Question

Function floor can be used to round a number to a specific decimal place. The statement y=floor( x * 10 + .5 ) / 10; rounds x to the tenths position (the first position to the right of the decimal point). The statement y=floor( x * 100 + .5 ) / 100; rounds x to the hundredths position (the second position to the right of the decimal point). Write a program that defines four functions to round a number x in various ways: a) roundToInteger( number ) b) roundToTenths( number ) c) roundToHundredths( number ) d) roundToThousandths( number ) For each value read, your program should print the original value, the number rounded to the nearest integer, the number rounded to the nearest tenth, the number rounded to the nearest hundredth and the number rounded to the nearest thousandth.

Short answer

The program defines four functions to round a number x to the nearest integer, tenth, hundredth, and thousandth place, respectively, and outputs the original and rounded values.

Step-by-step solution

Define the roundToInteger function

This function uses the floor function to round a number to the nearest integer. The mathematical operation is as follows: \( y = \text{floor}(x + 0.5) \). This rounds the number to the nearest integer by adding 0.5 and then using the floor function to eliminate all decimal places.

Define the roundToTenths function

This function uses the same principle as the integer rounding function but adapts it to round to one decimal place: \( y = \frac{\text{floor}(x * 10 + 0.5)}{10} \). By multiplying by 10 and adding 0.5 before applying floor, we round off to the nearest tenth, and then dividing by 10 shifts the decimal place back.

Define the roundToHundredths function

To round to two decimal places or the nearest hundredth, the function looks very similar to the one for rounding to tenths: \( y = \frac{\text{floor}(x * 100 + 0.5)}{100} \). The number is first scaled by a factor of 100, 0.5 is added, the floor function is applied, and then the result is divided by 100.

Define the roundToThousandths function

To round to the nearest thousandth, or three decimal places, follow the same steps but adjust the scaling to 1000: \( y = \frac{\text{floor}(x * 1000 + 0.5)}{1000} \). The number is multiplied by 1000 before adding 0.5 and applying the floor function, and then it is divided by 1000 to obtain the rounded number.

Output the results

For each input number x, compute the rounded values using the four functions defined. Then print the original number and the rounded results for integer, tenths, hundredths, and thousandths accordingly.

Key concepts

C++ Floor Function

In the world of C++, the floor function plays a vital role when it comes to rounding numbers downwards to the nearest integer. To put it simply, the floor() function takes a decimal number and chops off all the digits after the decimal point, resulting in the largest integer less than or equal to the original number.

For example, if you apply the floor function to the number 3.8, it will return 3. The same goes for -3.8, which would be rounded down to -4, as -4 is the nearest integer that is less than -3.8.

How does this apply to rounding to specific decimal places? Instead of directly using floor() on the number, we modify the number in such a way that when we apply floor, the result is effectively rounded to the desired decimal place. Multiplying the number by a power of ten, adding 0.5 (to handle the rounding correctly), and then using floor before dividing it back down by the same power of ten. This process shifts the decimal point, rounds the number, and then shifts the decimal point back to its original position.

C++ Function Implementation

Implementing a function in C++ requires a clear understanding of what the function is supposed to achieve and the logic that will get it there. When crafting functions to round numbers to various places, you'll need to consider the mathematical operations involved.

The typical structure follows declaring the function, which includes giving it a name and defining its parameters, then implementing the logic inside its body. For rounding functions, the main logic revolves around scaling the number to move the desired decimal digit to the ones place, rounding that number, and then scaling back.

For instance, roundToTenths will multiply the input by 10, apply the floor function after adding 0.5 for rounding purposes, and finally divide by 10 to get the rounded value at the tenths place. Each of the four functions (roundToInteger, roundToTenths, roundToHundredths, roundToThousandths) follows a similar pattern, tailored for the specific decimal precision.

Number Rounding in C++

Knowing how to round numbers in C++ is essential for ensuring your program outputs data in a format that is accurate and useful. Rounding is not about just chopping off digits; it's about increasing the preceding digit if the leading truncated digit is five or more.

In C++, this is often achieved through a combination of multiplication, the floor function, and division. When the goal is to round to a certain decimal place, this combination ensures that rounding happens accurately and adheres to the common mathematical rules of rounding numbers.

Essentially, by scaling the number (multipling by 10, 100, 1000, etc.), adding 0.5 to handle the rounding, applying the floor function, and then scaling back (dividing by the same number), we round the number to the desired decimal precision. Each place—integers, tenths, hundredths, thousandths—is addressed using this pattern, making it a versatile method for dealing with different rounding resolutions.