Find the error in the following 'proof' that 2 < 1.

Verified step by step guidance

1. Step 1: Start by assuming two numbers, say $a$ and $b$, where $a=b$.
2. Step 2: Multiply both sides of the equation $a=b$ by $a$, resulting in $a^2=ab$.
3. Step 3: Subtract $b^2$ from both sides to get $a^2-b^2=ab-b^2$.
4. Step 4: Factor both sides: $(a-b)(a+b)=b(a-b)$.
5. Step 5: Divide both sides by $a-b$, which is where the error occurs because $a-b=0$, leading to division by zero, an undefined operation.

Key Concepts

Inequalities

Inequalities are mathematical expressions that compare two values, indicating whether one is less than, greater than, or equal to the other. Understanding the properties of inequalities is crucial for analyzing statements like '2 < 1'. In this case, recognizing that 2 is greater than 1 is fundamental to identifying the error in the proof.

Proof Techniques

Proof techniques are methods used to demonstrate the validity of mathematical statements. Common techniques include direct proof, proof by contradiction, and counterexamples. In the context of the question, evaluating the proof's logic and identifying any fallacies or incorrect assumptions is essential to uncovering the error in claiming '2 < 1'.

Logical Fallacies

Logical fallacies are errors in reasoning that undermine the validity of an argument. They can occur in various forms, such as misapplication of rules or incorrect assumptions. Recognizing these fallacies is key to analyzing the 'proof' in question, as it allows one to pinpoint where the reasoning fails and why the conclusion '2 < 1' is incorrect.