What is the significance of the different sets of numbers such as real, rational, irrational etc.?
1 Answer
A few thoughts...
Explanation:
There is way too much that could be said here, but here are a few thoughts...
What is a number?
If we want to be able to reason about numbers and the things that they measure or provide the language to express then we need firm foundations.
We can start from whole numbers:
When we want to expression more things, we come across the need for negative numbers too, so we expand our idea of numbers to the integers:
When we want to divide any number by any non-zero number then we expand our idea of numbers to rational numbers
Then we come across inconveniences like the fact that the diagonal of a square with rational sides has a length we cannot express as a rational number. To fix that we have to introduce square roots - a type of irrational number. Square roots allow us to solve equations like:
#x^2+4x+1 = 0#
Often when we deal with irrational numbers like
Note that the numbers we have talked about so far have a natural total order - we can place them on a line in such a way that any two numbers can be compared.
What about the whole line?
It is commonly known as the real number line, with each point of the line being associated with a number.
How can we reason about numbers on this line in general?
We can use the total ordering, arithmetic properties and characterise real numbers in terms of limits. In general, reasoning about real numbers involves more of that kind of thinking.
So does mathematics get more complicated as we go from reasoning about natural numbers to reasoning about real numbers? No, it gets different - very different. For example, an unsolved problem in mathematics is:
Are there an infinite number of prime pairs - i.e. pairs of numbers
#p# and#p+2# such that both are prime.
It sounds simple enough, but about the best we can do so far is to show that there are an infinite number of prime pairs of the form
