Question

{ Rounding Numbers) Write a program that prints the value } 100.453627$ rounded to the nearest digit, tenth, hundredth, thousandth and ten thousandth.

Short answer

100; 100.5; 100.45; 100.454; 100.4536

Step-by-step solution

Understanding Rounding

Rounding is the process of reducing the digits in a number while keeping the value close. The concept depends on determining the place value to which you wish to round.

Rounding to the Nearest Digit

Identify the digit in the ones place (0.453627) of 100.453627, which is 0. The digit to the right of this, in the tenths place, is 4. Since 4 is less than 5, we don't round up and the number remains 100.

Rounding to the Nearest Tenth

Look at the tenths place, which is 4, and the number to its right is 5. Since 5 is equal to 5, round up the tenth place by 1. Thus, 100.453627 rounds to 100.5 at the tenth place.

Rounding to the Nearest Hundredth

Examine the hundredths place digit, which is 5. The number immediately to the right is 3. Since 3 is less than 5, the hundredths place doesn't change, rounding 100.453627 to 100.45.

Rounding to the Nearest Thousandth

The thousandths place is 3, and the next digit (2) is less than 5. Therefore, we keep the thousandths place as it is, which makes it 100.454.

Rounding to the Nearest Ten Thousandth

Now, focus on the ten-thousandths place, which is 2, while the hundred-thousandths place is 7. As 7 is more than 5, increase the 2 by 1, rounding the number to 100.4536.

Key concepts

Place Value

Place value is a fundamental concept in mathematics that helps us determine the value of a digit based on its position in a number. In a number like 100.453627, each digit has a different place value. To see how important place value is when rounding, consider each digit from the left:

• The number 1 is in the hundreds place, meaning it represents 100.
• The digit 0 sits in the tens place, making it worth 0.
• The next 0 is in the ones place, carrying no value in this context.
• Moving into decimals, 4 is in the tenths place, representing 0.4.
• The digit 5 is in the hundredths place, translating to 0.05.
• Following this, 3 is in the thousandths place, equal to 0.003.
• Finally, 2 is in the ten-thousandths place, adding 0.0002.

Recognizing these positions is critical because the value of a number changes based on which digit you round.

Mathematical Concepts

Rounding is an essential mathematical concept that simplifies numbers. It's particularly useful for making complex calculations simpler or when you don't need extreme precision. To round a number, you identify the place value you're interested in, then evaluate the digit immediately to the right.

Here's a quick guide to help you round effectively:

• If the digit next to your place value is 5 or greater, you round up the targeted digit.
• If it's less than 5, you round down and keep the targeted digit the same.

For instance, when rounding 100.453627 to the nearest tenth, you look at the tenths place, which is 4. The following digit, 5, prompts you to round the tenths place up to 5.

Precision and Accuracy

Precision and accuracy are integral when dealing with measurements or detailed calculations. While these terms are related, they address different aspects when rounding numbers.

• Precision: Refers to how detailed a measurement is, often denoted by the number of decimal places. More precise numbers have more decimal places, allowing for a finer level of detail.
• Accuracy: Describes how close a rounded value is to the true value. The more the original value is distorted by rounding, the less accurate the rounding.

Rounding can impact both precision and accuracy. For example, when you round 100.453627 to the nearest thousandth to get 100.454, the precision has been reduced but might still hold sufficient accuracy for many practical purposes. This balance is essential to understand when deciding how much detail to include in your calculations.