Portuguese Rounding Rule for Multiplication and Division
This rounding rule requires one to count numbers
whose leading digit is 5 or greater as having an extra significant digit and then to apply the
standard rule.
Background on the Portuguese Rule
There has been some controversy about its use: some Portuguese teachers and professors think that we
should all drop it and follow the standard rule in order to achieve normalization, since this rule is
not seen in international literature. ...
The origin of this rule has been hard to find. The oldest reference I found was in the notes of the late
Amorim Ferreira, Full Professor at the Faculty of Science (University of Lisbon), "Trabalhos Práticos de Física",
1ª Ed., Livraria Sá da Costa, 1929. In these notes, he talks about this rule as something that he learned from
others - so it is probably even older. This information was given to me by Prof. Jorge Valadares (...)
from the Open University at Lisbon, who owns the book.
In the first volume of the book (Mechanics) in a chapter entitled Metrology, we can read: "É costume
considerar com mais um algarismo os números cujo primeiro algarismo significativo é igual ou maior do
que 5. Assim, considera-se o número 0,0830 com 4 algarismos significativos e não 3." (It is usual to take
one more significant figure in numbers whose first significant figure is equal or greater than 5.
So, 0.0830 is considered to have 4 significant figures and not 3). Prof. Amorim Ferreira justifies this
rule by saying that the significant figures can give us an idea of the precision of a measurement: an
uncertainty of 1 in 800 gives us a precision of about 0,001 while the same uncertainty in 120 gives us
a precision of about 0.01 (ten times smaller). If we want this difference to be translated in terms of
significant figures, then 800 must have one more than 120 and since the origin of this difference comes
from the figure "8", because it is greater than 5, it must be counted twice.
This rule has been taught in Portugal in high schools and universities for a long time: those, like me,
who studied in the sixties and seventies learned it from a Guide of Practical Work in Physics,
written by Mrs. Helena Côncio, who justified it more or less in the same manner. Nowadays it is more
commonly taught in schools, since most of the authors of textbooks learned it while graduating. I have
been told that it is also widely used in Applied Mathmatics (but I am not sure).
The rule can be found in many Portuguese textbooks of Physics and Chemistry like my own "Técnicas
Laboratoriais de Física - Bloco 1", Edições ASA, 1995, but without a full explanation. It can also be
found in (at least) a university manual "Física Experimental - Uma Introdução", by M.C. Abreu, L. Matias
and L.F. Peralta, 1994. These three authors used to teach at the faculty of Science (University of Lisbon);
the first author, Prof. Maria da Conceição Abreu (mabreu@ualg.pt) is now at the Algarve University (High
Energy Physics).
...
Background information kindly provided by:
Adriano Sampaio e Sousa
Physics Department
Oporto University
Portugal
