The obelus in the elementary education
In Taiwan, as I remember, elementary school instructors and textbooks notate division as \(a\)\(\,\div\,\)\(b\), which is known as the obelus. Even as a kid, I learnt by myself, from more advanced books, that people used \(a\)\(\,/\,\)\(b\), the solidus, or slash, to denote division. Strangely, when I was about in the 7th grade, the textbook announced that from now on, the obelus was to be discouraged, just as \(\,\times\,\) was replaced by \(\,\cdot\,\). It maintained that we would always use fraction, and that when the space didn’t allow, we would use a solidus. Except in textbooks and exercise books in basic arithmetic, and on some calculators, I have never seen the obelus anywhere else, in analysis, algebra, geometry, nor in any scientific publication. Actually, the ISO 80000-2 standard for mathematical notation recommends only the solidus or fraction bar for division, saying that the obelus “shouldn’t be used” for division.
Origin of the obelus
Why is the obelus taught to kids, though it virtually never occurs elsewhere? I see no advantage in first teaching them a very rare notation, the obelus, and later abandoning it and enforcing the use of another one, the solidus. It is also mysterious why obelus stands for division, which Wikipedia doesn’t say. On the contrary, a solidus is a fraction “lying down” due to limited space, which appears rather intuitive.
Allow me here back up a quote from Wikipedia, ‹Obelus›:
Although previously used for subtraction, the obelus was first used as a symbol for division in 1659 in the algebra book Teutsche Algebra by Johann Rahn. Some think that John Pell, who edited the book, may have been responsible for this use of the symbol.
Maybe before the presence of the solidus, the obelus was first used for division. (By then, the convention of algebra equations was still being formed, and equations in «Principia Mathematica» were mainly written verbally.) Afterwards, the solidus probably grew more pervasive, but elementary school textbooks, somehow, didn’t follow the trend, leaving the relic of old notation.
Obelus versus solidus
At least thus I concluded myself. Then, after I asked this on Facebook [in Aug 2017], I realized that, quite to my surprise, some people still use the obelus in their life, and consider that they are somehow different. Facebook friend Xinbo held that \(8\)\(\,/\,\)\(6\) was intended for \(1\) with remainder \(2\), and such division with remainder was to be deprecated later curriculum. Piano instructor Chen suggested that there is some purpose in using a more verbose notation, as kids may confuse \(\,/\,\) with \(1\). This does suggest people perceive the obelus and the solidus differently.
Actually, some time after, there was an internet meme weirdly relevant to the present topic, asking the smart reader to vote what among the precedence below were correct, which went like (I am making up numbers now):
Obelus_eager:
\(8\)\(\,\div\,\)\(\left(\vphantom{4 \,+\, 2}\right.\)\(4\)\(\,+\,\)\(2\)\(\left.\vphantom{4 \,+\, 2}\right)\)\(\,\times\,\)\(3\)\(\;=\;\)\(4\)\(;\,\)
Obelus_lazy:
\(8\)\(\,\div\,\)\(\left(\vphantom{4 \,+\, 2}\right.\)\(4\)\(\,+\,\)\(2\)\(\left.\vphantom{4 \,+\, 2}\right)\)\(\,\times\,\)\(3\)\(\;=\;\)\(\dfrac{4}{9}\)\(;\,\)
Solidus_eager:
\(8\)\(\,/\,\)\(\left(\vphantom{4 \,+\, 2}\right.\)\(4\)\(\,+\,\)\(2\)\(\left.\vphantom{4 \,+\, 2}\right)\)\(\,\cdot\,\)\(3\)\(\;=\;\)\(4\)\(;\,\)
Solidus_lazy:
\(8\)\(\,/\,\)\(\left(\vphantom{4 \,+\, 2}\right.\)\(4\)\(\,+\,\)\(2\)\(\left.\vphantom{4 \,+\, 2}\right)\)\(\,\cdot\,\)\(3\)\(\;=\;\)\(\dfrac{4}{9}\)\(;\,\)
Fraction_eager:
\(\dfrac{8}{4 \,+\, 2}\)\(\,\cdot\,\)\(3\)\(\;=\;\)\(4\)\(;\,\)
Fraction_lazy:
\(\dfrac{8}{4 \,+\, 2}\)\(\,\cdot\,\)\(3\)\(\;=\;\)\(\dfrac{4}{9}\)\(;\,\)
Clearly, FracEager is right, and SolLazy, I think, is the norm. Certainly, SolEager and FracLazy are unanimously rejected, and they are displayed here just for comparison. However, for the rest, it isn’t so clear. While I absolutely take it for granted that ObLazy and SolLazy are correct, the consensus seems to be ObEager and SolLazy. It appears therefore that people equate the solidus with the fraction, but not with the obelus; that is to say, people regard \(a\)\(\,/\,\)\(b\) the same thing as \(\dfrac{a}{b}\), but different with \(a\)\(\,\div\,\)\(b\). I really don’t understand why they think that way; to me it will be more consistent if we always consider obelus the same as solidus and adopt conventions ObLazy and SolLazy.
Why spend so much time writing on such trivial matters? As I said, it is curious how the choice of notation has some bearing on the perception of precedence, and the underlying human psychology is worthy of examination.
❧ August 9, 2017; appended July 7, 2021