I was pleased to see that my recent post on the interrobang generated a good deal of excitement for this long neglected piece of punctuation. I’ve heard that there will even be a compose key sequence for the interrobang in future version of Debian’s X! It’s inspired me to do another little report from my explorations of Unicode.
I can not claim to be an expert in math(s) and I welcome clarifications and corrections. That said, I find the mathematical symbols in Unicode to be some of the most interesting. I have found these useful in the past when I wanted to concisely express that something is very much greater than (⋙) something else.
Recently, I have been confused by the "neither less-than nor greater-than" (≸) and its companion "neither greater-than nor less-than" (≹) glyphs.
In the past, I have (naively I’m told by people who are better at math than I) eschewed Unicode entirely and used the ASCII equals (=) character every time I wanted to express this relationship. I’m told (although I have yet to meet someone who can give me an example or explain why) that the relationship between numbers need not be equal to, less than, nor greater-than in some forms of math.
I’m willing to accept that. But wouldn’t that also require a "neither greater-than nor less-than nor equal to" symbol? Wouldn’t the "neither greater-than nor less-than" symbol really be implying "neither greater-than nor less-than but possibly equal to or not equal to" which would be something different?
Another character I’m still confused by is the "strictly equivalent to" symbol (≣). I understand =, ≠, ≡, and ≢ but my complexity threshold seems to be breached when the fourth bar is introduced. I also don’t understand why there is not a "not strictly equivalent to" character.
By the definitions I use, ≸ and ≹ seem strictly equivalent to me. Would be it fair to say that ≸ ≣ ≹‽
For an example of a number system where a number is neither less than, greater than or equal to another number, look no further than our favourite IEEE 754 floating-point numbers (the ones that our trusty i386 processors natively support). You see, there is a set of floating-point numbers collectively called NaNs; they compare neither less than, greater than nor equal to any oter floating-point number.
One example is the complex numbers, which are often expressed as x + yi (where i is the square root of -1). You can also think of these as just pairs of numbers (x, y) or points on a 2 dimensional plane (x across and y up).
While there are various ways you an compare these numbers (distance from (0, 0) or “magnitude” is one way) the relation “less than” or “greater than” doesn’t hold because there’s no obvious way to represent them on a single line.
I once heard a lecturer describe this as “we gain the ability to find the square root of every number, but we lose less than” (for real numbers, you can’t take the square root of negatives, but you can find the square root of -3 + 0i (-3, 0)).
Now, because there’s no less than or greater than, -4 + 6i ≸ 7 + 23i, but that doesn’t mean that -4 + 6i = 7 + 23i. (If you like points better, (-4, 6) ≸ (7, 23) but that doesn’t mean (-4, 6) and (7, 23) are the same points on a graph!)
There’s probably more interesting examples in transfinite numbers (Cantor’s theory of having different sizes of infinity involved calling them “transfinite”), although even then they talk about two different ways of thinking of numbers (“ordinals” and “cardinals”).
So one way you can think of this ≸ thing is that when there’s numbers which can’t be represented meaningfully on a single line, you can have different numbers that aren’t obviously left or right of the other. Like a lot of mathematical definitions it seems like a sleight of hand: you’re defining less than into and out of existence. Fortunately that isn’t the interesting part of mathematics: there are two. The first is coming up with definitions that are consistent and which seem meaningful and useful for whatever your goal is (a philosophical sleight of hand, but many things are) and the second is exploring the unexpected things that arise from the definitions. (A definition not rich enough to give rise to interesting things is useless and will be abandoned.)
Oh and while I don’t know (for functional purposes I should be regarded as having maths to first year undergrad level) I’m willing to bet there are examples of number systems where x > y doesn’t imply that y < x, so ≸ and ≹ might well be different things.
Wow Mary! That helps alot. :)
If you ever investigate combinatorial game theory and the surreal numbers, you’ll see that the ≸ relationship is important. Many games in CGT are numbers, and it is true that for two numbers, a ≸ b implies a = b. However, there are also games that are not numbers, such as the game called *={0|0}, which is neither less than, greater than nor equal to 0, and yet satisfies *+*=0 (Don’t blame me, I didn’t make this up ;-)
As for ≸ and ≹, I’d probably think that they will mean the same if you only use them in the sense proposed.
Aesthetic aspects aside (compare leq, leqq, and leqslant in AMSTeX, mathematicians can be very particular about such things), there are two uses Yours truly (holding a maths degree and studying for a Ph.D. in maths) can see two uses:
Pedagogical: If a ≸ b it might be nice to write b ≹ a, particularly if you have complicated expressions that swap sides while being transformed, so the reader can get some clue.
Economical: When writing two formulae in one:
1 ∓ 1 ≹ 1 ± 1 might make more sense than
1 ∓ 1 ≸ 1 ± 1.
Mind you, economics isn’t everything, neither in life nor in mathematical notation.
BTW: What did you think about “⋚” “⋛”? Yeah, the partial order guys will say something about “comparable”.
Here’s a list of common unicode stuff I’ve used:
http://www.pixelbeat.org/docs/utf8.html
There may not be such a pre-composed character (yet?), but you can make one yourself by using ‘GREATER-THAN EQUAL TO OR LESS-THAN’ (U+22DB) and ‘COMBINING LONG SOLIDUS OVERLAY’ (U+0338). E.g., in OS X, with the Unicode Hex Input input method activated and selected, type
⌥22db0338.