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12 Proof 1 = 2
- 1. A Misguided Review of Mathematics YouTube Channel AHMEDMATICS Facebook Page AHMEDMATICS WhatsApp Group AHMEDMATICS
- 2. 1 − 1 0 = 𝟏 − 𝟏 + 𝟏 − 𝟏 + 𝟏 − 𝟏 + 𝟏 − 𝟏 + ⋯ 0 = 1 + −𝟏 + 𝟏 + −𝟏 + 𝟏 + −𝟏 + 𝟏 + ⋯ −1 + 1 −1 + 1 −1 + 1 0 = 1 + −1 + 1 + −1 + 1 + −1 + ⋯ Associative Law 0 = 1 + 0 + 0 + 0 + ⋯
- 3. 0 = 𝟏 − 𝟏 + 𝟏 − 𝟏 + 𝟏 − 𝟏 + 𝟏 − 𝟏 + ⋯ 0 = 1 + −𝟏 + 𝟏 + −𝟏 + 𝟏 + −𝟏 + 𝟏 + ⋯ 0 = 1 + −1 + 1 + −1 + 1 + −1 + ⋯ Associative Law 0 = 1 + 0 + 0 + 0 + ⋯ 0 = 1
- 4. 0 = 𝟏 − 𝟏 + 𝟏 − 𝟏 + 𝟏 − 𝟏 + 𝟏 − 𝟏 + ⋯ 0 = 1 + −𝟏 + 𝟏 + −𝟏 + 𝟏 + −𝟏 + 𝟏 + ⋯ 0 = 1 + −1 + 1 + −1 + 1 + −1 + ⋯ Associative Law 0 = 1 + 0 + 0 + 0 + ⋯ 0 = 1 0 = 1 =
- 5. 𝐿𝑒𝑡 = 1 𝑥 𝑥 − 1 = (𝑥 + 1)(𝑥 − 1) = ( + 1) 𝑥 = (𝑥 + 1) =
- 6. 𝐿𝑒𝑡 = 1 𝑥 𝑥 − 1 = (𝑥 + 1)(𝑥 − 1) = ( + 1) 𝑥 = (𝑥 + 1) = 𝐿𝑒𝑡 = 1
- 9. 𝟏 = 𝒊. 𝒊 + 𝟑 𝟐 𝟏 = −1. −1 + 𝟑 𝟐 𝟏 = −1 × −1 + 𝟑 𝟐
- 10. 𝟏 = 𝒊. 𝒊 + 𝟑 𝟐 𝟏 = −1. −1 + 𝟑 𝟐 𝟏 = −1 × −1 + 𝟑 𝟐 𝟏 = 1 + 𝟑 𝟐
- 11. 𝟏 = 𝒊. 𝒊 + 𝟑 𝟐 𝟏 = −1. −1 + 𝟑 𝟐 𝟏 = −1 × −1 + 𝟑 𝟐 𝟏 = 1 + 𝟑 𝟐 𝟏 = 1 + 𝟑 𝟐 𝟏 = 4 𝟐 = ± 𝟏 = −1. −1 + 𝟑 𝟐 𝟏 = −1 × −1 + 𝟑 𝟐 𝟏 = 1 + 𝟑 𝟐 𝟏 = −1 + 𝟑 𝟐 𝟏 = 2 𝟐 = 𝟏 = 1 + 𝟑 𝟐
- 12. < 𝟒 × × 𝟒 < 𝑢𝑛𝑖𝑡 𝑢𝑛𝑖𝑡
- 13. < 𝟒 × × 𝟒 < 𝑢𝑛𝑖𝑡 𝑢𝑛𝑖𝑡 < > =
- 14. < 𝟒 × × 𝟒 < 𝑢𝑛𝑖𝑡 𝑢𝑛𝑖𝑡 𝑐𝑚2 𝑐𝑚
- 16. 𝒅𝟐 = 𝟏𝟐 + 𝟏𝟐 𝒅𝟐 = 𝟐 𝒅 = 𝟐
- 17. 𝟏 + 𝟏 2 𝟏 + 𝟏 2 𝟏 + 𝟏 2
- 18. 𝟏 + 𝟏 2 𝒅𝟐 = 𝟏𝟐 + 𝟏𝟐 𝒅𝟐 = 𝟐 𝒅 = 𝟐
- 19. 𝟏 + 𝟏 2 𝒅𝟐 = 𝟏𝟐 + 𝟏𝟐 𝒅𝟐 = 𝟐 𝒅 = 𝟐
- 20. 2 𝒅𝟐 = 𝟐 𝒅 = 𝟐 𝟐 𝟐 =
- 21. 𝟏 + 𝟏 2 𝟏 + 𝟏 2 𝟏 + 𝟏 2
- 22. 𝟐 𝟐 𝟐 𝟐 𝟐 𝟐 𝟐 𝟐 𝟐 𝟐 𝟐 𝟐𝒙 = 𝟐 − 𝟏 + 𝟐 𝟑 − 𝟏 𝟐 + 𝟐 𝟓 − 𝟏 𝟑 + 𝟐 𝟕 − 𝟏 𝟒 + ⋯
- 23. 𝟐 𝟐 𝟐 𝟐 𝟐 𝟐 𝟐 𝟐 𝟐 𝟐 𝟐 𝟐𝒙 = 𝟐 − 𝟏 + 𝟐 𝟑 − 𝟏 𝟐 + 𝟐 𝟓 − 𝟏 𝟑 + 𝟐 𝟕 − 𝟏 𝟒 + ⋯ 𝟐𝒙 = 1 + 2 𝟑 − 𝟏 3 − 1 2 + 𝟐 𝟕 − 𝟏 𝟒 + ⋯
- 24. 𝟐 𝟐 𝟐 𝟐 𝟐 𝟐 𝟐 𝟐 𝟐 𝟐 𝟐 𝟐𝒙 = 𝟐 − 𝟏 + 𝟐 𝟑 − 𝟏 𝟐 + 𝟐 𝟓 − 𝟏 𝟑 + 𝟐 𝟕 − 𝟏 𝟒 + ⋯
- 25. 𝟐 𝟐 𝟐 𝟐 𝟐 𝟐 𝟐 𝟐 𝟐 𝟐 𝟐 𝟐𝒙 = 𝟐 − 𝟏 + 𝟐 𝟑 − 𝟏 𝟐 + 𝟐 𝟓 − 𝟏 𝟑 + 𝟐 𝟕 − 𝟏 𝟒 + ⋯ 𝟐𝒙 = 1 + 2 𝟑 − 𝟏 3 − 1 2 + 𝟐 𝟕 − 𝟏 𝟒 + ⋯ 𝟐 𝟓
- 26. 𝟐 𝟐 𝟐 𝟐 𝟐 𝟐 𝟐 𝟐 𝟐 𝟐 𝟐 𝟐𝒙 = 𝟐 − 𝟏 + 𝟐 𝟑 − 𝟏 𝟐 + 𝟐 𝟓 − 𝟏 𝟑 + 𝟐 𝟕 − 𝟏 𝟒 + ⋯ 𝟐𝒙 = 1 + 2 𝟑 − 𝟏 3 − 1 2 + 𝟐 𝟕 − 𝟏 𝟒 + ⋯ 𝟐𝒙 = 𝟏 + 1 3 − 𝟏 2 + 2 7 + 𝟏 𝟓 + 2 7 − 1 4 + ⋯ 𝟐 𝟓
- 27. 𝟐𝒙 𝟐 𝟐 𝟐 𝟐 𝟐 𝟐 𝟐 𝟐 𝟐 𝟐𝒙 = 𝟐 − 𝟏 + 𝟐 𝟑 − 𝟏 𝟐 + 𝟐 𝟓 − 𝟏 𝟑 + 𝟐 𝟕 − 𝟏 𝟒 + ⋯ 𝟐𝒙 = 1 + 2 𝟑 − 𝟏 3 − 1 2 + 𝟐 𝟕 − 𝟏 𝟒 + ⋯ 𝟐𝒙 = 𝟏 + 1 3 − 𝟏 2 + 2 7 + 𝟏 𝟓 + 2 7 − 1 4 + ⋯ 𝟐 𝟓
- 28. 𝟐 𝟐 𝟐 𝟐 𝟐 𝟐 𝟐 𝟐 𝟐 𝟐 𝟐 𝟐𝒙 = 𝟐 − 𝟏 + 𝟐 − 𝟏 + 𝟐 − 𝟏 + 𝟐 − 𝟏 + ⋯
- 29. 𝟐𝒙 𝟐𝒙 = 1 + 2 𝟑 − 𝟏 3 − 1 2 + 𝟐 𝟕 − 𝟏 𝟒 + ⋯ 𝟐𝒙 = 𝟏 + 1 3 − 𝟏 2 + 2 7 + 𝟏 𝟓 + 2 7 − 1 4 + ⋯ 𝟐 𝟓 𝟐𝒙 𝟐(1) =
- 30. 𝟐 𝟐 𝟏 + + 𝟏 =
- 32. =
- 33. = = = =
- 34. =
- 35. Here’s a strange question? Here’s a strange answer.
- 36. Suppose N is a number, not “1”
- 37. Suppose N is a number, not “1” This leaves “1” as the only option for the largest counting number.
- 38. Here’s a strange question? Here’s a strange answer.
- 39. I was recently at a shop and came across a I wanted only one item, so I asked the shop clerk how much a single item would be. came the reply I checked vouched the clerk