Question

Approximate each number (a) rounded and (b) truncated to three decimal places. $$ 9.9985 $$

Short answer

Rounded: 9.999, Truncated: 9.998

Step-by-step solution

- Understanding Rounding to Three Decimal Places

Rounding involves looking at the fourth decimal place. If the fourth decimal place is 5 or higher, increase the third decimal place by 1. If it's 4 or lower, keep the third decimal place as is.

- Apply Rounding Rule

For the number 9.9985, the fourth decimal place is 5. Since 5 is equal to or greater than 5, we increase the third decimal place (8) by 1. Thus, 9.9985 rounded to three decimal places is 9.999.

- Understanding Truncating to Three Decimal Places

Truncating involves simply cutting off all digits after the third decimal place without rounding up.

- Apply Truncating Rule

For the number 9.9985, if we truncate, we cut off after the third decimal place. Therefore, 9.9985 truncated to three decimal places is 9.998.

Key concepts

Rounding Numbers

Rounding is a common way to simplify numbers. It helps to make them easier to understand or use in calculations. When rounding to a specific number of decimal places, you focus on one digit beyond the last place you want to keep.
For example, if you are rounding to three decimal places, look at the fourth decimal place:

• If this fourth digit is 5 or greater, you round the third decimal place up by one.
• If the fourth digit is less than 5, the third decimal place remains the same.

In our example, we rounded 9.9985. The fourth decimal place is 5, so we increased the third decimal place by one, making it 9.999. This rule ensures that the number stays as close as possible to its original value.

Decimal Places

Decimals are numbers that have a fractional part, separated from the whole number by a decimal point. The position of each number after the decimal point is called a decimal place. The first digit after the decimal is the tenths place, the second is the hundredths, and the third is the thousandths.
When we talk about rounding or truncating to three decimal places, we are focusing on the thousandths place. For instance, in the number 9.9985:

• 9 is the whole number.
• The first decimal place (tenths) is 9.
• The second decimal place (hundredths) is 9.
• The third decimal place (thousandths) is also 9.
• The fourth decimal place (ten-thousandths) is 5.

Identifying these places is crucial for accurately rounding or truncating numbers.

Truncation in Mathematics

Truncation is another method to simplify numbers, similar to rounding, but without modifying the digits. When you truncate a number to a certain number of decimal places, you simply cut off any digits beyond that place.
For example, to truncate 9.9985 to three decimal places:

• Keep the first three digits after the decimal point.
• Remove any remaining digits without rounding up or down.

Therefore, 9.9985 truncated to three decimal places is 9.998. Truncation is useful when an exact, unrounded representation is required.
It's simpler than rounding because you only need to remove extra digits, but it might not always give the most accurate representation of the original number.