Problem 53 - Thomas Calculus

Question

a. If \(b\) is any nonzero real number, prove that \(|1 / b|=1 /|b|\) b. Prove that \(\left|\frac{a}{b}\right|=\frac{|a|}{|b|}\) for any numbers \(a\) and \(b \neq 0\)

Step-by-Step Solution

Answer

Both properties hold due to the nature of absolute values equalizing signs, thus, \(|1/b|=1/|b|\) and \(|a/b| = |a|/|b|\).

Step 1: Understand the Properties of Absolute Value and Division

To solve problems involving absolute values and fractions, we first need to recall the basic property of absolute value: For any real number \(x\), the absolute value is defined as \(|x| = x\) if \(x \geq 0\), and \(|x| = -x\) if \(x < 0\). Additionally, absolute values tend to equalize signs, meaning \(|x/y| = |x|/|y|\).

Step 2: Prove \(|1/b| = 1/|b|\)

First, consider \(b\) as a nonzero real number. The absolute value \(|b|\) of \(b\) is defined as:1. \(b\) if \(b > 0\).2. \(-b\) if \(b < 0\).Looking at \(1/b\), the absolute value would be calculated as:- \(|1/b| = 1/b\) if \(b > 0\), making it equal to \(1/|b|\).- \(|1/b| = -1/b\) if \(b < 0\), in which case \(1/|b| = -1/b\).Thus, in both cases, \(|1/b| = 1/|b|\) holds true.

Step 3: Prove \(|a/b| = |a|/|b|\)

For any real numbers \(a\) and \(b eq 0\), consider:- If both are positive or both are negative, the signs cancel out; hence, \(|a/b| = |a|/|b|\).- If one is positive and the other negative, \(|a/b|\) again equals \(|a|/|b|\) because the negative signs effectively cancel the effect of division by the absolute values. In all scenarios, this intuitive approach confirms the property \(|a/b| = |a|/|b|\).

Key Concepts

Division and Absolute Value

The relationship between division and absolute value is a fundamental concept in mathematics. When dividing two real numbers, their absolute values maintain a consistent property. For example, when dealing with the expression \(|1/b| = 1/|b|\)\, you're essentially looking at how absolute values handle division.Here's a simplified breakdown:

• If \(b\) is a positive number, then \(|b| = b\), thus \(|1/b| = 1/b\), and naturally \(1/|b| = 1/b\).
• If \(b\) is negative, then \(|b| = -b\), meaning \(|1/b| = -1/b\), and also \(1/|b| = -1/b\).

The principle here is that the absolute value of a division respects the order of operations, simplifying negatives to positives. This mirrors how mathematics handles sign changes, ensuring that any negative is flipped to a positive in a way that simplifies problems consistently.

Real Numbers and Absolute Value

Real numbers encompass every number that can be found on the number line, including fractions, integers, and irrational numbers. The absolute value of a real number is its distance from zero, disregarding direction, which means it’s always non-negative. This concept is widely used in proving mathematical identities and equations.To illustrate:

• For any real number \(x\), the absolute value \(|x|\) is \(x\) if \(x \geq 0\) and \(-x\) if \(x < 0\).
• As seen in expressions like \(\left|\frac{a}{b}\right| = \frac{|a|}{|b|}\), you are comparing signs or directions of division to ensure outcomes stay in positive form.

When you use real numbers with absolute value, you’re applying these properties to real-world distance and magnitude calculations. It's about determining not just the value, but how the number interacts within operations, maintaining a clear positive outcome irrespective of sign.

Proofs in Calculus

Proofs in calculus often incorporate various properties and rules to deduce more complex aspects of mathematics. The absolute value properties provide a sturdy foundation for proofs involving real numbers and division.

• The proof deals with demonstrating equalities, such as \(|\frac{a}{b}| = \frac{|a|}{|b|}\), by systematically breaking them down into logical steps.
• They show how the properties maintain their truth across different numeric circumstances, ensuring stability in mathematical expressions.

Whether it’s proving theorems or verifying equations, calculus relies on these rigorous proof structures to assert factual mathematical relationships. Through logical decomposition and articulate explanation, proofs ensure that every mathematical step builds on a verified foundation, optimizing understanding and application across problems. They reaffirm the predictability and reliability inherent in mathematical operations.