mazdatweaker
+1y
Shalom, I hope you are having a good Day of rest, as I M.
As I was seeking the definition of "congruence," I found this:
Congruence is the state achieved by coming together, the state of agreement. The Latin congruere means to come together or agree. As an abstract term, congruence means similarity between objects. Congruence, as opposed to equivalence or approximation, is a relation which implies a kind of equivalence, though not complete equivalence.
-------
That is why I can read the Scripture in one place, and reflect what I have learned into this thread.
Just like any other life experience.
This is why Jesus could say, "I and Abba, I"
Here is some interesting news for all you Internet_savvy professionals . . . I guess that includes anyone who finds US out here in a virtual universe
The following cut and paste shows the Internet version of the real time game "Operator."
TAKEN DIRECTLY ON A CUT AND PASTE FROM
" target="_blank" target="_blank
If you go to that site, you will find all the information that did not transfer verbatim to this site. I say all that to indicate this:
I still teach.
--------------
Congruence arithmetic is perhaps most familiar as a generalization of the arithmetic of the clock. Since there are 60 minutes in an hour, "minute arithmetic" uses a modulus of . If one starts at 40 minutes past the hour and then waits another 35 minutes, , so the current time would be 15 minutes past the (next) hour.
Similarly, "hour arithmetic" on a 12-hour clock uses a modulus of , so 10 o'clock (a.m.) plus five hours gives , or 3 o'clock (p.m.)
Congruences satisfy a number of important properties, and are extremely useful in many areas of number theory. Using congruences, simple divisibility tests to check whether a given number is divisible by another number can sometimes be derived. For example, if the sum of a number's digits is divisible by 3 (9), then the original number is divisible by 3 (9).
Congruences also have their limitations. For example, if and , then it follows that , but usually not that or . In addition, by "rolling over," congruences discard absolute information. For example, knowing the number of minutes past the hour is useful, but knowing the hour the minutes are past is often more useful still.
Let and , then important properties of congruences include the following, where means "implies":
1. Equivalence: (which can be regarded as a definition).
2. Determination: either or .
3. Reflexivity: .
4. Symmetry: .
5. Transitivity: and .
6. .
7. .
8. .
9. .
10. .
11. and , where is the least common multiple.
12. , where is the greatest common divisor.
13. If , then , for a polynomial.
Properties (6-8) can be proved simply by defining (3)
(4)
where and are integers. Then (5)
(6)
(7)
so the properties are true.
Congruences also apply to fractions. For example, note that (8)
so (9)
To find (mod ) where (i.e., and are relatively prime), use an algorithm similar to the greedy algorithm. Let and find (10)
where is the ceiling function, then compute (11)
Iterate until , then (12)
This method always works for prime, and sometimes even for composite. However, for a composite , the method can fail by reaching 0 (Conway and Guy 1996).
Finding a fractional congruence is equivalent to solving a corresponding linear congruence equation (13)
A fractional congruence of a unit fraction is known as a modular inverse. A fractional congruence can be found in Mathematica using the following function:
FractionalMod[r_Rational, m_Integer] := Mod[
Numerator[r]PowerMod[Denominator[r], -1, m], m]
or using the undocumented syntax PolynomialMod[r, m] for an explicit rational number.
------------------
IF, you have read this far, thank you. If what you have read provokes thought, wonderful, please leave quietly.
/s