Problem 54 - College Algebra and Calculus: An Applied Approach

Question

Evaluate the expression.\(|4-\pi|\)

Step-by-Step Solution

Answer

The value of \(|4-\pi|\) is 0.8584.

Step 1: Subtract \(\pi\) from 4

First, subtract \(\pi\) from 4. This gives us \(4 - \pi\). It's known that \(\pi\) is approximately 3.1416. So the subtraction looks like this: \(4 - 3.1416 = 0.8584\) (rounded to 4 decimal places).

Step 2: Extract the Absolute Value

Next, find the absolute value of 0.8584. The absolute value of a positive number is just the number itself, so \(|0.8584| = 0.8584\).

Key Concepts

Understanding Pi (\(\backslash)pi\)

Pi, represented by the Greek letter \(\pi\), is one of the most intriguing numbers in mathematics. It's a constant defined as the ratio of the circumference of a circle to its diameter, and it's approximately equal to 3.14159. Pi is an irrational number, meaning it cannot be expressed as a simple fraction and its decimal representation goes on infinitely without repeating.

In certain math problems, like the one we're discussing, \(\pi\) is used in its approximate decimal form for calculations. This is common when we are not dealing with exact sciences or when an approximate result is acceptable. In such cases, \(\pi\) can be rounded to a certain number of decimal places, typically to 3.1416.

• \(\pi\) is used in formulas involving circles and other geometric shapes that involve curvature.
• It is imperative to remember that \(\pi\) is an approximate value in most calculations unless specified otherwise.
• \(\pi\) shows up in some unexpected places, such as number theory, probability, and even the famous Euler's identity.

Knowing when to use the precise or an approximate value of \(\pi\) is essential for accurate calculations in mathematics.

Mathematical Operations

Mathematical operations are procedures such as addition, subtraction, multiplication, and division, which are used to manipulate numbers and expressions to reach a result or to simplify a problem. These operations are fundamental to algebra and they follow a specific order called the order of operations (PEMDAS/BODMAS).

When we carry out these operations, it's crucial to understand the rules and properties that apply, such as commutative, associative, and distributive properties. In the case of the exercise involving \(4-\pi\), we're performing a subtraction:

• Subtraction is a binary operation that represents the operation of removing objects from a collection.
• Subtraction is anti-commutative, meaning that changing the order changes the sign of the answer (\(a - b \eq b - a\) for almost all \(a\) and \(b\)).
• In the context of the exercise, the subtraction is the first step before evaluating the absolute value.

A firm grasp of these operations allows students to efficiently solve algebraic and arithmetic problems and progress to more complex algebraic concepts.

Algebraic Expressions and Absolute Value

Algebraic expressions are combinations of variables, numbers, and mathematical operations that represent a specific value or set of values. Absolute value, which is the distance of a number from zero on the number line, is used in algebra to always yield a non-negative result, regardless of the sign of the number involved.

The absolute value function, denoted by two vertical bars (|...|), turns negative numbers into their positive counterparts. For positive numbers and zero, the absolute value remains the same:

• For any positive number \(a\), \(|a| = a\).
• For any negative number \(b\), \(|b| = -b\), which makes it positive.
• The absolute value of zero is zero, \(|0| = 0\).

In our exercise, after performing the subtraction \(4 - \pi\), we apply the absolute value to get the final result. The subtraction yields a positive result, so the absolute value of this positive number is the number itself. Understanding this concept is pivotal for grasping the basics of algebra and tackling problems that involve distance, magnitudes, and real-life applications where direction does not matter, but size does.