| Re .0:
There are real numbers which are not the quotient of any two integers,
or any two rational numbers. For example, there do not exist any
rational numbers p and q such that p/q equals the square root of two.
Numbers of that sort are called irrational, meaning "without ratio".
You asked about "non irrational" numbers; the real numbers can be
partitioned into the rational numbers and the irrational numbers.
Every real number is one or the other, but not both. Every rational
number can be expressed in the form p/q, where p and q are integers.
Every irrational number cannot be expressed in that way.
Here is a hierarchy of types of numbers:
If you count, starting at zero, you get the numbers 0, 1, 2, 3, 4, and
so on. These numbers are closed under addition. That means you can
add any two of the numbers and you will get another of those numbers.
They are not closed under subtraction; if you subtract 7 from 3, you do
not get any of those numbers. So we construct the negative integers,
which are the "additive inverses" of the positive integers.
The integers are closed under addition, subtraction, and
multiplication, but not division. There is no integer that is equal to
2/3. So we construct the ratios of all pairs of integers, and that
gives us the rational numbers. They are closed under addition,
subtraction, multplication, and division.
However, if we want to take the square root of two, no rational number
will be a correct answer. If you'd like, I can show you a fairly
simple proof of that. Using the rational numbers, we can construct a
larger set of numbers called the real numbers. I will skip that
construction for now, since it gets a bit technical, but basically it
involves partitioning the rational numbers into two sets. If you would
like to see it, just ask.
The set of real numbers that we get from that construction actually
contains several types of numbers. Obviously, it contains all the
integers and rational numbers. It also contains a type of numbers that
are called algebraic. If you write an equation such as ax^3+bx^2+cx+d
= 0, where you have a polynomial with rational coefficients equal to 0,
then the real solutions to such an equation are algebraic numbers. For
example, a solution to x^2-2 = 0 is the square root of two. That is
not a rational number, but it is an algebraic number.
There exist real numbers that are not the solution to any equation of
that sort. The number pi is such a number. Those types of numbers are
called transcendental numbers.
There is one more type of number. The real numbers are still not
closed when we consider solutions to polynomial equations and
operations such as exponentiation with fractional exponents. Thus, we
construct complex numbers in the form a+bi, where a and b are real
numbers and i is an imaginary number.
The complex numbers are algebraically closed; every solution to any
polynomial equation which has complex coefficients is a complex number.
Every algebraic operation on complex numbers -- addition, subtraction,
multiplication, division, and exponentiation -- has a result that is a
complex number.
-- edp
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