Problem 83 - Calculus with Concepts in Calculus

Question

Show that \(|a+b|=|a|+|b|\) if and only if \(a b \geq 0\) (which means that \(a=0, b=0\), or \(a\) and \(b\) have the same sign).

Step-by-Step Solution

Answer

\(|a+b|=|a|+|b|\) holds if and only if \(ab \geq 0\).

Step 1: Understand the Absolute Value Equality

We need to show that the equation \(|a+b|=|a|+|b|\) holds if and only if \(ab \geq 0\). This means that the equation holds true when \(a\) and \(b\) have the same sign or one or both are zero.

Step 2: Analyze the Case When \(ab \geq 0\)

Assume \(ab \geq 0\). This means that either both \(a\) and \(b\) are positive, or both are negative, or at least one is zero. So, if \(a\geq 0\) and \(b\geq 0\), then \(|a+b| = a+b = |a|+|b|\). If \(a\leq 0\) and \(b\leq 0\), then \(|a+b| = -(a+b) = |a|+|b|\). If one is zero, then the absolute value conditions still hold, ensuring \(|a+b| = |a| + |b|\).

Step 3: Verify the Reverse Implication

Assume \(|a+b|=|a|+|b|\). We want to show that this implies \(ab \geq 0\). If the given is true, then either \(a+b = |a|+|b|\) or \(a+b = -|a|-|b|\). In both situations, these are true when \(a\) and \(b\) are either both non-negative or both non-positive. Therefore, in these cases, \(ab \geq 0\) is satisfied.

Step 4: Conclusion

Given these analyses, we have shown: \(|a+b|=|a|+|b|\) holds if and only if \(ab \geq 0\). This concludes the required "if and only if" proof.

Key Concepts

Absolute Value Equality

The property of absolute value equality is a fascinating concept in mathematics. It states that the absolute value of the sum of two numbers, \(|a+b|\), is equal to the sum of their absolute values, \(|a|+|b|\), under certain conditions. These conditions occur if and only if both numbers have the same sign or at least one of them is zero. This concept is crucial because it helps us understand how absolute values behave differently from regular addition when negative numbers are involved. For example, consider two positive numbers, 3 and 2. Here, \(|3+2| = 5 = 3 + 2 = |3| + |2|\). However, if we look at a positive and a negative number, say 3 and -2, the absolute value equality doesn't hold: \(|3-2| = 1\), which is not \(|3| + |-2| = 5\).

Thus, absolute value equality is unique and only manifests when both numbers contribute safely, either as zeros or sharing the same positive or negative energy, leading to the condition \(ab \geq 0\).

Sign Analysis

Analyzing the sign of numbers is an essential step in understanding when absolute value equality holds. When dealing with the product \(ab \geq 0\), we mean that the numbers \(a\) and \(b\) must "agree" in terms of their sign, whether positive, negative, or zero. Let's break this down:

• Both Positive: If both \(a > 0\) and \(b > 0\), then \(ab > 0\), and their sum is unaffected in magnitude by their sign, hence \(|a+b| = a+b\).
• Both Negative: Similarly, if \(a < 0\) and \(b < 0\), \(ab > 0\). However, in this situation, \(|a+b| = |a| + |b|\) manifests because the subtraction effect negated by absolute values transforms the sum to non-negative.
• Involving Zero: When at least one number is zero, such as \(a=0\) or \(b=0\), the multiplication yields \(ab = 0\). In this case, absolute values function straightforwardly to show that \(|a+b| = |a| + |b|\).

This analysis is basically the lens through which we verify that the absolute value equality statement continually holds true based on how \(a\) and \(b\) interact through their signs.

Mathematical Proof Steps

Mathematics flourishes on clear proofs, and demonstrating absolute value equality via a proper proof helps solidify understanding. The proof uses an 'if and only if' logical structure, meaning we must prove the statement in both directions: one assuming \(ab \geq 0\) and the other assuming \(|a+b| = |a| + |b|\).

Here’s a breakdown of the proof steps:

**Assume \(ab \geq 0\):**
When \(ab \geq 0\), \(a\) and \(b\) share signs, enabling us to express \(|a+b|\) as \(a+b\) when positive, or \(-(a+b)\) when negative. Thus, the equation \(|a+b| = |a| + |b|\) naturally holds.

**Assume \(|a+b| = |a| + |b|\):**
On this side, deconstructing \(|a+b|\) into \(a+b\) or \(-(a+b)\) depending on positivity or negativity, logically results in \(|a| = a\) and \(|b| = b\) or their negatives. These situations imply that \(a\) and \(b\) don’t switch signs, asserting \(ab \geq 0\).

This organized approach ensures that all cases are considered to validate that absolute value equality simultaneously requires and confirms the non-negative product \(ab\). It underlines the robustness of correlating these mathematical ideas.