Symbols are used instead of an equal sign

[tony873004]

Sometimes the following symbols are used instead of an equal sign. I think the single one ~ means "is proportional to". One of the other ones means "is approximately". Which one is it? I'm guessing the second one because LaTex calls it approx. When are the others used?

$$\begin{array}{l} \sim \\ \approx \\ \simeq \\ \cong \\ \end{array}$$

 

[radou]

The first one sometimes denotes 'to be in a relation'. The second one, means 'approximately'. The third one can mean 'to be congruent'. I'm not sure about the fourth one. Do some google-ing, and you should find out easily.

 

[arildno]

The first is also used in the narrower sense "asymptotic to".

 

[Matthew Rodman]

Doesn't the first one mean "goes as", i.e. is functionally similar, but not neccessarily asymptotically?

 

[AlbertEinstein]

I think
first-similar to(as in similar triangles)
second-approximately equal to
third-asymptotically equal to
fourth-congruent to

 

[tony873004]

I found these pages that lists all the symbols
http://www.dessci.com/en/support/mathtype/tech/encodings/mathpi3.htm
http://www.gomath.com/htdocs/ToGoSheet/Algebra/mathsymbols.html

1st: similar to
2nd: approximately equal
3rd: asymptotically equal
4th: congruent

I'm not quite sure what asymptotically equal refers to. I Googled it but I'm still confused.

If I have a formula:
$$P_{KOZ} \simeq P_1 \left( {\frac{{m_0 + m_1 }}{{m_2 }}} \right)\left( {\frac{{a_2 }}{{a_1 }}} \right)^3 \left( {1 - e_2^2 } \right)^{3/2}$$
And plugging in the numbers I find that the answer is asympototically equal to 330,000, but observations reveal a value of 220,000, is this too much of a descrepancy to be considered asymptotically equal?

 

[matt grime]

The only symbols there that have a (reasonably) unique meaning are the 2nd and 4th. The second means approximately, as in pi is approximately 3.14, and the 4th means isomorphic. The first and the third have many meanings, ranging from 'relates' (are in the same equivalence class) to 'is homotopic to' respectively.

 

[tony873004]

I think I might know what "asymptotically equal" refers to in this situation.

After plotting a graph from the results of a celestial mechanical simulation, I get a sinusoidal graph. But the spacing between peaks is not equal. That formula above tries to compute the period of oscillation. But since each period is different, the formula can't possibly give a correct answer. But as more and more cycles are simulated, the averaged period approaches the value given by the formula. It's like an asymptote on a graph, approaching but never reaching a value. Maybe that's why he used the "asymptotically equal" symbol. Just my guess, any thoughts?

Edit**
Here's a graph illustrating my thought. Notice that the peak-to-peak distances are not the same from cycle to cycle. But perhaps if you had an infinite number of them their averaged distances would equal the formula's answer, hence "asympototically equal to"
http://www.orbitsimulator.com/gravity/images/acGraph01.GIF