Can 1/3 and 1/3 = 2/6? It seemed so

> I still can't see anything about the first one that indicates the point that infix notation is a bad idea.

I didn't make a very good argument for it. I really intended that to be more of a throwaway rant than a serious critique. But since you ask...

There are two problems with infix:

1. It's hard to parse. It requires precedence rules which are not apparent in the notation. In actual practice, the precedence rules vary from context to context and this causes real problems. It's an unnecessary cognitive burden that pays very little in the way of dividends (a few less pen strokes or key strokes).

2. It obscures the fact that infix operators are just syntactic sugar for function applications. It leads people to think that there is something fundamentally different about a+b that distinguishes it from sum(a,b) and this in turn leads to a ton of confusion.

> that's clearly not what you meant

Indeed not. I meant the successor operator as defined in the Peano axioms.

> you quoted

Yeah, sorry about that. When I first replied, I thought you were the same person who posted the grandparent comment. My first draft response turned out to be completely inappropriate when I realized you were a different person, but some of my initial mindset apparently leaked into the revised comment. My apologies.

It's easy to see that this cannot be a valid definition for fraction addition because applying it along with the rule that a/b = (ak)/(bk) gives a contradiction:

3/7 = 2/4 (+) 1/3 = 1/2 (+) 1/3 = 2/5, absurd because 2/5 != 3/7

Once this is realized, it's easy to see that the correct formulation for the "table joining" operation used here is a weighted arithmetic mean, i.e. a(/)b (+) c(/)d = weighted arithmetic mean of a/b and c/d with relative weights b and d = ((a/b) * b + (c/d) * d) / (b + d) = (a + c)/(b + d).

On the other hand, fraction addition is clearly determined as a/b + c/d = ad/bd + bc/bd = (ad + bc)/bd given the a/b = (ak)/(bk) and (a/b + c/b) = (a + c)/b axioms.

You've never seen a textbook present the series form of e^x?

    exp(x) = summation(x^n/n!, n=0, inf)

exp(0) = 1 is perfectly valid but the identity above only holds for x=0 only if we define 0^0 = 1. And we do!

Fair, infinite series are pretty esoteric. How about derivatives. The power rule:

    d/dx x^n = n x^(n-1)

This identity doesn't hold for n = 1 and x = 0 unless 0^0 = 1.

Eh maybe not, programmers don't use calculus that often. But surely statistics!

    (1 + x)^n = summation((n choose k)x^k, k=0, n)

Take x = 0 and n = 0

    (1 + x)^n = (1 + 0)^0 = 1 = 0^0 = x^0 = x^0 + ... + x^n

0^0 is undefined in general but locally we sometimes need to define it.

I love this discussion and want to throw in my two cents. Why not teach beginning students group theory as the foundation to addition and subtraction?

While it wouldn't solve this particular problem since fractions require fields, it would give teachers the tools to explain precisely why addition of two tables is not a group or field operation; they are from different sets and require measurement with units to define a new inclusive set.

My own understanding of arithmetic and algebra was enhanced immensely by a book outlining the derivation from Peano axioms.

Ultimately the concepts are very simple and kids intuitively do the steps without having the names for the mental processes that they're doing. The educational hurdles to me seem to be 1) identifying the memorized names of the sequence of natural numbers with a procedure for generating them; 2) distinguishing between procedures, objects, numerals, and numbers; 3) basics of set theory; 4) natural induction.

To fill the quiet, I said to the class, “When thinking about fractions, it’s important to keep your attention on what the whole is.”

This is basic. There was no need to write entire blog of how you misunderstood this.