Mathematical Fallacies: What is wrong in this proof that 2+2 = 5?
The proof that (2 + 2 = 5) is a classic example of a mathematical fallacy. The specific steps you provided don't directly lead to a well-known fallacy, but similar proofs often involve violations of basic mathematical rules or operations. Here's a breakdown of a common type of fallacy that might be used to "prove" such an equation:
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Violating BODMAS Rules: One common mistake is not following the order of operations (Brackets, Orders, Division, Multiplication, Addition, and Subtraction). For example, in the video you mentioned, the calculation might involve incorrect grouping or cancellation of terms, leading to a false conclusion4.
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Incorrect Cancellation: Another fallacy involves cancelling terms that are equal to zero. For instance, if you have an equation where you multiply both sides by a term that is zero, you can end up with a false equation. However, the specific steps you provided don't explicitly show this.
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Misapplication of Algebraic Identities: Sometimes, proofs like these misapply algebraic identities or manipulate equations in ways that are not mathematically valid. For example, incorrectly factoring or expanding expressions can lead to false conclusions.
To illustrate, let's consider a hypothetical example similar to the one you might be referring to:
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Hypothetical Example: [ 4 - 4 = 10 - 10 ] [ 2^2 - 2^2 = (5 \cdot 2) - (5 \cdot 2) ] [ (2 + 2)(2 - 2) = 5(2 - 2) ] [ 2 + 2 = 5 ]
What's Wrong: The mistake here is in the step where you "cancel" the (2 - 2) on both sides. While (2 - 2 = 0), you cannot cancel these terms because they are zero, which would imply dividing by zero in a sense. This is a form of the division by zero fallacy, where you're effectively treating (0) as if it were a non-zero value that can be cancelled out.
In summary, the fallacy in such proofs usually involves violating basic mathematical rules or operations, such as incorrect application of algebraic identities or ignoring the rules of arithmetic operations.