Notation Stifles Arithmetic Intuition Building

There are a multitude of ways to represent additive and multiplicative operations. Typically students will learn something like the following for addition, subtraction, multiplication and division respectively. Where $a,b$ are real numbers. $a+ b, a - b, a \times b, a \div b$

Here I am just going to enumerate the problems with these basic notations and why similar notations do not work either, and then conclude with modern notations that are well suited to these operations.

Firstly $a + b$ is actually fine. The important property of addition, which is that addition is commutative, is respected by this notation. $a+b = b+a$

In the abstract we say a binary operation (two arguments), $a \ast b = c$ is commutative on a set $S$ if for every $a,b \in S$, we have that $a \ast b = b \ast a$. This is important for some objects were certain subsets are commutative with an operation, but the whole set is not.

In the case of real numbers, and complex numbers, all real and complex numbers commute with all others under addition and multiplication. We can however write notations for commutative operations that don’t commute when we consider naively ‘swapping’ their arguments.

Consider subtraction as written. $a - b$ is the same as $a+ -b$

But if we just swap the arguments, we get the following $b-a$ is not the same as $-b +a$

The first notation also obscures the fact that subtraction doesn’t exist per se, and is just a domain extension of addition, considered as a binary operation.

Division is more interesting and more troublesome.We have the following standard notations for $a$ divided by $b$. $a \div b, \frac{a}{b}$

These two have the same problem as subtraction notation, but it’s harder to write in such a way that the arguments do commute naively. $a \div b \neq b \div a$ and $\frac{a}{b} \neq \frac{b}{a}$

If I were to choose between these two notations , I’d choose the second. It is much clearer how to distribute operations over it. It is for example unintuitive that $a\times (c \div b) = (a \times c) \div b$ but $(a \times c) \div b \neq a \div (c \times b)$

This is what leads to the arbitrary seeming, and didactic, order of operations, which simply follow from the definitions of various operations without need for rote memorization or arguments about order of operations. For reference, I could not tell you what the order of operations is, but I don’t make mistakes because of this lack of knowledge.

The second notation makes this clearer as the following is much more intuitive. But again it does not commute if we swap arguments. $a \times \frac{c}{b} = \frac{c \times a } {b} = \frac {c \times a} {b}$

The best notation for division, at least for understanding what is going on mathematically, especially in a pre-algebra or pre-calculus context, is the following for $a$ divided by $b$. $a \div b = \frac{a}{b} = a\times b^{-1} = b^{-1} \times a$

Normally I’d just write $ab^{-1} = b^{-1}a$. The $'\times '$ symbols here are just illustrative.

It would seem that “naively” this new notation doesn’t commute either. however $b^{-1}$ should be thought of as a number, rather than the outcome of an operation on $b$, although strictly speaking, these are the same.  The operation here should be thought of as multiplication. $b^{-1}$ is just the number $latex \frac{1}{b}$, so naively ‘swapping’ the arguments here gives $a \times b^{-1} = b^{-1} \times a$ as written above.

This is illustrative that division by a number bigger than $1$ is just multiplication by a number smaller than $1$.

This notation isn’t always practical, but I’d argue teaching it to students learning algebra initially will cut down drastically on the mistakes they make, and bypass having to rote learn order of operations altogether.