Paradoxical Prove?

Algebra Level 2

My friend finds a way to prove 4=5:

  • Step 1: 2 + 2 = 4 \ \ 2 + 2 = 4
  • Step 2: 2 + 2 = 4 − 9 2 + 9 2 \ \ 2 + 2 = 4-\dfrac{9}{2}+\dfrac{9}{2}
  • Step 3: 2 + 2 = ( 4 − 9 2 ) 2 + 9 2 \ \ 2 + 2 = \sqrt{\left(4-\dfrac{9}{2}\right)^{2}} +\dfrac{9}{2}
  • Step 4: 2 + 2 = − 20 + ( 9 2 ) 2 + 9 2 \ \ 2 + 2 = \sqrt{-20 + \left(\dfrac{9}{2}\right)^{2}} +\dfrac{9}{2}
  • Step 5: 2 + 2 = 25 − 45 + ( 9 2 ) 2 + 9 2 \ \ 2 + 2= \sqrt{25 - 45 + \left(\dfrac{9}{2}\right)^{2}} +\dfrac{9}{2}
  • Step 6: 2 + 2 = 5 2 − 2 ( 9 2 ) ( 5 ) + ( 9 2 ) 2 + 9 2 = ( 5 − 9 2 ) 2 + 9 2 \ \ 2+2= \sqrt{5^{2} - 2\left(\dfrac{9}{2}\right)(5)+\left(\dfrac{9}{2}\right)^{2}} +\dfrac{9}{2} = \sqrt{\left(5-\dfrac{9}{2}\right)^{2}} +\dfrac{9}{2}
  • Step 7: 2 + 2 = 5 − 9 2 + 9 2 = 5 \ \ 2 + 2 = 5-\dfrac{9}{2}+\dfrac{9}{2} = 5
  • Step 8: ⟹ 4 = 5 \implies 4 = 5

Which step is incorrect? ( Bonus: Why?)

Clarification: The answer is the main reason which leads to a paradoxical result of 4 = 5 4=5 .

Step 3 Step 6 Step 2 Step 8 Step 7 Step 5 Step 1 Step 4

Tin Le by Tin Le , 15, Vietnam

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2 solutions

Chew-Seong Cheong
May 23, 2019

In Step 2: 4 − 9 2 = − 1 2 4 - \dfrac 92 = - \dfrac 12 but Step 3: ( 4 − 9 2 ) 2 = 1 2 \sqrt{\left(4-\dfrac 92\right)^2} = \dfrac 12 . Therefore, Step 3 is incorrect.

1 Helpful 0 Interesting 1 Brilliant 0 Confused

@Tin Le , you don't need to separate the equations with pairs of \ ( \ ). Just use one \ ( \ ) for the entire equation. It is easier and the font and spacing are the same throughout.

Chew-Seong Cheong - 2 years ago

Square root of a number is non-negative definite.

0 Helpful 0 Interesting 0 Brilliant 1 Confused

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