AI Tutor
If every algebraic step looks completely legal on paper, how can a proof end up concluding that 1=21 = 2?
Flawed proofs rarely make obvious arithmetic mistakes — instead, they disguise logical fallacies, circular logic, or unstated domain restrictions inside routine algebraic manipulation.
What happens when a proof secretly uses the very statement it is trying to establish?

Circular Reasoning: Starting with the Target

Circular reasoning (also called begging the question) occurs when a proof assumes the truth of the conclusion in order to prove that same conclusion.
A common student trap is starting with the identity to be proved, such as sin⁡x1+cos⁡x=1−cos⁡xsin⁡x\frac{\sin x}{1 + \cos x} = \frac{1 - \cos x}{\sin x}, cross-multiplying to get sin⁡2x=1−cos⁡2x\sin^2 x = 1 - \cos^2 x, and declaring the proof complete because 1=11 = 1 is true.
Starting from A=BA = B assumes the equality already holds — a valid proof must start strictly from known identities or transform the Left-Hand Side independently until it matches the Right-Hand Side.
What happens when an algebraic operation silently breaks the laws of arithmetic?

The Zero Trap and Unstated Assumptions

An unstated assumption is a condition taken for granted without justification, such as assuming an expression is non-zero before dividing, or assuming a variable is positive before taking a square root.
In the 19th century, logician Augustus De Morgan warned that algebraic symbols will happily perform nonsense operations unless mathematicians explicitly enforce their domain limits.