Lies, Damn Lies

Maths part 4- functions and maps

So, now we have our numbers. The Natural numbers, the integers, the rational, the real, and the complex. And hopefully you all have a vague idea of what they are. Remember that each collection of numbers contains the one preceding it, so whenever I say the complex numbers, that includes all other kinds of numbers as well.

So what next? Well I'm rather eager to talk about infinity, but along the way I think we had better talk about what a function is. You will have undoubtedly encountered them if you have done any mathematics at all, although they may not have been labeled as such.

So what is a function/ map (the words are essentially interchangable. I am not aware if there is any significant distinction between them)? Well simply put, it's something, that, given an inputted number, give an outputed number. For example f(x)=x^2 is one. You give me a value, x, and f(x) gives you back the square of that number. You will most probably have seen them in the form y=x^2. y and f(x) here are interchangable. Generally we use f(x) when talking just because it is clear that f is dependent on x, where y by itself gives you very little information.

Now, there are certain rules that functions follow. When you define a function, you generally define the "domain" that it works on. This will often be the real numbers, but could be the integers or even the complex numbers. The domain is just the set of numbers you put into the function- all the values your x can take. So for example f:R->R x->x^2 means that our f(x)=x^2 takes all the real numbers and squares them, sending them into R. Notice that becuase we are squaring we will never get negative numbers (unless we are squaring complex numbers), so we could just write f:R->[0,infinity) where [0,infinity) is the set of all real numbers from 0 to infinity. However, we can write R quite happily, as while our map never goes to the negative numbers, it's still mapping to a smaller part of the reals.

The choice of domain is quite important. For example f(x)=1/x is simply undefined at x=0 (it is NOT infinity), so the domain can't include x=0.

Another important rule for a function is that it is well defined. What does this mean? Well if I put the same number in twice, we get the same answer. For example f(x)=2/-2 can give both answers. This is quite sensible, as it means my function will do predictable things.

So, lets recap. We have a function, f:D->S where D and S are some collection of objects. Notice that these don't have to be one dimensional- for example f(x,y)=x+y takes from a two dimensional space to a one dimensional space, and oddest of all do not have to be numbers. For example the function capital(country) could give capital(france)=paris. This meets all our requirements for functions.

I think we might get to infinity next time. I've just got to talk about sets first......

posted by Mr K @ 3:39 pm