Current Events > Simplify y = 6*x/6*x

(+) Yeah! (+)

show your work/explain?

also, you can pick and option and specify domain, if you want.

Something else

y = 1, x does not equal 0

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I just took a great dump

Damn_Underscore posted...

Something else

y = 1, x does not equal 0

Yup, y=1 when x does not equal zero.

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Multiplications and divisions have the same priority. It is written 6*x/6*x, not (6*x)/(6*x)
6*x = 6x
6x/6 = x
x*x = x^2

if This was written as 6x/6x (which it often is on paper) the parentheses are implied


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Damn_Underscore posted...

if This was written as 6x/6x (which it often is on paper) the parentheses are implied

people keep saying this, but i still haven't seen a source that suggests that this is actually true mathematically (i.e., through formal written convention by mathematicians or in any textbook). Is this just something some people have decided on?

As a mathematician -- what you're complaining about here is an artifact of standard typefonts being terrible for math. Have we formally defined this as something we would use in a textbook? Fuck no, because in a textbook I'm going to have access to some way of using a standard fraction bar instead of coopting the slash for lack of better options. If I'm writing out my math? Again, this issue doesn't arise. Even if I'm using a diagonalized fraction bar I'm going to be writing the numerator visibly above the denominator. Sort of like this, except better (and without a period to force proper alignment):
6x/
./6x

Have mathematicians generally decided to treat terms like 6x as having implied parentheses when typing in a standard type font? Yes, because we're lazy fuckers and because it makes sense to us to treat "6x" as one term, not two. Is it some formalized mathematical convention sanctioned from the high muckety-mucks of mathematical theory? No, because any such formalizing agency doesn't give a crap about trying to do formal math within the limitations of standard type fonts.

To summarize, math is a language and TC's equations have a heavy accent.

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tiornys posted...

As a mathematician -- what you're complaining about here is an artifact of standard typefonts being terrible for math. Have we formally defined this as something we would use in a textbook? Fuck no, because in a textbook I'm going to have access to some way of using a standard fraction bar instead of coopting the slash for lack of better options. If I'm writing out my math? Again, this issue doesn't arise. Even if I'm using a diagonalized fraction bar I'm going to be writing the numerator visibly above the denominator. Sort of like this, except better (and without a period to force proper alignment):
6x/
./6x

Have mathematicians generally decided to treat terms like 6x as having implied parentheses when typing in a standard type font? Yes, because we're lazy fuckers and because it makes sense to us to treat "6x" as one term, not two. Is it some formalized mathematical convention sanctioned from the high muckety-mucks of mathematical theory? No, because any such formalizing agency doesn't give a crap about trying to do formal math within the limitations of standard type fonts.

finally, a reasonable response. it's not an actual rule, as a bunch of people seem to think / want to yell at you to tell you is right and you're stupid for not knowing this mathematical rule (which doesn't exist).

It may not be a rule, but keep in mind that if you try to seriously argue that 6x/6x should be treated like (6x/6)*x, no mathematician is going to take you seriously.

tiornys posted...

It may not be a rule, but keep in mind that if you try to seriously argue that 6x/6x should be treated like (6x/6)*x, no mathematician is going to take you seriously.

i'll make sure not show a mathematician a calculator or a computer algebra system, then!

Eh, computers aren't mathematicians.

So here's the thing. Multiplication is commutative, and division is just multiplication by the reciprocal. In other words, if I have a whole lot of terms that are multiplying and dividing each other then I can write that as a whole lot of terms that are multiplying each other and then I can order them however I want. This is reflexive knowledge for mathematicians, not something we would need to really think about.

Barring some arcane reason, I wouldn't ever write 6/6. 6/6 is obviously 1, so I would just get rid of that term entirely. So let's use numbers that don't just conveniently go away, say 7/5. Similarly, I would never write xx to represent multiplying x by x. I would either write x^2 or x*x. So, if I saw an expression written as 7x/5x, I would assume that 5x was the denominator because if the x is not in the denominator then I would expect to see 7x^2/5--except standard fonts suck, so if I wasn't able to superscript the 2 I would actually use parentheses to avoid confusion between (7x^2)/5 and 7x^(2/5).

An easier example to work with in standard fonts is 7x/5y. Again, if the y is not part of the denominator, then I would never write it this way. I would write 7xy/5. So if I see 7x/5y, my expectation is that 5y is the denominator. No parenthesis required.

Why are there so many topics about a math equation being written poorly so there can be an argument about how to properly read it?

Wouldn't the best way to write this be y = 6*(x/6)*x?

Did anyone else get 36/x^2, or am I just smarter than the rest of you?

Nah, the best way to write that would be y = x/x. The 6's cancel so there's no reason to keep them. The x's do not cancel because there actually is a meaningful difference between 1 and x/x, specifically at x = 0.

This whole topic has helped me understand why I've always kind of disliked the way pemdas and similar mnemonics are taught. Specifically, the part about having to do things in order from left to right is unnecessary. More accurately, doing mixed addition and subtraction or mixed multiplication and division in order is necessary only if we treat addition as being different from subtraction and multiplication as different from division. In fact, subtraction is just addition of the additive inverse and division is just multiplication of the multiplicative inverse.

To illustrate what I mean: when I look at the sum 1 + 17 - 5 - 3, my brain interprets that as 1 + 17 + (-5) + (-3), and with that interpretation it doesn't matter what order I evaluate in. (-8) + 18 is 10. 1 + 9 is 10. (-2) + 12 is 10. etc.

Similarly, if I look at something like 5 * 7 / 6 / 3 * 2, the way I interpret it is 5 * 7 * (1/6) * (1/3) * 2. And again, with that interpretation I can evaluate the expression in whatever order I want, and I will get the same answer. 2 * 35 * (1/18) = 35/9. 7/3 * 5/3 = 35/9. etc.

tiornys posted...

only if we treat addition as being different from subtraction

addition is different from subtraction though, subtraction isn't associative

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I CAN'T do MATH

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tiornys posted...

This whole topic has helped me understand why I've always kind of disliked the way pemdas and similar mnemonics are taught. Specifically, the part about having to do things in order from left to right is unnecessary. More accurately, doing mixed addition and subtraction or mixed multiplication and division in order is necessary only if we treat addition as being different from subtraction and multiplication as different from division. In fact, subtraction is just addition of the additive inverse and division is just multiplication of the multiplicative inverse.

Doesn't that still have to be from left to right though? Using your example of 5*7/6/3*2 is different whether you go left to right or 5*7/(6/3)*2 for example. The latter would be 35/(2)*2 = 35

Which is why something like this equation is written poorly , because you can't tell if its 6x/6x or 6*x/6*x or 6*x/6x

Garioshi posted...

addition is different from subtraction though, subtraction isn't associative

If you treat subtraction as adding the additive inverse, it is associative.

Proto_Spark posted...

Doesn't that still have to be from left to right though? Using your example of 5*7/6/3*2 is different whether you go left to right or 5*7/(6/3)*2 for example. The latter would be 35/(2)*2 = 35

Yes, but also no. Given the absence of parentheses, the correct way to parse 5*7/6/3*2 is as 5 * 7 * (1/6) * (1/3) * 2. More generally in a string of multiplication and division, each term preceded by a * is treated normally and each term preceded by a / is treated like * (1/whatever). Once correctly parsed, the evaluation can proceed in any order. Order only matters if you don't parse the division symbols as inverse multiplication.

In other words, this sort of problem is a symptom of the way in which we teach subtraction and division. It's something we learn in grade school and then have to unlearn in higher math (unless you're lucky like I was, and were taught how to think about these things correctly outside of school by engineers who were friends of your parents--honestly this is probably a lot of why I was always ahead of my class in math). If we taught that subtraction is the inverse of addition up front, and that multiplication was the inverse of division up front, I think people would have an easier time with handling fractions, transitioning to algebra, etc.