Why is division not associative?

Do it in order, from left to right, just like multiplication.

for the case x/y/z, first divide x by y, then divide that by z.

Then throw a book at whoever wrote the problem for being an ambiguous *****.

In practical cases, this sort of conundrum never comes into play unless somebody really screws up. Even in most textbooks and stuff, the divisor signs are different sizes, which will imply priority.

Heh, you already answered your own question: division is not associative because 0.25 does not equal 4.

If you have multiple divisions, then the problem usually has one fraction bar that's clearly bigger than the others, e.g.

1

---

2

-------

4

which would be 0.5/4 = 0.125.

If you have fraction bars of the same size, e.g.

1

---

2

---

4

... then the problem is ambiguously written. You can't really tell what the answer should be, and you're fully justified in whining at any instructor that gives you such poorly formatted problems.

If everything's written out on a line, e.g.

1 / 2 / 4

... then you should do the operations in the order they're given to you. However, it's bad form to write problems that way; good mathematicians don't do it.

Hopefully that helps!

You NEED parentheses. No other way of telling whether 4/3/2 = (4/3)/2 or 4/(3/2). But if thereis a multiplication involved then the division comes first i.e. 4*3/2*5 = 4*(3/2)*5

If you treat division as multiplication by the reciprocal, then it is!!

Same thing for subtraction: treat it as adding the negative and then it's associative (and communitative)

Biggest things I've noticed since starting grad school in math: division and subtraction have just about disappeared. And then there's 'divisors of zero"!

There's something else to keep in mind: those are binary operators. They take two items and return one result. When we write

a*b*c it really doens't make sense. We can only combine two items at once. We've agreed on conventions to allow such sloppyness.

(1/1)/4 is equal to (1/1)*(1/4) = 1/4

You turn the denominator upside down and multiply. 4 is 4/1 and flip over is 1/4

If there were no parenthesis then you would do it left to right.

1/1/4 = 1/4

Same answer.

1/(1/4) would be (1/1)*(4/1) = 4

Order of operations:

Parenthesis

Exponents

Multiply

Divide

Add

Subtract

Please excuse my dear aunt Sally is a helpful mnemonic. No parenthesis or exponents, work left to right.

you shouldn't have to... people tend to write it in such a way as to make it clear. i actually don't know the ooo on something like 4/3/4.

but you have just demonstrated yourself why it's not associative. there is nothing more to it except conventions.

Please Excuse My Dear Aunt Sally, but please follow the order of operations. It will tell you the answer to all of you mathematical troubles and if you get used to using it now, it won't be nearly as confusing when you get into advanced levels of Mathematics

I don't know why. But you need to work from the parentheses out to solve any math problem.

well.. x/y/z = (x/y)/(z/1) dividing by fractions you multiply by the reciprocal of the bottom

= (x/y)*(1/z) by multiplication = (x*1)/(y*z) = x/yz