Question

What is the value of x after each of the following statements is executed? a) x = Math.abs( 7.5 ); b) x = Math.floor( 7.5 ); c) x = Math.abs( 0.0 ); d) x = Math.ceil( 0.0 ); e) x = Math.abs( -6.4 ); f) x = Math.ceil( -6.4 ); g) x = Math.ceil( -Math.abs( -8 + Math.floor( -5.5 ) ) );

Short answer

a) 7.5, b) 7, c) 0.0, d) 0.0, e) 6.4, f) -6, g) -14.

Step-by-step solution

Calculate Math.abs(7.5)

The function \( \text{Math.abs}(x) \) returns the absolute value of \( x \). For the statement \( x = \text{Math.abs}(7.5) \), the value of \( x \) will be \( 7.5 \) because 7.5 is already non-negative.

Calculate Math.floor(7.5)

The function \( \text{Math.floor}(x) \) returns the largest integer less than or equal to \( x \). For the statement \( x = \text{Math.floor}(7.5) \), the value of \( x \) will be \( 7 \), as this is the greatest integer less than 7.5.

Calculate Math.abs(0.0)

The function \( \text{Math.abs}(x) \) returns the absolute value of \( x \). For the statement \( x = \text{Math.abs}(0.0) \), the value of \( x \) will therefore remain \( 0.0 \), since the absolute value of zero is zero.

Calculate Math.ceil(0.0)

The function \( \text{Math.ceil}(x) \) returns the smallest integer greater than or equal to \( x \). For the statement \( x = \text{Math.ceil}(0.0) \), the value of \( x \) will be \( 0.0 \), since 0.0 is an integer and no greater integer is needed.

Calculate Math.abs(-6.4)

The function \( \text{Math.abs}(x) \) returns the absolute value of \( x \). For the statement \( x = \text{Math.abs}(-6.4) \), the value of \( x \) will be \( 6.4 \), as it converts the negative number to a positive.

Calculate Math.ceil(-6.4)

The function \( \text{Math.ceil}(x) \) returns the smallest integer greater than or equal to \( x \). For the statement \( x = \text{Math.ceil}(-6.4) \), the value of \( x \) will be \( -6 \), since -6 is the smallest integer greater than -6.4.

Calculate Math.ceil(-Math.abs(-8 + Math.floor(-5.5)))

First, evaluate \( \text{Math.floor}(-5.5) \), which is \( -6 \). Then, compute \(-8 + \text{Math.floor}(-5.5) = -8 - 6 = -14\). Next, evaluate \( \text{Math.abs}(-14) \), which is \( 14 \). Finally, compute \( -14 \) (the negative of \( 14 \)) and apply \( \text{Math.ceil}(-14) \), resulting in \( -14 \) because it is already an integer.

Key concepts

Math.abs Function

The `Math.abs` function in Java is a useful mathematical method that returns the absolute value of a given number. The absolute value of a number is its distance from zero on the number line, without considering its sign, meaning it will always be a non-negative value.
To better understand how `Math.abs` works, let's take a few examples:

• For a positive number, such as `7.5`, `Math.abs(7.5)` is `7.5`.
• For zero, `Math.abs(0.0)` gives `0.0` because the distance of zero from zero is zero!
• For a negative number, like `-6.4`, `Math.abs(-6.4)` results in `6.4` as it changes the sign to positive.

Understanding `Math.abs` is fundamental to Java programming as it is often used in algorithms that require magnitude comparison without regard for the sign of the number.

Math.floor Function

The `Math.floor` function in Java is designed to round a given number down to the nearest whole integer, that is less than or equal to the original number. This function is particularly useful when we need whole numbers.
If you apply `Math.floor` to different numbers, you get results like:

• For `7.5`, `Math.floor(7.5)` returns `7`.
• Numbers that are already integers, such as `5` or `0`, will remain the same as in `Math.floor(0.0) = 0.0`.
• For negative numbers, e.g., `-5.5`, `Math.floor(-5.5)` gives `-6`, because that is the largest integer not greater than `-5.5`.

The `Math.floor` function is crucial in programming where truncating to base integer values is needed, such as graphics rendering or pagination in applications.

Math.ceil Function

The `Math.ceil` function in Java is almost the opposite of `Math.floor`. It rounds a number up to the nearest whole integer, meaning that it will take a number and turn it to the smallest integer greater than or equal to it.
You can see how `Math.ceil` operates with these examples:

• For a positive decimal, `Math.ceil(7.5)` results in `8`.
• When applied to zero, `Math.ceil(0.0)` simply remains `0.0`, as zero is already an integer.
• With negative numbers, `Math.ceil(-6.4)` returns `-6`, because `-6` is the smallest integer not more than `-6.4`.

Programmers often rely on `Math.ceil` when implementing algorithms that depend on gathering the whole numbers accurately and rounding up as needed, such as pricing systems, billing calculations, or statistical analysis.

Problem Solving in Java

Problem-solving in Java involves breaking down complex tasks into simple, manageable parts by leveraging the language's robust set of built-in functions and its object-oriented principles.
When you tackle problems in Java, you should consider the following approach:

• **Understand the Problem:** Always begin by thoroughly understanding what the problem is asking. Identify the inputs and the desired outputs.
• **Plan a Solution:** Think about the functions or algorithms you can use. For the given exercise, functions like `Math.abs`, `Math.floor`, and `Math.ceil` are essential in manipulating and working with numeric data.
• **Break The Problem Into Steps:** Decompose into smaller tasks or steps. Take advantage of Java’s functions to handle these effectively.
• **Write and Test Code:** Write the Java code, test it frequently and refine the solution.
• **Optimization:** Finally, optimize your solution for better efficiency if necessary.

Using these practices, Java programmers can solve a wide variety of problems efficiently, exploiting the power of the language's mathematical and logical constructs.