What we're never taught explicitly

Question

I would like to make a complaint really. School math(s) can be the most boring way to learn: sitting down and rote learning binomial expansion or the volume of a cylinder is just not interesting. It seems that schools don't teach the interesting way, with plenty of variety and proof. I have some very basic facts that students are never taught explicitly, perhaps because of the rigidity of education. Instead, teachers seem to count on a mistake being made. My specific example is the fact that ab is not equal to a*b, but (a*b).

We are all taught of the order of operations, however the rule stating that ab can be expressed as (or is shorthand for) a*b, is wrong. I was pulled up when I tried to solve a linear-style expression like 20/2a=4 like a smart-alec by first assuming that this is equal to 20/2*a=4. This is wrong, right? Apparently, terms are always in their own little group, and this effects the order of operations. 20/2a=4 is the same as 10/(2*a)=4. The mistake would not have been made by using TeX style, maths notation.

My questions are: Am I correct in saying ab = (a*b)? Why are these things generally overlooked? Are there any other typical errors involving the order of operations, especially when using linear notation? Are these things simplified by teachers as to avoid bombarding the students with 'special cases'?

Answer 1

You're right that $a/2c$ is ambiguous, if you were only taught the rule that multiplication and division come before addition and substraction. But so is $a/2\cdot c$ then. There are multiple ways to deal with this

1. You could require parenthesis to be used whenever the meaning would be unclear otherwise

2. You could decree that operations with the same precedence level are to be performed left-to-right. This is what most programming languages do, I think.

3. You could add the rule that multiplication comes before division.

4. You could, as you suggest, understand the concatenation of two variables as in $ab$ as a different operation which yields the same result as multiplication but has a higher precedence. This is what your idea of interpreting $ab$ as $(ab)$ amounts to.

Unfortunately, people don't universally agree on which of the above they pick. Most programming languages use (2), I think. Mathematical texts will often write division as a fraction, and thus avoid this issue - the positioning than makes the meaning clear (compare $\frac{a}{2c}$ and $\frac{a}{2}c$). If using a fraction isn't possible for layout reasons, they will hopefully use (1), since that avoids confusion. But in a less formal setting than a printen text, I wouldn't be surprised to find your option (4). One can only hope, thought, that such a thing won't appear on exam questions! And if it does, I'd say that's a reason to complain.

What I hope you do take away from this is that mathematical notation isn't always a 100% precise. The actual math is precise, but people are sometimes sloppy about how they write it down. And, also, about how they talk about it.

Answer 2

The notation $x/2a$ is ambiguous, and when used without parentheses the meaning is expected to be obvious from the context. That being said, it seems that more often then not $x/2a$ is meant to be $x/(2a)$.

Answer 3

There is much less ambiguity to $xyz/2abc$ than first seems to be the case, because of the heuristics (basic to conversation and literacy) such as

where a very different alternative meaning (that is common knowledge to writer and audience as a syntactically correct reading) could have been stated unambiguously by a trivial rewriting but was not, it is probably not the intended meaning

and

when there is a unique reasonable rule for parsing otherwise ambiguous abbreviations, use that rule

and

minimize the amount of hidden structure (parentheses) in the answer and the types of alternative answers considered in reaching the answer

If somebody means $\frac{xyzc}{2ab}$ and wants to write on one line without parentheses, they can very easily write $xyzc/2ab$, instead of $xyz/2abc$ and relying on guesswork. But there is no way to express $\frac{xyz}{2abc}$ in that mode except as $xyz/2abc$. Hence the default interpretation should be $(xyz)/(2abc)$.

Answer 4

$ab$ is shorthand for $a*b$. In fact, first $a*b$ is shorthanded to $a\cdot b$, then the dot is completely dropped. This speeds up algebraic manipulations and once one is used to this convention one sees it is very convenient. In general, one relies of using parenthesis in order to clarify any ambiguity (or potential ambiguity) in algebraic expressions (or any other kind of expression).

As to the problem of teachers simplifying things too much (such oversimplifications are warned against by Einstein's words that "everything should be made as simple as possible, but not one bit simpler"), unfortunately that happens too often. My favourite example of oversimplification at the level of very fundamental mathematics is the way real numbers are taught. The emphasis is on their decimal representation and the students are just bombarded with tricks and algorithms to compute with real numbers until they are under the illusion they know what they are. However, the ubiquity of the disbelief of the fact that $0.999\cdots =1$ shows the dangers of this oversimplifications.

Other oversimplifications include the rule that the product of two negative numbers is positive. This is often just dropped on students without any explanation as to why this is in fact a necessary consequence of algebra. It creates the illusion that mathematics is a just a set of arbitrary rules for computation, obstructing the inherent logic of mathematics.

Other oversimplifications include computing limits at $\infty$ by plugging in $\infty$ into the equation and computing with it according to some silly set of rules. This creates in the mind of the student the false idea that $\infty $ is somehow a number. Another example is teaching integrals by bombarding the students with integration techniques, without really understanding what integrals are. Similarly, computing eigenvalues/eigenvectors methodically without really understanding what is going on. Plenty of other examples are common-place.

Answer 5

The problem is that division is not associative and often leads to ambiguity. Division is really multiplication by the inverse. x/2a should be written as $x{2^{ - 1}}a$ then there can be no confusion. Of course parentheses can make all things clear. We often cause confusion in our attempt to minimize typesetting.

Answer 6

$ab=(a*b)$ is technically incorrect (there is no such rule in conventional mathematics) but you can get away with using it anyway because it is very hard for any normal human being to interpret it in any way other than the correct one.

Part of the reason is probably how the human mind parses text - we tokenize, if you will, expressions by first splitting at spaces and special characters, and if that doesn't work, by capitalization and numbers, and if that doesn't work, deeper semantic inspection. $ab$ has no delimiter, so it looks like a single word - even though we are used to variables being letters and not words, our mind is too trained to look for words because of all the text we read.

To interpret $a/bc$ as $\frac{a}{b} \cdot c$ requires a very convoluted sort of thinking. I'd be very surprised to meet a human adult who did it naturally. If you ask thousand people what $\pi /2 \pi$ equals, I'd wager not a single one would answer $\pi ^2/2$.

Consider even something as simple as $a/b*c$. Ask a thousand people what $\pi / \pi * 2$ is, you will probably get just as many saying $2$ as $1/2$, and then again that many who refuse to answer until you specify whether you meant for the $*2$ to be a part of the fraction or not. This is why you'll rarely see $a/b*c$ in a good book, and when you do see it, the book can be improved appreciably by replacing it with a clearer expression.

Another example is $f^r(x)=y$ meaning ${(f(x))}^r=y$ except when $r=-1$, in which case it means $g(y)=x \mbox{ such that } f(x)=y$ despite $f^{-2}(x)$ being $\frac{1}{(f(x))^2}$ and not:

• The inverse function applied twice - $f^{-1}(f^{-1}(x))$ (even though for matrices, $A^{-2}$ does mean "the inverse of $A$ applied twice")
• The square of the inverse function - $(f^{-1}(x))^2$
• The function inverted twice (!!!)

For this one, there is actually a lot of support from literature, so you can't dismiss it - but again, it is rare to misinterpret an inverse function as a reciprocal, especially in a context dealing with function inversion and vice versa. One exception is trigonometric functions, where you often mix numeric and function reciprocals - and the $sin^{-1}$ notation can become extremely annoying in that case (there are even some questions here about it).

In practice, whenever misinterpretation is possible, people just give a different name to the inverse. With probability distributions, the inverse of the CDF is called the quantile function and given a separate symbol where relevant. For trigonometry, inverse functions get the prefix "arc".