A field, in abstract algebra, is a system where:

  1. Addition works the way you expect.
  2. Multiplication also works the way you expect.
  3. Multiplication distriutes over addition

We can make this more precise.

A field obeys the following axioms:

A1: Addition is commutative: for all x and y, x + y = y + x
A2: Addition is associative: for all x and y, (x + y) + z = x + (y + z)
A3: The field contains an additive identity: there exists a member 0 such that for all x, x + 0 = x
A4: Each member has an additive inverse: for each x there is a -x such that x + -x = 0

M1: Multiplication is commutative: for all x and y, x * y = y * x
M2: Multiplication is associative: for all x and y, (x * y) * z = x * (y * z)
M3: The field contains an multiplicative identity not equal to the additive identity: there exists a member 1 ≠ 0 such that for all x, x * 1 = x
M4: Each member other than the additive inverse has an multiplicative inverse: for each x ≠ 0 there is an x-1 such that x * x-1 = 1

D1: Multiplication distributes over addition: for all x, x, and z, x * (y + z) = (x * y) + (x * z)

That’s the whole thing. The most familiar fields are the rational numbers, real number, and complex numbers [1]. There is also a collection of finite fields. For each prime number P, the field consists of the numbers 0 through P-1. Addition and multiplication are normal, except that if the answer would be P or greater, you replace it with the remained you’d get dividing it by P. This is called modulo arithmetic. For instance, in the field with members 0, 1, 2, 3, and 4, 2 * 3 = 1, because 6 divided by 5 leaves the remainder 1. Here are the addition and multiplication tables for this field. You can verify all the axioms, if you have the patience.

+ 0 1 2 3 4
0 0 1 2 3 4
1 1 2 3 4 0
2 2 3 4 0 1
3 3 4 0 1 2
4 4 0 1 2 3
* 0 1 2 3 4
0 0 0 0 0 0
1 0 1 2 3 4
2 0 2 4 1 3
3 0 3 1 4 2
4 0 4 3 2 1


Note one oddity of the above: in axiom M3, we have to specifically assume that 0 ≠ 1, because there is no way to prove it. In fact, if it were true, we’d get a system which is is one way interesting and in another trivial:

Lemma: For all x, x * 0 = 0
We know that 0 + 1 = 0, so x = x * 1 = x * (1 + 0)
Using D1, x = (x * 1) + (x * 0) = x + (x * 0)
Substituting equals for equals into 0 = -x + x,
we get 0 = -x + x + (x * 0) = 0 + (x * 0) = x * 0

Using the lemma, if 0 = 1 then for all x, x * 1 = 0, or x = 0. So all numbers are equal to zero, and thus to each other. The result is a very complete system in which there is one answer to almost all questions. Is there a square root of two? Yes, it’s 0. Is there a square root of -1? Yup, 0. Can you divide by 0? Sure, and the result is 0.

Any resemblance to your least favorite ideology, which considers every issue a nail to be attacked with the same sledgehammer, is purely coincidental.

1. There’s a four-dimensional analog of the complex numbers called the quaternions, where in addition to 1 and i you have j and k, but it’s not quite a field because multiplication isn’t commutative. i * j = k, but j * i = -k

Mike Schilling

Mike has been a software engineer far longer than he would like to admit. He has strong opinions on baseball, software, science fiction, comedy, contract bridge, and European history, any of which he's willing to share with almost no prompting whatsoever.


  1. So if everything equals nothing, and nothingness is pure consciousness (consciousness without an object), then everything is pure consciousness? Duuude.

    • Nothing is everything
      Everything is nothing is
      Please the people, audiences
      Break the fences, nothing is

  2. A post about equality? When did we start getting political on MD?

    • And the way in which you draw a false equivalency between the left side of an equation and the right side of the equation is… troubling.

  3. Surely it’s possible, in theory, to prove that 1 and 0 aren’t equal.

    Or is that something that only a n00b would say?

    • This is pure mathematics; check your intuition at the door. If you can create a model where A1-An are true but An+1 is false, then you can’t prove An+1 from the others.

      • Nonsense. I’m sure the Fifth Postulate can be proved from the others. Why, it even looks like a theorem.

        Actually, hyperbolic geometry would be kind of fun. I like the equivalents to the Fifth Postulate, the ones that seem so obvious but turn out to be false in hyperbolic geometry. A rectangle exists. There is no upper limit to the area of a triangle. Pythagoras’ Theorem.

  4. This does not count as mindless diversions.
    I must be very mindful to understand what you’re talking about.

    Nonetheless, I approve.

    • Any requests? The real difficulty with these is coming up with topics.

      • Explain and refute the incompleteness theorem in no more than 100 words.

        • Heh… “Provide your own proofs of the Riemann hypothesis, using at least 3 different methods, with diagrams.”

        • Heinlein used to do stuff like this. Anything he disapproved of (Cantor’s theory of transfinites, uncertainty, incompleteness) would have been proved to be complete nonsense sometime before a story was set, and some character would laugh at the stupid 20th-Centuryites for believing that crap.

        • When I was in the first semester of my second year of grad school, I TA’d for a cognition course taught by a guy who was infamous among undergrads: one of those professors who so scares the shite out of students on the first day of class that by the end of the first week the course’s enrollment is half of what it was that day. I don’t want to say too much about him because it’d be pretty easy to figure out who he was, but suffice it to say his style and the style psych majors were used to weren’t quite compatible.

          Anyway, one day he calls me at my lab office and says something like, “I’m sick, can you give tomorrow’s lecture? I’ve emailed you the topic.” Now, this dude was old school: no power point, maybe a few overhead slides, but mostly just lecturing from memory, so what I got in my email was two words: “Information theory.” Why he didn’t just say those two words over the phone I don’t know, and why he expected me to know shit about information theory I definitely don’t know, but that’s all I got. So I panicked, and read everything I could find online about information theory, went to the library and got journal articles about information theory and read them, talked to professors about it, and then wrote up a whole lecture on the topic basically cribbed from those sources. I went to bed at like 3 am.

          When I woke up and checked my email the next morning, I had another email from him with an attachment. It said, “Oh, I forgot to include this. Feel free to just read directly from my notes,” and contained word file that was basically his entire lecture on information theory. Apparently he wrote up his lectures, memorized them, and didn’t bring his notes with him, something I really would have liked to have known the day before.

      • I’m studying to be a math teacher, so I’d love to see posts on the higher math that ties into the stuff I’ll be teaching high school students. This Abstract algebra post is a great example of that. As an RPG/Board/Card gamer, I love probability theory. But I’ll enjoy pretty much any math thing that you’d like to write about.

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