Errata
On p.108, middle column, second paragraph, the statement "numeracy in developing countries shows steady decline" whereas I actually meant to write "developed countries." I have no data on numeracy in developing countries, but there is evidence (see below) for such decline in some developed countries, including my own (see for instance Kevin Donnelly's Australia needs to think again about its unis or section 4.1 of Changes in Academic Work (pdf))
Ralph Place has taken issue with my use of accuracy in "the
calculator's accuracy greatly exceeds the slide rule's," where I had
meant "the accuracy I can achieve with a calculator greatly exceeds
what I can achieve with a slide rule."
See Sphere
for more on accuracy and precision of slide rules,
this introduction
for more on slide rules,
here for details of Otis King's cylindrical slide rule, and,
closely related to slide rules, nomograms are described in the excellent
Blood, Dirt, and Nomograms by Thomas L Hankins and published in
Isis, vol.90, 1999, pp.50-80.
By way of contrast to these analogue calculators, the little remembered
Curta calculators
competed with slide rules in their heyday,
and still have a fan club.
Incidentally, IBM's first electronic digital computer, the
701,
was commonly referred to as a
calculator.
The Operational Calculator
Several e-mails have led me to believe that my explanation of how
one aspect of clarity is to be achieved, namely, how the operational keys
are to be directly linked to the operation marked on the key, was inadequate.
Thimbleby's Study of Calculators
Harold Thimbleby has made a thorough and persistent study of calculators,
as the study cited in the essay shows.
But he has gone further and designed an excellent basic calculator which
works on an algebraic or declarative basis rather than an arithmetic or
operational one
(A True Calculator,
A New Calculator and Why it is Necessary).
The declarative calculator maintains an equation, the operational calculator
carries out arithmetic operations.
The arithmetic of the declarative calculator is inexact, that of the operational
calculator is exact.
It seems to me that the operational calculator is ideally suited to support
the kind of arithmetic I learnt in primary school, whereas the declarative
calculator is ideally suited to the algebra and trigonometry I learnt in
secondary school.
Editorial Keys
The essay mentions editorial keys for the basic calculator.
Such keys are not strictly necessary for a basic calculator, but four keys,
marked say with up- down- left- and right-pointing arrows, are sufficient to
provide a capability much greater than that of the four M keys of the
typical commodity calculator.
The following description assumes that the basic calculator can be in either
of edit state or history state.
In edit state there is a cursor active in the line where
the next second operand is to be entered.
In edit state, the cursor is either in insert mode, in which case the
cursor appears as a vertical line between character spaces, or in replace
mode, in which case the cursor appears as an overlay of the current displayed
character.
Keyed digits and signs either insert or replace at the cursor, depending on the
mode. Any operation key triggers an operation, after which the cursor is in
insert mode.
The arrow keys have the following effect:
In history state one of the previous operand values is highlighted in about
the middle of the display area.
Displayed values cannot be edited in history state.
Actions in history state may be in scan mode, wherein the highlight is an underline
and no alteration to the stored values is possible, or in remove mode, wherein the
highlight is a video reversal and the highlighted value can be removed.
Keyed digits move the highlight the appropriate number of extra values (that is,
one plus the value of the key) in the direction of the previous movement, deleting
the previously highlighted value if in remove mode.
Any operation key (and prefixes may be keyed) causes the highlighted value to be
copied to the edit line and the keyed operation to be carried out immediately.
The arrow keys have the following effect:
Basic Operations
Some e-mailers have slated the calculator as I described it for
having unfamiliar advanced mathematical operations, referring in
particular to the count operation.
However, the count operation is both basic and arithmetic, and
is neither advanced nor mathematical.
Counting is not only widely useful in everyday arithmetic,
but is the basis for the long division that is, or was, taught
in elementary schools.
Thus, if 7 is to be divided into 654321 then five counts of 7s
must be carried out.
Note that this is associated with four or five modulus
or remainder operations, the other operation that I
associated with the octothorp key in my essay.
It is perhaps no coincidence that Snyder and Mitchell suggest
in their paper Is
Integer Arithmetic Fundamental to Mental Processing?
that the similar operation of equi-partitioning is behind the
remarkable arithmetical capabilities of many lightning calculators.
One of the reasons why counting is unfamiliar is that it is
so basic that there is no primary symbol identified with it.
Even the vertical bar associated with the remainder operation in
high school mathematics is not used in primary school.
Other Relevant URLs
up arrow to history state in scan mode, highlighting most recent number
down arrow to history state in remove mode, highlighting most recent number
left arrow move cursor one position left, deleting the character
at the previous position if in delete mode
right arrow move cursor one position right, deleting the character
at the previous position if in delete mode
left & right arrows toggle between insert and replace modes
up arrow move the highlight one value up (back), removing the previously highlighted value if in remove mode
down arrow move the highlight one value down (forward), removing the previously highlighted value if in remove mode
up & down arrows toggle between scan and remove modes
left arrow to edit state in replace mode, with the cursor to the
extreme left
right arrow to edit state in insert mode, with the cursor to the
extreme right