# Different use of approximate equality symbols

I have been wondering for a long time whether there is a unequivocal way to define and use the symbols commonly adopted for an approximate equality between two quantities. I am a physicist, and I often see them used interchangeably and more or less only accordingly to the taste of the author or lecturer.

But what is a reasonable and clear way to differentiate between these symbols?

I refer to:

$$\approx\ \simeq\ \sim\$$

• None of those have precise meanings until you specify that meaning (the middle one is even sometimes used for isomorphisms). – Tobias Kildetoft Apr 12 '16 at 10:17
• What would be a possible choice of the different meaning that makes use of all three? Thanks! – usumdelphini Apr 12 '16 at 10:20
• I can think of probably at least 10 different things I might use these to denote, so I don't see any obvious choices. – Tobias Kildetoft Apr 12 '16 at 10:21
• Assuming you are only using these for numbers, I would use $\sim$ as "approximately", $\approx$ as "approximately equal" and never use $\simeq$. For example "The table is $\sim 4$ feet in length" or "$\pi\approx 3.1415$". So, when saying two numbers are almost the same $\approx$ when saying one number is almost the desired quantity $\sim$. – Sean English Apr 12 '16 at 10:29

## 1 Answer

The symbol $\sim$ is usually used for asymptotic equivalence for functions.

One says that $f\sim_a g$ if $$\lim_{x \to a} \frac{f(x)}{g(x)} =1.$$ If no $a$ is precised (which is the most usual case), the limit is taken at infinity.

On the other side, $\approx$ and $\simeq$ are used for decimal approximation for numbers. Example $$\pi \approx 3,14.$$ We prefer $\approx$ to $\simeq$ since $\simeq$ denotes often an isomorphism (for example between two groups, two rings).

In conclusion, the difference between these symbols is made thanks to the mathematical objects for which there are defined.

• Functions : $\sim$
• Numbers : $\approx$
• Groups, rings (in general : category theory) : $\simeq$

NB : These are the conventions that I often read in the literature, no doubt that can be others.

• Thank you very much! What about Taylor series and approximate values of variables. Let's say,respectively: $f(x)\simeq f(x_0)+f'(x_0)(x-x_0)$; and $r=aB/2\simeq 2$? – usumdelphini Apr 12 '16 at 10:26
• For each $x$, $f(x)$ is a number so I recommend to use the number approximation $f(x) \approx f(x_0)+f'(x_0)(x-x_0)$. – C. Dubussy Apr 12 '16 at 11:02