Unsolvable Math Conjectures

In the last month, the entire math community has been shaken by revelations in proof theory. These three simple but revolutionary proofs have changed mathematics forever, bringing light to secrets that disprove millennia of postulates and theorems we thought were true. (In all seriousness, none of these are true. It’s a fun challenge to try to figure out why!)

Let’s start with a proof that 1=2. Warning: this includes calculus. First, we can start off with  x^2 = x + x+...+x with x amount of x’s on the right. This is true by the definition of multiplication. Next, we can take the derivative of both sides. In calculus, the derivative of a function is the rate of change at a point and is written d/dx f(x) (meaning the derivative of a function of x). You can think of dx as a really really tiny change in x, and d as the change in f(x). So, for example, when f(x) = x^2, dx^2/dx means how much x^2 changes for a tiny, microscopic change in x. (For any equation, you can take the derivative of both sides and the equation will remain true.) 

So, we have dx^2/dx=d/dx (x+x+...+x). The derivative of x^2 is 2x, giving 2x=d/dx (x+x+...+x) = d/dx (x)+d/dx (x) + ... +d/dx (x). The derivative of x is 1 because the change in x over the change in x is just 1. Thus, we have 2x = 1+1+...+1 = x. Divide by x and you get 2 = 1.

Next is a newly discovered proof that all natural numbers can be described in 14 English words or fewer! We can prove this by contradiction. First, we assume that the statement is false. That means a number exists that cannot be described in 14 words or fewer. Since these numbers exist, there must also be a smallest number that cannot be described in 14 words or fewer. Call this number n. However, n can be described as “The smallest natural number that can not be described in fourteen words or fewer,” which is, in fact, 14 words. Thus, n does not exist since the definition of n does not hold, meaning that all natural numbers can be described in 14 words or fewer. In addition, since there are a finite number of English words, there are a finite number of 14-word arrangements of words. So, this proof shows that there is also a finite number of natural numbers! 

The final proof involves an equation whose answer can either be positive or negative infinity! We start with the arithmetic sequence S = 1-2+3-4+5-6+7-8+9- .... If we insert parentheses, we get S=(1-2)+(3-4)+(5-6)+(7-8)+ .... Now, this can be simplified to S = (-1)+(-1)+(-1)+(-1)+ ..., and since it’s an infinite series, it is equivalent to - ∞. Now, if we group the terms in a different way, shifting the parentheses one number farther, we get S = 1+(-2+3)+(-4+5)+(-6+7)+ .... This can be simplified to S=1+(1)+(1)+(1)+(1)+ .... This is +∞. We’ve created a paradox just by grouping numbers in different ways! 

Next time you encounter a math enthusiast or teacher, make sure to show them your newly acquired knowledge! We’re sure that they will recognize your incredible problem-solving skills and creative abilities!