I ask this in relation to ” /(x/y) ” = ” *(y/x) ”
My mathematical ignorance does not allow me to perceive exactly what it is that confuses me about these manoeuvres and so perhaps my question is vague.
I have no difficulty with it as a technique; as something through which I can put an expression, and out at the other end the right result will appear. What I am trying to understand is *why it works*, contrasted with remembering it as a kind of magical spell.
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**EDIT:**
It was very rewarding for me to read all of your comments. Thank you most kindly for enlightening me.
For those interested in the cause of my previous confusion:
The gaps in my understanding of going from y*x=z to y=z/x were definitions of the equal sign and division.
I can see now that I previously considered the = sign to mean «result» or «answer» in some sort of final sense, like a conclusion; I now see that it only states that this is equal to that.
Following this fundamental piece of knowledge, I can belatedly understand what an equation is. From there, via the definition of division as the opposite of multiplication, I can see that if I divide something while also multiplying it with the same number, these actions cancel each other out.
And so the magical spell between y*x=z and y=z/x is the logic above expressed mathematically as x/(y*x)=z/x.
In: 25
I have y marbles in each cup, and I have x cups, so I have z marbles all together ( y*x=z ). Now I take all z marbles and divide them out again into x cups. Each cup has y marbles. ( z/x=y )
I have 3 marbles in each cup, and I have 4 cups, so I have 12 marbles all together ( 3*4=12 ). Now I take all 12 marbles and divide them out again into 4 cups. Each cup has 3 marbles. (12/4=3 )
Both “arrangements” of marbles are equally valid. The marbles continue to exist without losing nor gaining any marbles.
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