Hi everybody! Seth here with another cool math article. I discovered these proofs recently, and thought that they were really cool! So, sit back and enjoy!

So, here are two math proofs that are essentially proving the same thing. The first one proves 1=0, and the second one proves 4=5. Neither one of them is accurate, but they are seemingly possible.

### 1=0

Step 1. Set a variable a equal to a variable b. We have a=b.

Step 2. Multiply both sides by a. We have a^2=ab.

Step 3. Subtract b^2 from both sides. We have a^2-b^2=ab-b^2.

Step 4. Factor both sides, using the difference of squares on the left. We have (a+b)(a-b)=b(a-b)

Step 5. Divide both sides by (a-b). We have a+b=b.

Step 6. Subtract b from both sides. We have a=0.

Step 7. Divide both sides by a. We have 1=0.

### 4=5

Step 1. We know 2+2=4.

Step 2. We know 2+2= 4-(9/2)+(9/2)

Step 3. We know 2+2= sqrt((4-(9/2))^2)+(9/2).

Step 4. We know 2+2= sqrt(16-(2)(4)(9/2)+(9/2)^2)+(9/2).

Step 5. We know 2+2= sqrt(16-36+(9/2)^2)+(9/2).

Step 6. We know 2+2= sqrt(-20+(9/2)^2)+(9/2)

Step 7. We know 2+2= sqrt(25-45+(9/2)^2)+(9/2)

Step 8. We know 2+2= sqrt((5^2)-(2)(5)(9/2)+(9/2)^2))+(9/2).

Step 9. We know 2+2= sqrt((5-(9/2))^2)+(9/2)

Step 10. We know 2+2= 5-(9/2)+(9/2).

Step 11. We know 2+2=5.

Step 12. We know 4=5.

So, both of these proofs seem to work. However, they don’t. 1 is not equal to 0, and 4 is not equal to 5. So, if you would like to figure out the errors by yourself, go ahead. But below this I have a clue and a hint for each one, if you want to figure them out with some help. The numbers match the proof number. Or, if you just want the answers, scroll down.

Clue 1: If a=b, then what is a-b?

Clue 2: Is the square root of x^2 equal to x?

Hint 1: The error is in line 5.

Hint 2: The error is in line 3.

Answer 1: If a=b, then a-b equals zero. In step 5, we are told to divide by a-b. It is impossible to divide by 0, so it is likewise impossible that 0=1. This is my favorite disproof of being able to divide by 0.

Answer 2: The square root of x^2 is not x, it is the absolute value of x. This wouldn’t be an issue if 4-(9/2) was positive, but it isn’t. In doing this, the proof is adding 1 (going from -0.5 to 0.5) The rest of the proof is just making sure to hide this impossible move.

#### Seth Casel

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