In the world of classical computers, everything boils down to bits. A bit is the simplest possible piece of information, a tiny switch that is either OFF or ON. We label these two states 0 and 1. There are no other options. A classical bit has a definite state. It's one or the other, never something in between.

Quantum computing starts by throwing this simple rule out the window. Its basic unit of information, the qubit, is much more flexible. Instead of being a simple switch, a qubit's state is described by a vector. For a single qubit, this vector lives in a special two-dimensional space. Think of it like a point on a standard 2D graph, with an x-axis and a y-axis. The state of the qubit is an arrow pointing from the origin to that point.

To make sense of this new landscape, we need some landmarks. In our 2D qubit space, we define two special vectors that correspond to the classical 0 and 1. These are called the computational basis states and they form the fundamental building blocks for all other qubit states. We give them a special notation, called Dirac notation or bra-ket notation, which is the standard language of quantum mechanics.

The state corresponding to the classical '0' is written as $|0\rangle$. The state for '1' is written as $|1\rangle$. That vertical bar and right angle bracket make up a 'ket', which is just a fancy name for a column vector in this context. These kets are our fundamental reference points.

These are the two 'pure' states. If a qubit is definitely in state $|0\rangle$ or definitely in state $|1\rangle$, it behaves just like a classical bit. When you measure it, you'll get 0 or 1 with 100% certainty. The real power, however, comes from mixing them.

A qubit doesn't have to be just $|0\rangle$ or just $|1\rangle$. It can be a combination of both. This is where vector addition comes in. We can create a new state, let's call it $|\psi\rangle$ (the Greek letter psi), by adding together some amount of $|0\rangle$ and some amount of $|1\rangle$.

Mathematically, we write this general state as:

$|\psi\rangle = \alpha |0\rangle + \beta |1\rangle$

Here, $\alpha$ (alpha) and $\beta$ (beta) are just numbers that tell us how much of each basis state is in the mix. They are called amplitudes. By choosing different values for $\alpha$ and $\beta$, we can describe any possible state for our qubit.

This is the core idea. Instead of being stuck at (1, 0) or (0, 1), a qubit's state vector can point anywhere in this 2D space. The direction and length of this vector contain all the information about the qubit. We've moved from a simple on/off switch to a rich, continuous landscape of possibilities.

This vector representation is the starting point for everything else in quantum computing. It allows us to use the powerful tools of linear algebra to describe and manipulate quantum information.