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
If I understood the question correctly, it’s because of the “=” part of the equation. Suppose you have
x=y
Then you can freely manipulate these two **so long as you do it on both sides**. Twice X should be twice Y and half X should be half Y. Then if you add another element like
x=y*z
you just play around with that.
x=y*z (/y)
x/y) = y*z/y
x/y = z
or
x/y = a*b (*y)
(x/y)*y = a*b*y
x = a*b*y
The comment herein is a different approach, trying to respond to the reason why the question was asked rather than merely justify or explain the procedures involved. If you only care about math, please skip it.
>What I am trying to understand is *why it works*, contrasted with remembering it as a kind of magical spell.
It works as mathematical (logical) procedure because it is a kind of “magic spell”, and it is a kind of magic spell because it always works even if, and even though, we don’t know why it always works.
Consider the following equivalency:
Meta = above and beyond
Super = above and beyond
Physics = the universe
Nature = the universe
Therefore metaphysical isn’t logically different from supernatural.
Why mathematics works is a deeply philosophical and, in truth, unanswerable question. Descartes reasoned that it could only be answered by presuming that a benevolent God (a supernatural agency) was kind enough to provide a rational (logically consistent) universe for us to exist in.(This relies on an “inverse teleology” of God’s intention.) More contemporary mathematicians generally (but not exclusively) tend to believe that a better answer is the *anthropic principle*, a “reverse teleology” of natural selection: if we didn’t exist in a rational universe (one where math works) then we wouldn’t exist at all to begin with and wouldn’t be here to wonder why.
Most mathematicians (which might include Decartes, I think, if he had lived in a post-Darwin world) prefer the reverse teleology because it doesn’t require a supernatural agency, but it isn’t *logically* different from the inverse teleology, it simply assumes the rationality of metaphysics (that math always works, even when it isn’t human calculation but executed by automated computers) without a supporting supernatural explanation of *why* metaphysics is rational. This explains why many hyper-rationalists ascribe to a “brain in a jar” worldview of the universe as a simulation. This is supposedly a coherent and “logical” idea, but isn’t really. The “simulation theory” requires that our universe is ostensibly in a computational system *intentionally designed by an extra-universal [metaphysical/supernatural] agency*, so in reality it takes the weakest part of Descartes inverse teleology and combines it with the weakest part of the anthropic reverse teleology and still doesn’t address the original question of *why* mathematics is metaphysical in nature, or why metaphysics is mathematical in nature. It simply accepts the assumption as correct because it is “logic!” and leaves it at that, using circular logic to deny the fact it is circular logic. It works because everything in the objective universe (outside of humanity and our own thoughts, perhaps) behaves logically, according to mathematical laws of physics, which cannot be broken despite the absence of any enforcement mechanism or agency.
Ultimately, “why it works” has only one real answer, described as *the ineffability of being*. The meaning and purpose of it working is unknowable (the meaning is epistemically uncertain and the purpose is metaphysically uncertain,) all we can know is *that it does work*, every time, without fail, almost as if it is magic.
Thanks for your time. Hope it helps.
True ELI5 (not a thorough mathematical explanation but an explanation by example, as you’d teach a kid)
A good example for y*x=z might be:
I have four kids (y=4)
Each kid has five apples (x=5)
This totals 5+5+5+5 apples or 20 apples (z=20)
So y*x means 5 apples 4 times, or 4 kids with 5 apples.
Now when somebody asks “what is z/x?” they’re saying:. “You have 20 apples and you want to divide them so that each kid gets 4 apples. How many kids can you feed?”
It’s backwards-engineering the prior problem. Before, I knew the number of kids (y) and how much each kid had (x) and had to figure out the total (z). Now, I have the total (z) and know how much each kid gets (y) and need to figure out the number of kids (x).
Similarly, if they ask “what is z/x?” they’re asking: “I have 20 apples and would like to share them evenly between 4 kids. How many apples should I give each kid?”.
We (or at least I)’ve been taught that “you move it to the other side as it’s opposite”.
I understand it much easier as “you do the same thing on both sides so it’s still equal”.
In your example:
y*x=z
You want to clear Y, so you want to use the opposite of *x
y*x/x=z/x
And then you simplify
y=z/x
So far, I’ve only seen one ACTUAL “like I’m 5” answer, so here’s my stab at it:
Multiplication is just adding a certain number together as many times as you’re told to, so if you have 6 * 7, you would add 6 together 7 times.
6+6+6+6+6+6+6=42
Division is similarly SUBTRACTING a certain number of times until you reach zero, so like above, if you’re asked “What is 42 / 7” then you subtract 7 until you reach zero.
42 – 7 = 35 (1 time)
35 – 7 = 28 (2 times)
28 – 7 = 21 (3 times)
21 – 7 = 14 (4 times)
14 – 7 = 7 (5 times)
7 – 7 = 0 (6 times)
So 42 / 7 = 6.
6(y) * 7(x) = 42(z), and 42(z) / 7(x) = 6(y).
Main thing to understand is that you can add, subtract, multiple or divide an equation on both sideswith any constant, and the equality sign holds.
So when you go from y*x = z => z/x = y you are doing a few steps
First step is to divide both sides with x:
y*x/x =z/x
That cancels out to
y*1=z/x = y
and you get the equation. Remember that which side things are on in the equation sign doesn’t matter.
> ” /(x/y) ” = ” *(y/x) ”
If you take 7 minutes to run thousand meters and you want to know how many meters far you can run in 60 minutes you calculate 60 min / (7 min / 1000m) = 8571m OR you can calculate 60 min * (1000m / 7 min) = 8571m.
If you divide 7 by 1000, you get the minutes you need for *one* meter.
If you divide 1000 by 7, you get the meters you can run in *one* minute.
You can think of similar scenarios with buckets of paint that can be used to paint a certain amount of walls.
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I don’t know if that helps anything. It’s not a proof. Do you want a mathematical proof?
Do you accept `x / y = x * (1 / y)`, by the way?
It would be interesting to see your responses to all the answers.
Example where y=4, x=5, and z=20
If I have 4 equal groups of apples, with 5 apples in each group, then I have 20 apples in total. If I divide 20 apples into equal groups with 5 apples in each group, then I’d have 4 different groups.
Note: this only works for x that isn’t 0 because you can’t divide by 0.
Multiplication and division are opposites.
If I have 5 groups of 6 things, that’s 30 things because 5*6=30
If I have 30 things and divide them into 5 groups, that’s 6 things per group because 30/5=6
If I have the equation y*x=z, if I do the same thing to both sides of the equation, it would make sense that the equality of both sides wouldn’t change. y*x+1=z+1 is still a true equation. The same holds true if we multiply or divide. y*x*3=z*3 is still true. y*x/5=z/5 is still true. So logically, y*x/x=z/x should be true as long as x isn’t 0. x/x = 1, therefore is the same thing as y*1=z/x, and y*1=y, so y=z/x.
In a non mathematical description:
You have some apples. You decide to make groups of two apples and you are able to make three of those groups. Therefore you have (2*3) = 6 apples.
you then take your 6 apples and split them into groups of three. 6/3 = 2 groups.
Of course the theory answers are better for why it works from a mathematical perspective. Hopefully this helps from a practical perspective
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