# Factorial! and the commutative property of multiplication

In how many different ways can we arrange three letters A, B, and C? There are three candidates for the third position that leaves the two remaining letters for the second position and so 3 times 2 is 6 and One is the multiplicative identity I am astonished by The commutative property of multiplication because it seems as if the system were thought up in some very convenient way. I mean How do you get any two numbers when multiplied or added together in either order to be the same number? I am simply astonished! Math is so very elegant! How does the mathematics system know these things. Albert Einstein stated that nature reveals an intelligence of such superiority that compared with it all the systematic thinking and acting of human beings is an utterly insignificant reflection.
3 times 2 times 1 or 1 times 2 times 3 For 3! how does the commutative property apply?

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jedishrfu
Mentor
The commutative property is an outcome of how we defined the addition and multiplication operations. These definitions are a consequence of the way we view the world.

If I take 6 apples and place 2 to the right and 4 on the left hence (2 + 4 = 6) then someone facing you will see 2 on the left and 4 on the right hence (4 + 2 = 6) as we still have six apples. Hence 2 + 4 must equal 4 + 2.

Similarly for multiplication, if I organize my collection of 6 apples as 2 rows of 3 apples (2x3=6) then someone standing to my right might see 3 rows of 2 apples (3x2=6) since we both see the same number of apples then 2x3 must equal 3x2.

Nugatory, berkeman, fresh_42 and 1 other person
Drakkith
Staff Emeritus
I'm no mathematician, but I believe it is important to look not for grand, fantastic reasons that math works the way it works, but to go the other way and look at the basics. Look at how the basic operations are defined and how they themselves then define other operations and relationships. You'll find that it quite naturally leads to such rules as the commutative property.

phinds and jedishrfu
What comes to mind here is the Pascal triangle, and its many applications.
A few of them stressing the symmetries can be found on:
http://www.mathsisfun.com/pascals-triangle.html
Infinity commonly inspires feelings of awe, futility, and fear. Who as a child did not lie in bed filled with a slowly mounting terror while sinking into the idea of a universe that goes on and on, for ever and ever? Blaise Pascal puts this feeling very well: "When I consider the small span of my life absorbed in the eternity of all time, or the small part of space which I can touch or see engulfed by the infinite immensity of spaces that I know not and that know me not, I am frightened and astonished to see myself here instead of there... now instead of then." "I am no longer frightened by our situation here in the Cosmos but only astonished to see ourselves here instead of there now instead of then." Terrestrial officers Tyler and Denver On Process Being and the Absolute Infinite Twilight Zone the Movie goes really good with this quote from Blaise Pascal The part of the movie to watch is about the young boy who could wish people away or into a television and get eaten by a cartoon monster he has complete control the of universe and the absolute infinite!

Mark44
Mentor
Infinity commonly inspires feelings of awe, futility, and fear. Who as a child did not lie in bed filled with a slowly mounting terror while sinking into the idea of a universe that goes on and on, for ever and ever?
I didn't. If I had, my mother probably would have given me a dose of cod liver oil.

Terrestrial officer
jedishrfu
Mentor
As a child I had a fear that a robot would break into my room. I think this was after I saw the movie Target Earth on TV. Humanity eventually prevailed when it was discovered that a high frequency sound would shatter their vacuum tube computer. Hey it was the 1950/s before the advent of the transistor.

http://www.bmoviecentral.com/bmc/reviews/118-target-earth-1954-75-minutes.html

and the movie clip, still quite terrifying:

Also as a child, I had dreams that I was shrinking though now I believe I was in tune with the universe expanding. :-) Not!

Children don't think so deeply on such things as their focus is on the immediate world around them including what they see on TV. After 911, some really younger children were in great fear as the saw the repeated broadcasts of the World Trade Center collapse and thought Oh no, another building collapsed!" not understanding it was a recording being replayed.

With that note this thread has run its course and will now be closed.

Thanks to everyone for participating.

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berkeman