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
>the logic of why going from “y*x=z” to “z/x=y” is possible.
The answer y to the division problem z/x is defined to be the number y such that y*x = z. That is what division *means* period, without any reference to something else. It is defined to be the opposite operation to multiplication. The same is true about subtraction, for the problem z-x=y, y is the number such that y+x = z.
If it helps, addition and multiplication are things you can do on a more basic level, subtraction and division are made up based on analogous notions that they reverse multiplication and addition. Division and subtraction come with problems that you might be familiar with. Examples: Trying to divide 3 hard candies among 5 people, trying to buy 5 apples from a vendor that only has 3. Similar setups for addition and multiplication do not have issues of needing to resort to fractional hard candies or negative numbers of apples to explain away the problem.
>I ask this in relation to ” /(x/y) ” = ” *(y/x) ”
Multiplication (so long as you define fractions and division) admits a type of number called inverses, which is that if I have some number i and some number j that are inverses, then x*i*j = x. The inverse property is that multiplication by j undoes the multiplication by i, and vice versa. It’s a requirement of inverses that i*j = 1. Since x*1 = x, if you do the i*j multiplication first in x*i*j, then you ought to get 1.
Combining these properties, i*j = 1 implies by the definition of division, that 1/j = i. If we bring in some other number k, well then k*1/j = k/j ought to be the same as k*i. So inverses i and j have this property where multiplying by i is the same thing as dividing j.
Turns out the quantity x/y and the quantity y/x are inverses, so you can do this. Multiplying by one is the same as dividing by the other. You can see that they’re inverse by multiplying them together: (x/y)*(y/x) = (x*y)/(y*x) = 1
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