You probably have heard of the imaginary number i. i is the square root of -1, and one of the solutions to the equation x^2= -1. i has many interesting properties such as denoting a 90° rotation on the imaginary plane and also being in what is considered one of the most beautiful equations in math: e^i(pi)+1=0. The beauty in the equation is the fact that it combines all of the most prominent mathematical constants into one equation. Some people even considered it beautiful enough to be a proof of god. i can even be found in specific quantum mechanics formulas that have real world application, one of which allowed chemist Linus Pauling to discover orbital hybridisation.
Before the 16th century mathematicians thought i as a number could not possibly exist, as it seemed contradictory at the time, until mathematician Gerolamo Cardano broke the rules of 16th century math and created a new set of numbers: imaginary numbers. While the current definition of an imaginary number is a number with the term i in it, I hope by the end of this article you will accept a new definition.
While you probably have heard of i, a number you likely have not heard of is the imaginary number L. The general definition is that L is a solution to |x|=-1. A math teacher has probably told you |x|=-1 has no solutions. However, that is only due to the current mathematical definitions of what the absolute value function can output. If we define the absolute value function to be able to output negative numbers when the input is imaginary, we can then define a new constant L such that |L|=-1. We now have two solutions to the equation |x|=-1 where one solution is L and the other solution is -L similar to the equation x^2= -1 having the solutions i and -i.
Due to L not actually being proven to exist in formal mathematics, and the fact that it has barely been studied, there are not many properties of L that are generally known. However, if one assumes the existence of L, there are a few equations that can be derived from algebra such as :
If L were to exist it could have a lot of implications on mathematics, one of them being that imaginary numbers would have to be redefined to include L within the set—or else a new set of imaginary numbers would have to be created altogether. While there are no currently known real world applications of L, some are likely to exist that have simply not been discovered. It took the world hundreds of years before anyone was able to find a use for i, so who’s to say we won’t find a use for L? However, until L is proven to exist and becomes accepted in mathematics, it is probably best to write “no solution” to |x|= -1.
コメント