Question

Find the absolute value for each of the following (1) zero (2) 4 (3) \(-\pi\) (4) a, where a is a real number.

Short answer

The absolute values are as follows: (1) |0| = 0 (2) |4| = 4 (3) \[|\text{-}\pi| = \pi\] (4) \[|a| = \begin{cases} a &\text{if } a \geq 0 \\ -a &\text{if } a < 0 \end{cases} \]

Step-by-step solution

1) Find the absolute value of zero

Since zero is neither positive nor negative, its absolute value is also zero. So, the absolute value of zero is |0| = 0.

2) Find the absolute value of 4

Since 4 is a positive number, we follow the formula for positive numbers |x| = x. Therefore, the absolute value of 4 is |4| = 4.

3) Find the absolute value of \(-\pi\)

Since \(-\pi\) is a negative number, we follow the formula for negative numbers |x| = -x. Therefore, the absolute value of \(-\pi\) is \[|\text{-}\pi| = -(-\pi) = \pi.\]

4) Find the absolute value of a, where a is a real number

We need to consider two possibilities for a: a) If a is a positive number (a≥0), then the absolute value of a would be |a| = a. b) If a is a negative number (a<0), then the absolute value of a would be |a| = -a. We don't know which category 'a' falls into, so we write the solution as a piecewise function: \[|a| = \begin{cases} a &\text{if } a \geq 0 \\ -a &\text{if } a < 0 \end{cases} \]

Key concepts

Understanding Piecewise Functions

Piecewise functions are mathematical expressions that have different rules or formulas for different parts of their domain. This means that the function behaves differently based on the input value you decide to use. A common example is the absolute value function. It changes its formula depending on whether the input is positive, negative, or zero.
When talking about the absolute value function, we say:

• If the input is a positive number or zero, the output is the same as the input.
• If the input is a negative number, the output is the positive version of that number.

For example, the absolute value of a number 'a' can be expressed as a piecewise function like this: \[|a| = \begin{cases} a &\text{if } a \geq 0 \ -a &\text{if } a < 0 \end{cases}\] This function makes it easy to understand how absolute value works. It simply "turns" any negative input into its positive counterpart while leaving positive numbers and zero unchanged.

Exploring Real Numbers

Real numbers include all the numbers that you can find on the number line. This set contains both rational numbers (like integers and fractions) and irrational numbers (like \(\pi\) and \(\sqrt{2}\)). Real numbers are essential in everyday math as they help describe quantities that can be measured.
Here are some characteristics of real numbers:

• They can be positive, negative, or zero.
• Real numbers include both decimals and fractions.
• They cover an infinite set of numbers, without any gaps on the number line.

Understanding real numbers is helpful when considering functions like the absolute value because we need to know where on the number line we are in order to apply the correct piece of a piecewise function.

Positive and Negative Numbers in Context

Understanding the difference between positive and negative numbers is crucial, especially when working with functions like the absolute value. Positive numbers are those that are greater than zero, while negative numbers are less than zero. Zero itself is neutral, meaning it is neither positive nor negative.
Here are key points to remember:

• Positive numbers find themselves on the right side of zero on a number line.
• Negative numbers appear on the left side of zero.
• When applying absolute values, negative numbers are turned into positive ones, making the concept easier to manage during calculations.
• For example, if \(x = -5\), its absolute value \(|x|\) becomes 5.

This understanding helps in using the absolute value function correctly, as it always results in zero or a positive number, irrespective of the input being positive or negative.