Big-O notation is a symbolism used in complexity theory, computer science, and mathematics. It describes how fast a function grows or declines. Formally, it describes the asymptotic behaviour of functions.

Big-O notation is also called Landau's symbol, from the German theoretician Edmund Landau. The letter $\mathcal{O}$ stands for "order," which means the rate of growth of a function.

For example, an algorithm to compute a problem of size $n$ might take $T(n)$ steps to complete. This function might be $T(n) = 4n^2 - 2n + 2$. However, for $n\to\infty$ note that the behaviour of $T$ is only affected by the $4n^2$ term. Not even the coefficient has any behavioural change. Therefore we could say that $T(n)$ grows at the order of $n^2$, or

$$T(n) = \mathcal{O}(n^2)$$

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