Question

(Rounding) Write a program that prints the value 100.453627 rounded to the nearest digit, tenth, hundredth, thousandth and ten-thousandth.

Short answer

100.453627 rounded to the nearest digit is 100, to the nearest tenth is 100.5, to the nearest hundredth is 100.45, to the nearest thousandth is 100.454, and to the nearest ten-thousandth is 100.4536.

Step-by-step solution

Understanding Rounding Concepts

Before starting to write the program, understand the concept of rounding to different places. Rounding to the nearest digit means rounding to the nearest whole number. Rounding to the tenth means rounding to one decimal place. Rounding to the hundredth means two decimal places, to the thousandth means three decimal places, and to the ten-thousandth means four decimal places.

Initialize the Value

Start by defining the floating-point value that needs to be rounded. For this program, initialize a variable with the value 100.453627.

Round to Nearest Digit

Use a rounding function to round the number to the nearest whole number (no decimal places). Most programming languages provide a built-in function for this; for example, in Python, you would use round(value).

Round to Nearest Tenth

Round the value to one decimal place by specifying in the rounding function that you want to keep only one digit after the decimal point.

Round to Nearest Hundredth

Similarly, to round to the nearest hundredth, specify two decimal places in the rounding function.

Round to Nearest Thousandth

To round to the nearest thousandth, specify three decimal places in the rounding function.

Round to Nearest Ten-Thousandth

Finally, for rounding to the nearest ten-thousandth, specify four decimal places in the rounding function.

Printing the Results

Print the results of each rounding step separately to display the value rounded to the nearest digit, tenth, hundredth, thousandth, and ten-thousandth.

Key concepts

Floating-Point Arithmetic

In programming, floating-point arithmetic is essential for dealing with real numbers that have a fractional part. When you encounter a number like 100.453627, it's represented in the computer's memory as a floating-point number. This representation allows the storage and manipulation of a wide range of values with varying levels of precision.

Floating-point numbers use a binary equivalent of scientific notation, which has two parts: the significand (or mantissa) and the exponent. The significand represents the main digits of the number, while the exponent scales it, representing how many places to move the decimal point.

One of the challenges with floating-point numbers is that they cannot always precisely represent all real numbers. As a result, rounding errors can occur, and that's why rounding functions are vital for ensuring numbers are handled accurately, especially when presenting results or conducting financial transactions where precision is critical.

Rounding Functions

Rounding functions are tools in programming that allow you to modify the precision of floating-point numbers. They help in approximating a number to a specified number of decimal places or to the nearest integer.

Most programming languages come with built-in rounding functions. For example, the round() function in Python takes two arguments: the number you're rounding, and an optional second argument that specifies the number of decimal places to round to. If the second argument is omitted, the number is rounded to the nearest integer.

Different functions handle rounding in various ways. Some functions round up, some round down, and others may use round half up or round half even (also known as banker's rounding), where numbers that are equidistant between the two closest representations are rounded to the nearest even number. Selecting the appropriate rounding function depends on your specific use case and the requirements of the task at hand.

Decimal Places

Understanding decimal places is crucial when working with numbers in programming. Decimal places refer to the number of digits to the right of the decimal point in a number. For instance, in 100.453627, there are six decimal places.

Specifying the number of decimal places is important when you want to control the accuracy and precision of your calculations. For example, different applications may require rounding to varying degrees of precision — a currency calculation would typically require two decimal places, representing cents, whereas a scientific calculation might require many more.

When rounding to a specific number of decimal places, you effectively set a threshold for how detailed the number should be. By using a combination of rounding functions and an understanding of decimal places, you can control the output of your program to match your needs, whether that's rounding to the nearest whole number, to two places for currency, or to any other level of precision required by your application.