Question

Show the value of x after each of the following statements is performed: a) x = fabs( 7.5 ) b) x = floor( 7.5 ) c) x = fabs( 0.0 ) d) x = ceil( 0.0 ) e) x = fabs( -6.4 ) f) x = ceil( -6.4 ) g) x = ceil( -fabs( -8 + floor( -5.5 ) ) )

Short answer

a) 7.5, b) 7, c) 0, d) 0, e) 6.4, f) -6, g) 14

Step-by-step solution

Using fabs

The function 'fabs' computes the absolute value of a floating-point number. For a), we have x = fabs(7.5), which makes x the absolute value of 7.5. Since 7.5 is positive, its absolute value is 7.5.

Using floor

The 'floor' function returns the largest integer less than or equal to a given number. For b), x = floor(7.5), which means x will be the nearest integer less than or equal to 7.5. So, x will be 7.

Absolute value of zero

The absolute value of 0 is 0, hence for c), x = fabs(0.0), x will be 0.

Using ceil on zero

The 'ceil' function returns the smallest integer greater than or equal to a given number. For d), x = ceil(0.0), x will be the smallest integer greater than or equal to 0, which is 0.

Absolute value of a negative

For e), x = fabs(-6.4), x is the absolute value of -6.4, which is 6.4.

Using ceil on a negative

For f), x = ceil(-6.4), x will be the smallest integer greater than -6.4, which is -6.

Combining functions

The expression in g) combines multiple functions. First, calculate the floor of -5.5, which is -6. Adding this to -8 gives us -14. The absolute value of -14 is 14. Finally, the ceiling of 14 is 14 itself. Therefore, x = ceil(-fabs(-8 + floor(-5.5))) results in x being 14.

Key concepts

Understanding the fabs Function in Floating-Point Arithmetic

Floating-point arithmetic is essential in programming for representing numbers with fractional parts. The C library function fabs() plays a fundamental role in this field. The fabs function computes the absolute value of a floating-point number. Absolute value refers to the distance of a number from zero on the number line without considering the direction. For example, when you apply fabs to both 5.2 and -5.2, the result is the same: 5.2.

This function becomes especially useful in situations where you need to ensure that a variable is non-negative regardless of its initial sign. Returning to our textbook exercise, the operation x = fabs(7.5) simply reaffirms that x is 7.5, as the function removes any negative sign from the number. If we deal with a negative value, such as x = fabs(-6.4), the negative sign is discarded by fabs, converting it to a positive 6.4. It's an indispensable tool in calculations that only make sense with positive numbers, like in measuring distances or dealing with absolute error.

Exploring the floor Function for Rounding Down

The floor function is another key player in floating-point arithmetic. The fundamental purpose of the floor function is to return the largest integer that is less than or equal to the specified floating-point number. It effectively 'rounds down' to the nearest integer.

Consider the scenario given in our exercise: x = floor(7.5). Here, the floor function takes the number 7.5 and rounds it down to the nearest whole number, resulting in an x value of 7. Furthermore, for negative numbers, it's crucial to understand that the floor function will move further away from zero. So, in a case like floor(-5.5), it will produce -6 as the result because -6 is the largest integer less than -5.5.

Applications of the floor Function

Applications of the floor function include setting ranges for array indexes, determining loop counters in iteration processes, or truncating numbers to integer values for discrete systems in programming.

The ceil Function: Rounding Up Integers

Complementing the floor function, we have the ceil function – short for 'ceiling'. The ceil function performs the opposite action: it rounds a floating-point number up to the nearest integer. In essence, it works by finding the smallest integer that is greater than or equal to the argument given.

For a positive input such as x = ceil(7.5), this would result in x becoming 8. For negative numbers, like in x = ceil(-6.4), the function rounds up towards zero, which gives x a value of -6. The ceil function proves valuable in scenarios where overestimation is safer or required, such as in allocating resources or adjusting functions to meet certain constraints.

Understanding Ceiling with Complex Expressions

In composite expressions that include the ceil function alongside others like fabs and floor, the order of operations is vital. As the last step, ceil ensures the final result adheres to its rounding-up rule. This layered functionality can be seen in the exercise's final part, x = ceil(-fabs(-8 + floor(-5.5))), showcasing how these mathematical functions interact in programming to produce precise and predictable outcomes.