Checking commutativity property with addition table

[dalarev]

I saw the technique of using a mathematical system called the "Four minute clock". Basically, the clock has 4 numbers, 0, 1, 2 and 3, and only one hand. We can make a summation table which will look like the image attached.

According to the text, if there is symmetry (reflection) across the main diagonal, the system is commutative.


The problem came up when I attempted a 5 minute clock. Same clock, except this time we have 5 numbers: 0, 1, 2, 3, and 4. Filling out the table, we see that there is NO evident symmetry along the main diagonal, yet the commutative property still applies to this system.
Namely: a+(b+c) = (a+b)+c

The system does not pass the commutativity test, yet it still works? Am I missing something?

[matticus]

For one the property a+(b+c) = (a+b)+c is the associative property not the commutative property. The commutative property is ab = ba. I can't see your picture yet, but the table should look like

01234
12340
23401
34012
40123

Where the entry in the position i,j is i + j (mod 5).
This table is symmetric about the main diagonal, which verifies that it is commutative. It is also associative, however there is no quick check to determine this property.

[dalarev]

For one the property a+(b+c) = (a+b)+c is the associative property not the commutative property. The commutative property is ab = ba.

You're correct about that, as confirmed here:
http://tutorial.math.lamar.edu/Classes/LinAlg/PropsOfMatrixArith.aspx


01234
12340
23401
34012
40123

Where the entry in the position i,j is i + j (mod 5).
This table is symmetric about the main diagonal, which verifies that it is commutative. It is also associative, however there is no quick check to determine this property.

That is the correct table, but according to these notes (new image I have attached), there should be reflection ALONG the main diagonal, not only surrounding it.

[matticus]

when they say there is a reflection along the main diagonal they mean that entry i,j = j,i, (i.e. i + j = j + i, the definition of commutivity. it's not just magic!) which you can see is happening here.

[dalarev]

when they say there is a reflection along the main diagonal they mean that entry i,j = j,i, (i.e. i + j = j + i, the definition of commutivity. it's not just magic!) which you can see is happening here.

Ahh, I see. Good explanation, thank you for that.

/thread solved