Problem 45 - Thomas Calculus

Question

A proof of the triangle inequality Give the reason justifying each of the numbered steps in the following proof of the triangle inequality. $$ \begin{aligned}|a+b|^{2} &=(a+b)^{2} \\ &=a^{2}+2 a b+b^{2} \\ & \leq a^{2}+2|a||b|+b^{2} \\ &=|a|^{2}+2|a||b|+|b|^{2} \\ &=(|a|+|b|)^{2} \\\|a+b| & \leq|a|+|b| \end{aligned} $$

Step-by-Step Solution

Answer

The triangle inequality shows that \(|a+b| \leq |a| + |b|\).

Step 1: Expand the expression

The proof starts with the expression \(|a+b|^{2}\). By definition, this can be rewritten as the square of the sum of \(a\) and \(b\), which is \((a+b)^2\). This is purely an algebraic manipulation.

Step 2: Apply the distributive property

Now, apply the distributive property to expand \((a+b)^2\). This becomes \(a^2 + 2ab + b^2\). This is the standard expansion of a binomial expression.

Step 3: Use the properties of absolute values

Recognize that \(ab\) is less than or equal to \(|a||b|\). Thus, \(a^2 + 2ab + b^2\) is less than or equal to \(a^2 + 2|a||b| + b^2\). This step uses the property \(|ab| = |a||b|\) and the fact that \(-|a||b| \leq ab \leq |a||b|\).

Step 4: Rewrite using absolute values

Write \(a^2 + 2|a||b| + b^2\) as \(|a|^2 + 2|a||b| + |b|^2\), since \(|a| = a\) when \(a\) is positive and \((-a)^2 = a^2\) if \(a\) is negative. This bridges \(a^2 + b^2\) to \(|a|^2 + |b|^2\).

Step 5: Factor and simplify

Notice that \(|a|^2 + 2|a||b| + |b|^2\) is the expansion of \((|a|+|b|)^2\). This step shows that \(|a+b|^2\) can be simplified to \((|a|+|b|)^2\), linking back to matching terms.

Step 6: Take the square root

Finally, take the square root of both sides of the inequality \(|a+b|^2 \leq (|a|+|b|)^2\) to find \(|a+b| \leq |a|+|b|\). Taking the square root preserves the inequality as both sides are non-negative.

Key Concepts

Proof Techniques

Understanding different proof techniques is vital in mathematics. In proving the triangle inequality, a structured approach ensures logic and coherence.
We begin by assuming a statement is true and use logical steps to prove it — a technique known as direct proof. The process involves laying out each step in an equation or inequality and justifying it, ensuring clarity and correctness.
Another common technique is proof by contradiction, where we assume the opposite of what we want to prove and show that this assumption leads to a contradiction. While not used in the triangle inequality proof, it is a powerful tool every student should recognize.
Mathematical proofs often encompass multiple techniques. Understanding them helps in crafting precise arguments and can be applied to an array of mathematical concepts.

Absolute Value Properties

Absolute values represent the distance from zero, which is always non-negative. These properties are critical in manipulating expressions involving sums or differences.
In the triangle inequality proof, we see the property that \( |ab| = |a||b| \) being utilized. This assures us that the multiplication of absolute values is handled the same as individual values but without concerns for sign changes.
Additionally, the inequality \( -|a||b| \leq ab \leq |a||b| \) is a crucial concept. This captures how even if one of the numbers is negative in a product, the absolute value controls the potential disparity.

• Absolute values ensure non-negativity.
• They provide a universal approach to manage positive and negative numbers.

Such properties are foundational in solving complex equations, ensuring solutions remain logically sound.

Algebraic Manipulation

Algebraic manipulation involves rearranging and simplifying expressions to make them easier to work with. Throughout the proof of the triangle inequality, algebra serves as a backbone.
Starting with \( |a+b|^2 = (a+b)^2 \), we apply standard binomial expansion techniques. This leads to \( a^2 + 2ab + b^2 \) which forms the basis of further steps.
The trick is to skillfully use properties like distribution, factoring, or even substitution to transform expressions without altering their meaning. Learning to fluently move between different forms of expressions greatly enhances problem-solving ability.
Recognize that such manipulation maintains equality and helps isolate desired terms or simplify the problem structure for easier comprehension and solving.

Inequalities in Mathematics

Inequalities like the triangle inequality are essential tools in mathematics, providing boundaries within which values operate. They form the basis of many concepts in math and its real-world applications.
The triangle inequality, \( |a+b| \leq |a| + |b| \), is a vital property describing how the distance between two points is bounded by the sum of distances to a third point. This arises naturally in geometry but extends into more abstract fields like functional analysis.
When working with inequalities, a crucial aspect is recognizing equality cases and understanding how the total expression can span values within set limits.

• They help in determining maximum and minimum bounds.
• Provide logical constraints.

Mastering inequalities assists in understanding how mathematical entities interact, reinforcing the larger framework of mathematical reasoning.