Question

Approximate each number (a) rounded and (b) truncated to three decimal places. $$ 18.9526 $$

Short answer

18.953 (rounded); 18.952 (truncated)

Step-by-step solution

Identify the Significant Figures

To approximate to three decimal places, first identify the first four digits to the right of the decimal point in the given number, which are 9, 5, 2, and 6.

Rounding to Three Decimal Places

To round to three decimal places, look at the fourth decimal place digit (6). Since it is 5 or greater, increase the third decimal place digit (2) by 1. Thus, 18.9526 rounded to three decimal places is 18.953.

Truncating to Three Decimal Places

To truncate to three decimal places, simply remove all digits beyond the third decimal place without rounding. So, 18.9526 truncated to three decimal places is 18.952.

Key concepts

Significant Figures

In any number, significant figures include all the non-zero digits, any zeros between non-zero digits, and any trailing zeros that are considered to be precise. Significant figures help us understand the precision of a number.
For instance, the number 18.9526 has six significant figures.
This is because every digit in the number is meaningful in defining its precision.

Rounding

Rounding is a method used to reduce the number of digits in a number while keeping its value close to what it was originally.
When rounding to a certain decimal place, you must look at the digit immediately following the desired precision level.
For example, if you want to round the number 18.9526 to three decimal places, you look at the fourth decimal place (the digit '6').
If this digit is 5 or more, you increase the digit before it by 1.
So, 18.9526 rounded to three decimal places becomes 18.953.

Truncation

Truncation is another way of approximating a number which involves cutting off digits beyond a certain point without rounding.
When you truncate a number to three decimal places, you simply remove all digits that come after the third decimal place.
Taking our example number 18.9526, to truncate it to three decimal places, you discard everything beyond the third digit after the decimal point.
Thus, 18.9526 truncated to three decimal places results in 18.952.