Greetings,
Since there is some discussion around comparing using#< let me try to explain
the context of my thinking.
I am approaching this as a programmer and a user of mathematics, not as a
mathematician.
I prefer the Principle of Least Surprise.
I prefer simple, easy to explain rules and simple implementations.
I expect the following expressions all to answer true.
1/2 = 0.5.
1 = 1 asComplex.
1 = (1 + 0i). "same as the previous line"
0 < 1.
0 < 1 asComplex.
Every number is-equivalent-to a complex number because 1 = (1 + 0i).
[In the Scheme language every number _is_ a complex number, BTW].
So to me, saying that 0 and (1 asComplex) are not comparable is saying that 0
and 1 are also not comparable, because I consider them numerically the same
as (0+0i) and (1+0i).
My observation is that I don't want to lose properties. If A and B are
numbers, I want them to be comparable on the real number line -- no matter
how many other dimensions I add.
If B is to the right on the real number line from A, then I expect (A < B) to
answer true. I expect this to be true NO MATTER HOW MANY DIMENSIONS I ADD.
To put this in context, say that I have an object which is a "basket of
measurements" at a place and time (say position, density, temperature,
pressure). Now if I add a new dimension, say electrical conductivity, I do
not expect temperature comparisons to stop working. I can still compare
temperatures between objects which lack the conductivity slot/measurement and
those which have them.
OK. Now to the extension. I suspect this is the root of the (potential)
controversy.
I chose to say that if the real/x component of a 2d number is equal, then one
number may be considered less than another if the imaginary/y value is less.
This means that (0+0i) < (0+1i) holds.
These are the properties (and test cases) I care about.
Currently, the definition for Complex>>< is just this (see below). This
implies that (3+0i) < (3+1i) holds, even though the x/real components are
equal.
I think that this is OK. What do you think?
Are there simpler, least surprising definitions for Complex>>< ?
What useful properties/invariants/tests would you add? [As unit tests]
Thanks for your thoughtful input.
-KenD
--------------------------------Complex
< other
"self < other if other's real-part is to the right or the real-parts
are
equal and the oher's imaginary part is larger"
| otherCpx |
otherCpx := other asComplex.
^self real < otherCpx real
or: [self real = otherCpx real and: [self imag <
otherCpx imag]]
---------------------------------
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