2.6: The function [x]. the symbols "O", "o" and "∼"
We start this section by introducing an important number theoretic function. We proceed in defining some convenient symbols that will be used in connection with the growth and behavior of some functions that will be defined in later chapters.
The Function \([x]\)
The function \([x]\) represents the largest integer not exceeding \(x\). In other words, for real \(x\), \([x]\) is the unique integer such that
\[x-1<[x]\leq x<[x]+1.\]
We also define \(((x))\) to be the fractional part of \(x\). In other words \(((x))=x-[x]\).
Note
We now list some properties of \([x]\) that will be used in later or in more advanced courses in number theory.
- \([x+n]=[x]+n\) , if \(n\) is an integer.
- \([x]+[y]\leq [x+y]\) .
- \([x]+[-x]\) is 0 if \(x\) is an integer and -1 otherwise.
- The number of integers \(m\) for which \(x<m\leq y\) is \([y]-[x]\).
- The number of multiples of \(m\) which do not exceed \(x\) is \([x/m]\).
Using the definition of \([x]\), it will be easy to see that the above properties are direct consequences of the definition.
We now define some symbols that will be used to estimate the growth of number theoretic functions. These symbols will be not be really appreciated in the context of this book but these are often used in many analytic proofs.
The "O" and "o" Symbols
Let \(f(x)\) be a positive function and let \(g(x)\) be any function. Then \(O(f(x))\) (pronounced "big-oh" of \(f(x)\))denotes the collection of functions \(g(x)\) that exhibit a growth that is limited to that of \(f(x)\) in some respect. The traditional notation for stating that \(g(x)\) belongs to this collection is: \[g(x)=O(f(x)).\] This means that for sufficiently large \(x\),
\[\frac{\mid g(x)\mid }{|f(x)|}<M,\]
here \(M\) is some positive number.
\(\sin (x)=O(x)\), and also \(\sin(x)=O(1)\).
Now, the relation \(g(x)=o(f(x))\), pronounced "small-oh" of \(f(x)\), is used to indicate that \(f(x)\) grows much faster than \(g(x)\). It formally says that
\[\lim_{x\rightarrow \infty}\frac{g(x)}{f(x)}=0.\]
More generally, \(g(x)=o(f(x))\) at a point \(b\) if
\[\lim_{x\rightarrow b}\frac{g(x)}{f(x)}=0.\]
\(\sin(x)=o(x)\) at \(\infty\), and \(x^k=o(e^x)\) also at \(\infty\) for every constant \(k\).
The notation that \(f(x)\) is asymptotically equal to \(g(x)\) is denoted by \(\sim\). Formally speaking, we say that \(f(x) \sim g(x)\) if
\[\lim_{x\rightarrow \infty}\frac{f(x)}{g(x)}=1.\]
\([x] \sim x\).
The purpose of introducing these symbols is to make complicated mathematical expressions simpler. Some expressions can be represented as the principal part that you need plus a remainder term. The remainder term can be expressed using the above notations. So when you need to combine several expressions, the remainder parts involving these symbols can be easily combined. We will state now some properties of the above symbols without proof. These properties are easy to prove using the definitions of the symbols.
- \(O(O(f(x)))=O(f(x))\) ,
- \(o(o(f(x)))=o(f(x))\) .
- \(O(f(x))\pm O(f(x))=O(f(x))\) ,
- \(o(f(x)\pm o(f(x))=o(f(x))\) ,
- \(O(f(x))\pm O(g(x))=O(\max(f(x), g(x)))\) ,
There are some other properties that we did not mention here, properties that are rarely used in number theoretic proofs.
Exercises
- Prove the five properties of the \([x]\)
- Prove the five properties of the \(O\) and \(o\) notations in Example 24.
Contributors and Attributions
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Dr. Wissam Raji, Ph.D., of the American University in Beirut. His work was selected by the Saylor Foundation’s Open Textbook Challenge for public release under a Creative Commons Attribution ( CC BY ) license.