Want Pedantic Math Answer to Simple Question

[font=&quot]Since it seems hard to avoid a pedantic answer to any post that gets more than five replies (I mean--how many ways can you beat a dead horse!), so I have a simple math question (~4-5 grade) that is really bugging me and needs a pedantic answer--and it seem that CPF is the place to get it.

In my daughter's current math book, it tells how to evaluate math problems (that are missing parentheses) as simply "right to left" and evaluate division-multiplication (sometimes lists multi-division) first, subtraction and addition second.

But, for a compound fraction (written like examples a & b below), the book evaluates the fraction from the "bottom, up". To me, this suggests that the "right to left" method for "string" division problems would be correct.

The method described in my daughter's book is the way my wife remembers it from her (overseas college education--excluding the compound fraction evaluation) math classes (and she has a degree in math)--but it is not the way I remembered it.

My memory (from the mid 1960's) was that is was very specific that you evaluate from left to right, but did divisions from right to left. And it was very specific division first, then multiplication, subtraction, and finally addition.

Now, if everyone used paren's, it would not be a problem... But then, there would not be a chance for a thread like this.[/font]

So, some examples (division samples, as this is the heart of the question):

x=1/1/10

y=1/0.1


I know better than to argue with my wife--but what the heck.

So, evaluating right to left:

x=[(1/1)/10]= 0.1 = 1/10

y=(1/0.1)=10


[font=&quot]The method I remember being taught gives me the same answer for all of them (10)--the "book" method gives me different answers depending on how the problem is written and even on what the numbers are (0.1 vs 1/10)... And, if this were algebra, using the "right to left" evaluation rule for division--one would always get the right answer no matter the number.

So, what is the correct method/answer from some math professionals out there? Has the "aproved" evaluation method changed--or is just my mind going faster than I thought? Any links to an authoritive source would be nice.[/font]

-Bill

To me, if you don't follow the math syntax both answers are ok, it depends on how you place the symbols.

Grammar analogy:

Phrase: Yes dumb question.

Option1: Yes, dumb question...
Option2: Yes Dumb, question!

see same words, different meanings.


Pablo

In the absence of parentheses and if one excludes exponentiation, multiplication and division have equal order of priority and are performed as they occur left to right. Then, addition and subtraction which also have equal order of priority are performed as they occur left to right.

If you simply have a string of successive divisions, they are performed as they occur left to right, not right to left. Consequently: 1 divided by 1 divided by 10 is 0.1. End of story.

That being said, remember that in the case of quotients written as fractions, there are parentheses assumed around the numerator and around the denominator. In the case of a compound or complex fraction, some texts will write fraction bars of different lengths to indicate which is to be considered which. For example, in the case that you raise:

1
-----
1
---
10

with the longer fraction bar as indicated would be written to indicate that 1/10 is to be considered the denominator of the complex fraction around which there are parentheses assumed. Consequently 1/10 would be evaluated first in this case.

It depends on if you're evaluating 1 ÷ 1/10 or 1/1 ÷ 10...

In the case of:

you would evaluate it as 1 ÷ (1/10).

In the case of

you would evaluate it as (1/1) ÷ 10.

The shorter bar tells you that it's a fraction within a fraction.

Carrot,

a/b = a÷b. There is no distinction.

BB,

Left to right is the correct answer (in the absence of parentheses and taking into account the correct order for math operations). Evaluate the expression first and mentally place parentheses so that multiplication and division are done before addition and subtraction. M & D are equivalent with respect to heirarchy. If you hit multiplication first, it is done first. If division comes first, do it first. Similar deal with addition and subtraction.

For x=1/1/10, you get mentally (1/1)/10. (1/1) is 1 so now 1/10 is .1 (in decimal).

For y=1/0.1 the answer is 10. You're dividing one by a tenth. If you want to turn the 0.1 into 1/10 you MUST use parentheses to have an equivalent equation, i.e. y=1/(1/10).

Here are some links for you.

http://mathforum.org/dr.math/faq/faq.order.operations.html
http://mathforum.org/library/drmath/view/57199.html
http://www.mathgoodies.com/lessons/vol7/order_operations.html


Hope the links help.

Mike

P. S. My 17 year old 800 Math SAT, 36 ACT, soon to be a college math major son says one of the expressions to remember the order is Please Excuse My Dear Aunt Sally. P=parentheses comes first, E=exponents is second, M=multiplication & D=division are next (whichever comes first), A=addition & S=subtraction are done last (in whichever order they appear). You definitely scan left to right. Also, as a computer programmer I'll say that it works the same way in programming.

I agree with Bob.

Big Bob said:

a/b = a÷b. There is no distinction.

Click to expand...

Right, but in this case I'm using "/" to represent fractions, so 1 ÷ 1/10 and 1/1 ÷ 10 come out with different answers.

Mike, up here, the memory aid is BEDMAS, with B for "Brackets". Same idea, but an acronym instead of an expression.

carrot said:

Right, but in this case I'm using "/" to represent fractions, so 1 ÷ 1/10 and 1/1 ÷ 10 come out with different answers.

Click to expand...

Forgive my sensitivity Carrot, but I've reviewed too many math texts over the years whose authors have incorporated their own little notation conventions which have no universality whatsoever.

x=1/1/10 sure looks like nonsense to me.

If this is typical of the rot being put into your daughter's head, you can only hope she is doing poorly in school.

What ought to be taught in school, but isn't:

If someone says x=1/1/10, and asks what the value of x is, the most appropriate response is:

A. 1
B. 10
C. 100
D. Go away you fool!

Well, learning the order to solve problems is important to math... And, whether I was given those problems 40+ years ago in (a very poorly funded) public school, or a good math book through home schooling--it is something that we all have to put up with.

The example of x=1/1/10 is one I made up--her book has different numbers, but I just made a very simple example to show the issue.

As a kid, one has to put up with a lot of BS as a student to learn the basics before one can be taught the more complex issues. I always hated these types of problems as in real life (like engineering) the formulas have to have paren's/grouping or you got the wrong answer and something did not work.

I never cared much for math until I had a need for it (like chemistry, navigation of boat/plane or when constructing something) or in college when I was able to relate calculus with physical problems.

Anyways--Alphamicro's first link is probably the clearest... Math ordering problems are simply conventions and that if you want something different, use parenthesis.

For example:

x = 105 / ab

Is solved as:

x = (105/a)*b

and not as (105)/(ab) no matter how your eye would like to group them.

Perhaps I remembered incorrectly what I was thought those 40 years ago--and/or I was trying to rationalize how to reduce complex fractions from the bottom up vs. the complex fractions written out on one line. The conventions are different because the problems are not written in the same form.

In either case, after a few hand full of days during home schooling, my daughter will never run across these types of problems ever again--as they will all have either obvious order and/or paren's.

Regarding eluminator's suggestion of "go away...", I did do that in a thermodynamics class--I had a substitute teacher for the semester who got down into, essentially, spending 30 minutes in a lecture that proving 1/1 = 1. I told off the professor (he normally taught mechanics/vector calculus) and walked out of class. He was pretty cool about the whole thing--commented after the semester was over that he noticed I was very quiet the rest of the semester. He also told me that he was "only a few days ahead of the class in text book the whole time." Had him later too--always graded me fairly.

Thanks,
-Bill

It seems to me that clarity is important whether you are writing in your native language, writing arithmetic, or writing computer code.

I would have gotten the 105/ab wrong, I think many others would also. That is a good reason to write it differently.

If what you mean is 105b/a, then that's what you ought to write. And it seems that our schools ought to teach clarity, not how to decode confusing notation.

The fact is I've been around many years and I don't think I've ever suffered because I couldn't decode 105/ab. How often does it come up in your life?

Maybe one of the problems is that there were no computers around when I went to school. So we wrote in longhand and would have written:

Also your suggestion to use parentheses is a good one. Especially if you have to string it out on one line. If kids are taught how to decode 1/1/10, and I think it's a waste of time and brain cells, then they at least ought to be taught to not write such stuff themselves.

Okay, my head just exploded! <grin>.
I'm going to point this thread out to my wife tonight. She's a special-ed math teacher and should get a kick out of this.