# Pure math 3 : 2 = 1!

Calculus Level 3

Identify the first invalid step in the following proof.

1. $$|x| = \underbrace{1 + 1 + 1 + ..... + 1}_{|x| \, \text{ times}}$$
2. $$|x| \times |x| = \underbrace{|x + x + x + ..... + x|}_{|x| \, \text{ times}}$$
3. $$x^2 = \underbrace{|x + x + x + ..... + x|}_{|x| \, \text{ times}}$$
4. $$\dfrac{d}{dx}(x^2) = \dfrac{d}{dx}(\underbrace{|x + x + x + ..... + x|}_{|x| \, \text{ times}})$$
5. $$2x = \underbrace{1 + 1 + ..... + 1}_{|x| \, \text{ times}}$$
6. $$2x = x$$
7. $$2 = 1$$
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