# 13.8 Big-O {% embed url="" %} O (pronounced "Big-Oh") is similar to $$\Theta$$. Instead of being an "equality" on the order of growth, it can be though of as "less than or equal." For example, the following statements are all true: * $$N^3 + 3N^4 \in \Theta (N^4)$$ * $$N^3 + 3N^4 \in \text{O}(N^4)$$ * $$N^3 + 3N^4 \in \text{O}(N^6)$$ * $$N^3 + 3N^4 \in \text{O}(N^{N!})$$ ### **Formal Definition** $$R(N) \in \text{O}(f(N))$$ means that there exists positive constant $$k\_2$$ such that:\ $$R(N) \leq k\_2 \cdot f(N)$$ for all values of $$N$$ greater than some $$N\_0$$ (a very large $$N$$). Observe that this is a looser condition than $$\Theta$$ since O does not care about the lower bound. --- # Agent Instructions: Querying This Documentation If you need additional information that is not directly available in this page, you can query the documentation dynamically by asking a question. Perform an HTTP GET request on the current page URL with the `ask` query parameter: ``` GET https://cs61b-2.gitbook.io/cs61b-textbook-2025/13.-asymptotics-i/13.8-big-o.md?ask= ``` The question should be specific, self-contained, and written in natural language. The response will contain a direct answer to the question and relevant excerpts and sources from the documentation. Use this mechanism when the answer is not explicitly present in the current page, you need clarification or additional context, or you want to retrieve related documentation sections.