Maths notation is needlessly complex [video]
I don&#x27;t like it for a number of reasons.<p>One relatively simple one is that it doesn&#x27;t naturally convert to a typed out version. Outside of category theory, where big commutative diagrams really do a ton of work, we should try to avoid introducing too much notation you can&#x27;t type out, especially at the introductory level. Suppose two kids are trying to study together over facebook chat - how are they going to write these huge triangles out?<p>Second, it&#x27;s actually unecessarily complicated. It introduces three concepts (exponentiation, logarithms and roots) as though they were entirely separate. But actually, the easiest way to understand how to work with roots is to just define them in terms of exponentiation. E.g. the nth root of x is just x^(1&#x2F;n). All of the normal rules for roots follow from the rules for exponentiation (and fractions) immediately. You don&#x27;t need a third side to the triangle, that&#x27;s just adding extra complications - all you really need is exponentiation and logarithms and a way of representing that they&#x27;re inverse operations in a certain sense.<p>So there&#x27;s actually a simpler way to express all this in a notation that&#x27;s much more similar to what we&#x27;ve seen before. The trick is to draw out parallels between familiar operations like multiplication and division. Note: I made this up 10 minutes ago, apologies if there are very similar proposals, it&#x27;s just really obvious.<p>Most of the basic arithmetic operations can be written using binary infix operators, e.g. +, * and &#x2F;. It turns out, exponentiation and logarithms can be too. In fact, if you&#x27;re typing, exponentiation already is.<p>Let x raised to the nth power be written as (x ^ n) (note this is essentially exactly the way it&#x27;s already typed out, only I&#x27;m using some unusual extra white space to emphasize that we&#x27;re treating ^ as an infix operator). It&#x27;s a little upward arrow that says scale x up by n (exponentially).<p>And let log base n of x be written as (x v n) or possibly (x \&#x2F; n). It&#x27;s a little downward arrow that says scale x down by n (logarithmically).<p>This makes the two operations work in a way that&#x27;s fairly analogous to multiplication and division in a relatively neat way. For example, for positive integers, multiplication can be thought of as (linearly) scaling one number up by another, while division scales it (linearly) downward in an inverse way. As already noted The same holds for these two operators, only the scaling is non-linear.<p>Lots of familiar relationships carry over, e.g. Note that ((x * n) &#x2F; n) = x. Similarly ((x ^ n) v n) = x. And where (n * (x &#x2F; n)) = x, it&#x27;s also the case that (n ^ (x v n)) = x.<p>And where multiplication and division interact with addition and subtraction, those operators interact similarly with multiplication and division, e.g. where x * (n + m) = (x * n) + (x * m), similarly x ^ (n * m) = (x ^ n) * (x ^ m). And so on. You can derive all the relationships you need from a very small number of rules that are easy to remember because they&#x27;re structurally almost exactly like the rules for operations you&#x27;re familiar with. You already know those when you learn about exponentiation, so you don&#x27;t have to learn new and weird geometric relationships.<p>All of those nice properties there follow from the fact that exponentiation is just the next operation in the sequence of hyperoperations (see <a href="https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;Hyperoperation" rel="nofollow">https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;Hyperoperation</a> ) after multiplication.<p>Introducing this weird three place operator actually masks the underlying simplicity of exponentiation and its inverse.