Understanding the Quantum States of a Quantum Computer?

I was trying to understand (more usefully) the states in addition to zero and 1 found in a Quantum Computer and was having a little difficulty with that.

The easiest way for me to articulate what I understand would be to say that there is the regular digital zero and 1 that all silicon based computers use for calculations and symbolic representations at a machine language level before there are compilers or languages which are above that.

In a quantum computer there is also zero and one but also a SUPERPOSITION of zero and one and everything in between the two. As I think about this even though people say calculations can be made it doesn't seem very stable a proposition because it seems like the base line is not mathematical but only formulaic like an Algorithm. But, with this kind of algorithm how to do you get to elementary or even Algebraic mathematics and logic? This is where I can't quite understand how they plan to reconcile mathematics and logic with the quantum world which lends itself to formulas but which appears to me to leave the realm of logic and basic mathematics and algebra?

So, where is the reconciliation back into a logical rational mathematical world? OR maybe there is no reconciliation and we are going to be dealing with something else entirely.
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The Bloch sphere is a representation of a qubit, the fundamental building block of quantum computers.

Image used under the GNU Free Documentation License 1.2

Defining the Quantum Computer

The Turing machine, developed by Alan Turing in the 1930s, is a theoretical device that consists of tape of unlimited length that is divided into little squares. Each square can either hold a symbol (1 or 0) or be left blank. A read-write device reads these symbols and blanks, which gives the machine its instructions to perform a certain program. Does this sound familiar? Well, in a quantum Turing machine, the difference is that the tape exists in a quantum state, as does the read-write head. This means that the symbols on the tape can be either 0 or 1 or a superposition of 0 and 1; in other words the symbols are both 0 and 1 (and all points in between) at the same time. While a normal Turing machine can only perform one calculation at a time, a quantum Turing machine can perform many calculations at once.
Today's computers, like a Turing machine, work by manipulating bits that exist in one of two states: a 0 or a 1. Quantum computers aren't limited to two states; they encode information as quantum bits, or qubits, which can exist in superposition. Qubits represent atoms, ions, photons or electrons and their respective control devices that are working together to act as computer memory and a processor. Because a quantum computer can contain these multiple states simultaneously, it has the potential to be millions of times more powerful than today's most powerful supercomputers.

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http://computer.howstuffworks.com/quantum-computer1.htm