In mathematics there’s a concept called invalid proofs. These invalid proofs are mathematical fallacies which lead to an absurd result. The interesting part isn’t the result per se bu the clever way in which such invalid proof is presented. Usually hiding by means of astute arithmetic and algebraic tricks the invalid steps. The Angle fallacy is another invalid proof. This one is geometrically in nature and used by teachers to explain common error in the measurement of angles.

In order to make things even more interesting, the invalid steps are mixed in with valid steps, as to disorient the person trying to follow the logic of operations. These invalid steps may involve stealthy dividing by zero, like for example by incorrectly extracting a root, or, more common, when different values of a multiple valued function end up equated.

The most famous and popular example of an invalid proof is the teachers’ favorite 2 + 2 = 5. First we’ll present the whole invalid proof and then we’ll dissect it to see where the trick is:

2+2=2+2

(2+2)-5=(2+2)-5

-10-10=-10-10

-20=-20

16-36=25-45

16-36+81/4=25-45+81/4

(a-b)^2=a^2-2ab+b^2

16-36+81/4=(4-9/2)^2

25-45+81/4=(5-9/2)^2

(4-9/2)^2=(5-9/2)^2

4-9/2=5-9/2

4=5-9/2+9/2

4=5

2+2=5

Let’s take a closer look:

2+2 = 2+2

Let’s multiply both sides by -5
(2+2)-5 = (2+2)-5
-10-10 = -10-10
-20 = -20

Now let’s factor each number
16-36 = 25-45

We add 81/4 to each side
16-36+81/4 = 25-45+81/4

As we can see these are squares
(a-b) 2 = a 2 -2ab+b 2
16-36+81/4 = (4-9/2)
2
25-45+81/4 = (5-9/2)
2
(4-9/2)
2 = (5-9/2) 2

Let’s cancel the squares
4-(9/2) = 5-(9/2)

We move -9/2 to the other side of the equation
4 = 5-9/2+9/2

As we can see -9/2+9/2 = 0 so:
4 = 5
2+2 = 5

### So, where’s the invalid proof?

Have you detected it? It’s not tricky if you your basic rules of algebra. The mistake is here:

(4-9/2) 2 = (5-9/2) 2
Let’s cancel the squares
4-9/2 = 5-9/2

We just cannot cancel these squares because the square root of any number to the power of two is a module of said number!. The proper way of doing it would be:

(4-9/2) 2 = (5-9/2) 2
| 4-9/2 | = | 5-9/2 |
| -0,5 | = | 0,5 |
0,5 = 0,5

As we can see a completely different outcome, and with this outcome we maintain the arithmetic equality in both sides!

### Who came up with 2+2=5

It’s origin isn’t really clear. Some say it dates back to Pythagoras era, but we know for sure that the one who popularized it was Fibonacci in the 13th century, who, after studying the euclidean principles proclaimed that “It’s probable that 2 + 2 is closer to 5 than 4”. He tried for years to came up with a demonstration for this, but was unable to do so.

Nonetheless, we can trace the origins of the modern interpretation of this invalid proof to Riemann, who “proved” that 2+2=5. Another great mathematician, the legendary Gauss, also “proved” that 2+2=3.

Finally, Gottlob Frege was also one of the mathematicians who got involved in trying to demonstrate that 2+2=5