# What does the sign $\propto$ mean?

I am a computer scientist, and one of my professors today used the symbol $\propto$. I tried to search that using google, but it returns no results, and I do not even know its name.

So, I would like to ask, what does the operator $\propto$ mean? What is its name, and how is it used?

• The relation $\propto$ is an equivalence relation that relates from the set $A$ to $B$ and can be defined as the set $\{(a, b)\in A\times B:a=kb\}$ for some constant $k$. – Ali Caglayan Jan 26 '14 at 18:19

$A\propto B$ means that $A$ is directly proportional to $B$. This means that $A=kB$ for some constant $k$.
Since you say you're a computer scientist, it should be mentioned that Garey & Johnson use $A \propto B$ in their book Computers and Intractability – A guide to the Theory of NP-Completeness [1, page 34] to mean that there is a polynomial transformation from $A$ to $B$.
Besides direct proportionality, you might also see something like $$\frac{4n^2+1}{n}\propto n \text{ as }n\to\infty$$ which might rigorously be defined as: $$\exists C \forall \varepsilon>0 \exists N\forall n>N: \left|\frac{4n^2+1}{n}-Cn\right|<\varepsilon$$ meaning there exists a constant $C$ such that for large $n$, $\frac{4n^2+1}{n}$ is approximately equal to $Cn$ (and in this case $C=4$).
The definition I gave above is very strong, you might weaken it to $$\exists c_1,c_2\exists N\forall n>N: c_1n<\frac{4n^2+1}{n}<c_2n$$ in which case we would also have $\frac{4n^2+1}{n}+\sin(n)\propto n$ (which would make sense because the sine becomes less important as $n$ grows).
• This is very common especially in older books; new books tend to use big-O notation and its variants, so that the statement would be written as $\frac{4n^2+1}{n}\in\Theta(n)$. Physics also tends to use it for the behavior of e.g. a field strength. – Steven Stadnicki Jan 27 '14 at 17:15