again there is absolutely no ambiguity, there are only people that don't understand or follow the rules
from the link,
What is 2x/3y-1 if x=9 and y=2 ? the author argues 'It is not clear what the textbook had intended with the 3y' that is a specious argument as the solver must only consider what is written, again Math not ESP, 'One can however also argue that "3y" is a unit which belongs together.' again a specious argument designed to deflect responsibility from knowing the rules, 3y is a unit is nonsense, what is written is 3 multiplied by y

Further using the arguement that various computer programs give different results merely shows that the coder wrote faulty code

first note that there are no parenthesis, the only way to add a parenthesis where one did not exist is to use the substitution principle as outlined below or where by adding the parenthesis the sense of the problem is not changed
an exaple of the latter is to rewrite 3x as 3(x)

so step 1 is to substitute equivalent values for the variables, the rule to do that is to first put parenthesis where the variable was
2( )/3( ) - 1 then fill the parenthesis w/ the equivalent value or equivalent expression.
2(9)/3(2) - 1 now eliminate the parenthesis by multiplication, this is part of the P in PEMDAS
18/6 - 1 now divide, this is the D in PEMDAS
3 - 1 now subtract
2

there alternate pedagogies to PEMDAS that do the same thing
one is to always isolate terms from each other and to simplify the terms in isolation, terms quite simply are separated from each other by + or - which are not inside a parenthesis or by comparatives like =, > or < etc.

using this method 2x/3y is considered in isolation
from there the same substitution principle is used and then since there is nothing to be done inside the parenthesis and the exponent on the parenthesis is 1 the next step is to multiple to remove each parenthesis. There is certainly a lack of full uderstanding by many about the full extent of the p in PEMDAS, limiting the discussion to only Algebra and excluding tri or other higher functions

P alone has several steps that are often not considered
step 1 of P is to do what you can inside the parenthesis, again this is done in isolation
step 2 is to do any exponents on the parenthesis, there is always an exponent even if it is only 1 which doesn't change anything most of the time so like extra parenthesis is rarley written for simplicity.
step 3 is to multiply the parenthesis bu it's numerical coefficient, as w/ the exponent 1 there is always a numerical coefficient even if it is 1 which again is often not written but which serves the very important function of eliminating the parenthesis from the problem

This expression shows the usually unwrtitten but understood 1's, here the exponent 1 does notheing but the coefficient 1 when miultiplied by using the distributive property eliminates the parenthesis
http://forums.pelicanparts.com/uploa...1478058321.gif
it is equivalent to
1(x - 1) and (x - 1) and x - 1

there is a third one that would be considered if this was a ratioanl expression, but since it's not I've ommited discussion of that

then the 2 terms are integrated to get the final answer

What does the set of natural #s have to do w/ this discussion?

if one limits ones solution set to the set of naturals, that doesn't change any of the actual math involved, it merely adds the possibility of a null answer should the result be a rational # such as 1/4 or 0

Bill,

The set of natural numbers has some ambiguity depending on what discipline is defining it. Because of that, extra clarity is often called for to avoid confusion.

I have never said the "actual math" ever changes. Just how we write it and how we interpret what is written. Ambiguity exists otherwise there would not be pages and pages written about it all over the Internet and we wouldn't have these arguments all over the Internet about equations and their results. Just because there shouldn't be any ambiguity doesn't mean there isn't any.

the set of natural #s has no ambiguity it is a well defined set
it is commonly represented in set notation as {1,2,3,...}, the ellipsis(...) means to continue in the same, manner is also well defined and standard

whole #s is also a well defined set, {0,1,2,3,...}

integers is the set, {..., -2, -1, 0, 1, 2, ... }

Rational #s is another well defined set, this one has to be defined by rule rather than iteration, {any # that can be represented as an integer divided by a natural #} alternately {any # that can be written as the ratio of an integer and a natural #}

irrational #s is the next set that needs to be defined, again by rule, {any # that cannot be written as an integer divided by a natural #} or {any # that cannot be expressed as the ratio of an integer and a natural #}

next is the real #s, { set of rational #s + the set of irrational#} this last is what makes up a Real # line, such as an axis used in graphing

there are further # sets such as {Complex #s}, {Diophantine #s}, {Transcendental #s} etc. that can also be defined

Anyone that claims other wise is making their own idiosyncratic rule or definition.

Bill, you a super smart guy and explain things like few can. Does math come naturally for you? What I mean is...do you invariably arrive at the correct answer without conciously thinking about those silly acronyms? I don't recall ever needing the crutches, was always exceptional in math, but could never "show my work" if teachers required it :(...don't have a clue as to how my brain functions sometimes :). But I could arrive at the correct answer better than 99.99% of those that don't have "the gift".

Can't explain why 1+1=2. (Base 10).
Or very complex equations either....

It just is :)

Well, that equation works Base 3 and up! SmileWavy

<iframe width="560" height="315" src="https://www.youtube.com/embed/UIKGV2cTgqA?list=RDUIKGV2cTgqA" frameborder="0" allowfullscreen></iframe>

Local news, true story....

Kid gets a math problem wrong...

5 x 3 = x (must show your work)

Answer:

15 (kid shows 5 + 5 + 5)

WRONG answer...

Teacher wanted to see 3 + 3 + 3 + 3 + 3

Idiots :(

Huh???

5 x 3 = x

5 x 3 - x = 0

15 x - x = 0

14 x = 0

x = 0

Oh brother.....

5 • 3 = x

5 • 3 - x = 0

15 - x = 0

I'll let you simplify from there. You should be able to get the right answer now.

Relax Francis :p

Not only do I suck at explaining equations, I can't even write them clearly either :)

The story on the news was (as I recall)

5 x 3 =

Still is silly that the teacher said that 15 was the wrong answer....and yep, I remember hearing Common Core Math in the story :(

You explained fine. I was just being a wise-azz.:cool:

How about

5 x 3 = ?

So "question mark" equals 15 now? :D

The teacher needs to consult with Ma and Pa Kettle:
https://www.youtube.com/watch?v=t8XMeocLflc

Bud Abbott Weighs in:
https://www.youtube.com/watch?v=MS2aEfbEi7s

Yep, "?" is our new variable. ;)

So...in my world

5 ? 3 = ?

Shaddup already Esel :)