28.65319 truncated to three decimal places is 28.653.

While truncation can be quicker and simpler, it is important to recognize that it can sometimes lead to larger errors than rounding, especially for calculations where precision is paramount.

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