Re: Re: confused about == vs === in this equality

*To*: mathgroup at smc.vnet.net*Subject*: [mg103783] Re: [mg103750] Re: [mg103732] confused about == vs === in this equality*From*: Daniel Lichtblau <danl at wolfram.com>*Date*: Mon, 5 Oct 2009 13:57:58 -0400 (EDT)*References*: <20091003104738.LCJ3I.416659.imail@eastrmwml34> <200910051136.HAA28533@smc.vnet.net>

danl at wolfram.com wrote: >> ?=== >> lhs===rhs yields True if the expression lhs is identical to rhs, and >> yields >> False otherwise. >> >> ?== >> lhs==rhs returns True if lhs and rhs are identical. >> >> But looking at this example: >> >> a = ComplexInfinity; >> If[a == ComplexInfinity, Print["YES"]] >> >> Expecting it would print "YES", but it does not. it just returns the whole >> thing unevaluated? But >> >> If[a === ComplexInfinity, Print["YES"]] >> >> does return YES. >> >> I guess I am a little confused about the "expression" bit in the >> definition. >> >> So, when using the 3"=", it is looking at the _value_ of the expression, >> but >> when using the 2"=", it is looking at the expression _as it is_, i.e. >> without evaluating it? Is this the difference? I've always used the 2"=" >> for equality, now I have to be more careful which to use. >> >> --Nasser > > First some background. Usually SameQ (===) is more stringent than Equal > (==). That is to say, things might pass the Equal test but fail the SameQ > test. Here are some examples. > > In[62]:= 2 == 2.0 > Out[62]= True > > In[63]:= 2 === 2.0 > Out[63]= False > > In[64]:= Sin[2]^2 + Cos[2]^2 == 1 > Out[64]= True > > In[66]:= Sin[2]^2 + Cos[2]^2 === 1 > Out[66]= False > > In[68]:= GoldenRatio == 1/2 + Sqrt[5]/2 > Out[68]= True > > In[69]:= GoldenRatio === 1/2 + Sqrt[5]/2 > Out[69]= False > > There are exceptions, and DirectedInfinity[] (the InputForm of > ComplexInfinity) is one such. In this case it is SameQ, but not Equal, to > itself. To try to make sense of this, consider instead Indeterminate. I > think it is uncontroversial that Indeterminate should not be deemed Equal > to itself. But it is the same expression as itself, therefore SameQ to > itself. > > The decision, which I believe only goes back a few versions, was to also > treat infinities in that way. From a mathematical point of view this makes > sense, and it may well be useful in the innards of some Limit code in > avoiding what would be incorrect cancellations. (Though I have worked on > that code, I do not recall all specifics, but I think we always carefully > guarded against this.) > > This has its shortcomings. For example, it is not uncommon to check > whether precision is finite, and handle infinite precision input > differently from finite. At one time I had to change code that had > constructs like > If [prec==Infinity,...] > to use SameQ instead of Equal. It had been pointed out to me that only ComplexInfinity seems to behave in this way. I suspect there was a time when we changed Infinity==Infinity behavior (and a code change message I have seen seems to bear out this suspicion), but I cannot verify that such a change ever went into a released version of Mathematica. So I was probably just muddying the waters there. That said, playing with infinities is a tricky business, and considering them as equal when their directions agree is itself something of a design decision. > My point being, as a design decision this is something of a question call. > I think the choice we made is the better one, but I realize there are > arguments against it, and there are examples where it can cause trouble if > one is not aware of the issue (or has inconveniently forgotten the lurking > "gotcha"). > > Let me address, finally, your reading of the documentation. I can only > conclude that the problem is at your end: clearly you cannot tell the > difference between "identical" and "identical". I'll report this as a > documentation bug of some sort. I guess I should mention that this "problem is at your end" was intended as irony. Documentation that uses non-identical meanings of "identical", with otherwise virtually identical wording, is in need of some serious spanking. I have filed a bug report about this. Daniel Lichtblau Wolfram Research

**References**:**Re: confused about == vs === in this equality***From:*danl@wolfram.com