Problem 31 - Technical Mathematics with Calculus

Question

Round each number to one decimal place. $$13.98$$

Step-by-Step Solution

Answer

13.98 rounded to one decimal place is 14.0.

Step 1: Identify the Tens Place and the Neighboring Digit

Locate the digit in the tens place (1) and look at the neighboring digit to the right (9) to determine whether to round up or stay the same.

Step 2: Determine Whether to Round Up or Stay the Same

Since the neighboring digit (9) is 5 or greater, round up the tens place digit.

Step 3: Perform the Rounding

Increase the tens place digit by 1, resulting in 14.0, and drop all digits to the right.

Key Concepts

Mathematical Rounding

Mathematical rounding is a fundamental skill in numerous areas of mathematics and everyday calculations. It is the process of simplifying a number to make it easier to work with while still keeping it close to the original value. To round a number, you must look at the digits to the right of the desired place value to see if these digits warrant increasing the value at the place you're rounding to.

For example, when rounding the number 13.98 to one decimal place, you primarily focus on the first decimal place (9), and then consider the digit immediately to the right (8). Since the digit to the right is 5 or higher, you would round the 9 up to 10, which then increases the whole number part of our example to the next integer, resulting in a rounded value of 14.0.

This 'rounding up' helps in simplifying computations and allows for approximation that is useful in daily life applications, such as when dealing with money or measurements where exact precision is not critical.

Place Value

Understanding place value is essential to mastering the concept of rounding numbers. Every digit in a number has a specific place value, depending on its position relative to the decimal point. For whole numbers, place values go from units, tens, hundreds, and so on to the left of the decimal point. For decimals, they proceed from tenths, hundredths, thousandths, etc., to the right of the decimal point.

Take the number 13.98, for instance. The '1' holds the place value of tens, the '3' is in the units place, '9' is in the tenths place, and '8' occupies the hundredths place. Recognizing these place values helps determine which digit you should pay attention to when rounding a number. If you're asked to round to one decimal place, you know to look at the digit in the tenths place and its neighboring digit to decide on your rounding action.

Rounding Decimals

Rounding decimals can be slightly trickier than rounding whole numbers, but the fundamentals are the same. The goal is to simplify the decimal to make it shorter or to match a desired precision. When you are asked to round to a specific place value, you have to consider the digit in that place and the digit immediately following it.

In our example with the number 13.98, if we're rounding to one decimal place, we look at the first decimal place (tenths) which is '9'. The next digit ('8' in the hundredths place) influences how we adjust the '9'. Because it's 5 or greater, we round up. Rounding decimals is a crucial skill in fields that demand accuracy like science and engineering, yet also allowing for estimation when exact figures are not required.

Significant Figures

Significant figures are all about precision. They indicate how many digits in a number are meaningful in terms of contributing to its accuracy. This includes all digits except leading zeros, trailing zeros when they are merely placeholders, and any spurious digits.

When rounding to a certain number of significant figures, you need to determine which digits in your original number are significant. For the number 13.98, all digits are significant because there are no leading or trailing zeros acting as placeholders. Rounding to significant figures can affect both whole numbers and decimals, and it's particularly important in scientific measurements where the precision of numbers reflects the precision of the measurement. When numbers are rounded to significant figures, it presents data in a reliably understandable way without overestimating the actual precision of the measurements.