Symbols are used instead of an equal sign

In summary, the given symbols have different meanings in mathematics. The first one can denote 'to be in a relation' or 'to be similar to', while the second means 'approximately'. The third can mean 'to be congruent' or 'to be homotopic to', and the fourth means 'isomorphic'. These symbols are often used in equations and formulas to represent different relationships between variables. The term 'asymptotically equal' refers to a situation where the values approach but never fully reach the value given by a formula. This can be seen in a graph where the peak-to-peak distances are not equal, but the averaged distances approach the value given by the formula as more cycles are simulated.
  • #1
tony873004
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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?

[tex]
\begin{array}{l}
\sim \\
\approx \\
\simeq \\
\cong \\
\end{array}
[/tex]
 
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  • #2
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.
 
  • #3
The first is also used in the narrower sense "asymptotic to".
 
  • #4
Doesn't the first one mean "goes as", i.e. is functionally similar, but not neccessarily asymptotically?
 
  • #5
I think
first-similar to(as in similar triangles)
second-approximately equal to
third-asymptotically equal to
fourth-congruent to
 
  • #6
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:
[tex]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} [/tex]
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?
 
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  • #7
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.
 
  • #8
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
 
Last edited:

What is the purpose of using symbols instead of an equal sign?

The purpose of using symbols instead of an equal sign is to represent a mathematical relationship between two quantities. Symbols can be used to show equality, inequality, or other mathematical operations.

What are some common symbols used in place of an equal sign?

Some common symbols used in place of an equal sign include the greater than (>), less than (<), not equal to (!=), approximately equal to (~=), and proportional to (∝) symbols.

Why would someone choose to use symbols instead of an equal sign?

Someone may choose to use symbols instead of an equal sign because it allows for more flexibility and specificity in representing mathematical relationships. It also allows for more concise and efficient mathematical notation.

Are there any rules for using symbols instead of an equal sign?

Yes, there are rules for using symbols instead of an equal sign. These symbols must be used correctly and in the appropriate context to accurately represent the mathematical relationship between the quantities.

Can symbols be used interchangeably with equal signs in all mathematical equations?

No, symbols cannot be used interchangeably with equal signs in all mathematical equations. Different symbols represent different mathematical relationships and using the wrong symbol can result in an incorrect equation or solution.

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