Problem 35 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Evaluate the expression. $$ |\pi-1|+2 $$

Step-by-Step Solution

Answer

The short answer is: \(|\pi-1|+2 \approx 4.14159\).

Step 1: Calculate the absolute value of \(\pi-1\)

The mathematical constant \(\pi\) is approximately equal to 3.14159. To determine the absolute value of \(\pi-1\), we will subtract 1 from \(\pi\): \[\pi-1 \approx 3.14159 - 1 = 2.14159\] The absolute value of a number is its distance from 0 on the number line, and it is always a positive value. In this case, \(|\pi-1| \approx 2.14159\).

Step 2: Add 2 to the absolute value of \(\pi-1\)

Now, we just need to add 2 to the calculated absolute value in Step 1: \[|\pi-1| + 2 \approx 2.14159 + 2 = 4.14159\] The evaluated expression is approximately \(4.14159\).

Key Concepts

Mathematical Constants

Mathematical constants are special numbers that arise naturally in various mathematical contexts. They are the building blocks for more complex mathematical ideas. One of the most famous constants is \(\pi\), which represents the ratio of a circle's circumference to its diameter.

Here are some important points about \(\pi\):

• \(\pi\) is an irrational number, which means it cannot be expressed as a simple fraction. Its decimal representation goes on forever without repeating.
• \(\pi\) is approximately 3.14159, which is often sufficient for basic computations.

Understanding constants like \(\pi\) helps in evaluating and solving various mathematical expressions and problems, like finding distances, areas, and volumes related to circles.

Absolute Value

The concept of absolute value is a fundamental part of mathematics that measures how far a number is from zero on the number line, regardless of direction. The absolute value of a number \(x\) is denoted by \(|x|\).

For instance:

• The absolute value of a positive number remains unchanged. Thus, \(|5| = 5\).
• The absolute value of a negative number is its positive counterpart. Therefore, \(|-3| = 3\).

Applying this to our exercise, \(|\pi - 1|\) calculates the distance of \(\pi - 1\) from zero, resulting in a positive value like approximately 2.14159, promoting a clearer understanding of mathematical operations without negative signs.

Basic Arithmetic Operations

Basic arithmetic operations are the cornerstones of mathematics, involving addition, subtraction, multiplication, and division. Mastering these operations is crucial for solving more complex mathematical problems.

For our exercise, these skills were employed as follows:

• Subtraction: We first computed \(\pi - 1\).
• Absolute Value: Next, we took the absolute value of the result to ensure a positive outcome.
• Addition: Finally, we added 2 to the absolute value to complete the expression.

These steps showcase the simplicity and power of basic arithmetic in evaluating expressions, serving as a reminder of their importance in everyday mathematical endeavors.