Rounding the Number 1.414863894... to Three Decimal Places
Rounding the Number 1.414863894... to Three Decimal Places
The number 1.414863894... is an example of an irrational number, meaning it has a non-repeating, non-terminating decimal expansion. However, we can round it to a desired number of decimal places, such as three decimal places. This article will explore the process of rounding this number and discuss the general rules and techniques involved.
Understanding the Rounding Process
To round a number to a specific number of decimal places, you follow a set of straightforward steps. One such example is rounding 1.414863894... to three decimal places.
Step-by-Step Explanation
Identify the digit in the desired place: The number 1.414863894... has 1 in the tenths place, 4 in the hundredths place, and 1 in the thousandths place. This means the number to be rounded to three decimal places is 1.414. Look at the next digit: To determine how to round the thousandths place, look at the next digit (the ten-thousandths place). Here, the digit in the ten-thousandths place is 8. Apply the rounding rule: According to the rule, if the next digit is 5 or greater, round up to the next higher number. If it is less than 5, do not change the digit in the place you are rounding to. Round the number: Since the next digit (8) is greater than 5, we round up the thousandths place from 4 to 5. The result is 1.415.Generalizing the Rounding Rule
The general rule for rounding to the (n)th decimal place is as follows:
Identify the (n)th digit. In our case, we identified the third decimal place (4) as the digit to be rounded. Identify the (n 1)th digit. The fourth decimal place was 8, which was the digit to be examined. Apply the rounding rule: Since the fourth digit (8) is greater than 5, we increased the third decimal place by 1, changing 4 to 5. Result: Rounding 1.414863894... to three decimal places gives us 1.415.Accuracy and Precision
Let's compare the rounded number with the original number to see how close they are. The original number 1.414863894... minus its rounded form 1.414 results in 0.000863. The next rounded number, 1.415, minus the original number 1.414863894... results in 0.000137.
Comparing these differences:
0.000863 and 0.000137 show that 1.415 is a closer approximation compared to 1.414.Since 1.415 is closer to the original number, it is the more accurate rounded form.
Conclusion
Rounding the number 1.414863894... to three decimal places results in 1.415. This is because the fourth decimal place is 8, which is greater than 5. According to rounding rules, we round up the third decimal place to 5. The process can be applied to any irrational or rational number with a non-terminating, non-repeating decimal expansion. This method ensures accuracy and simplification in calculations.