User:Barnaby dawson/extention of numbers

The development of the concept of number
Mathematics did not start with the concept of the complex numbers. It took many years and much discussion to get this far. Roughly speaking over time mathematicians have broadened the definition of number. Opinions differ as to how to treat the complex numbers philosophically.

Many of the extentions of the concept of number can be seen as responses to equations that would otherwise have had no solution. In each of the extentions given below we start with an equation and then give the extention to the system which allows the equation to be solved. We start with the notion of positive integers and 0 although it should be noted that some ancient mathematics did not have the concept of 0. Also note that it was assumed that the normal algebraic operations $$+\ -\ *\ /$$ return only one value (division by zero is not defined).


 * $$X+1=$$ This equation requires the existence of negative numbers such as $$-1$$ for its solution. The word negative was originally used by those who opposed the introduction of such numbers.
 * $$5X=3$$ This equation requires the existence of fractional numbers for its existence. If we allow the solution of all equations of the form $$mX=n$$ then we get the rational numbers (m and n are both integers).
 * $$X*X-2=0$$ has no rational solution. Mathematicians responded by introducting radicals which allowed many polynomials to be solved.
 * $$X*X+1=0$$ is the equation that introduces us to the complex numbers. Many people argued that it was just an imaginary construct to solve the cubic and shouldn't be considered 'real'.  This is the origin of the terms imaginary and real.  However it was found that a whole new beautiful world of complex numbers opened up if you did allow them.  To represent a solution to this equation mathematicians chose the letter i.  Even with all of these extensions of the naturals we are still not finished.

In order to construct the complex numbers we need only one more assumption: Any set of real numbers with an upper bound has a least upper bound.  This fills in the real line with all of the irrational numbers that cannot be derived mearly from the equations above. It is worth noting that this is an entirely different type of extension. This is because of the cardinality of complex numbers is greater than that of the rationals. Once this is done all polynomial equations can be solved (although this can be done in smaller fields than the complex numbers).

Mathematicians today rarely view the development of the complex numbers in this way (the prefered teaching method does not emphasize this stepwise development) but it demonstrates the tention in mathematics between the rigorous and the creative which is the main power behind much of modern mathematics.

Interestingly the independence of the continuum hypothesis can be seen as an inability to prove whether or not certain real numbers should be thought to exist.