How To Teach Algebraic Closure

Thanks to Kellie D for this question: “How do I know if a set is closed under the rule of addition, same question for multiplication?”

Closure under addition or multiplication is a concept from Abstract Algebra that’s taught in not-so-abstract Algebra classes. It can be intuitive, if taught properly.

You first have to start with two things: a set and an operation. A set is a bunch of things. Okay, the definition is fuller than that, but for general purposes, it’s a basket full of goodies.

An operation is a way to “combine” any two things in that basket.

You can’t speak of closure without having both of these together. In Abstract Algebra, this pair of things is called an Algebraic System. That’s a fancy math term that isn’t necessary to teach, but might be helpful if you need to look things up.

We say a set is closed under an operation if you can combine any two things and the result is back in the set. Notice that the definition has BOTH the words set and operation. You can’t just be closed under an operation – you gotta have the set to put it in context.

Example 1 – Kitchen Goodies

Suppose your basket of goodies is this: {flour, sugar, chocolate chips, baking powder, egg, butter}.

And your operation is this: “mix two things together in a bowl with a spoon.”

It isn’t mathematical, per se, but it’s a set and an operation.

So if you combine two things from the basket by mixing them together, what do you get? Yummy goodness for sure, right? But the result of the blend is not found in the basket as one of the things!

Mix sugar with butter and then look in the basket. Is it there as a single blob of buttery sweetness? Nope. So the set of goodies is not closed under the operation of mixing.

Note: the opposite of closed is “not closed.” The word “open” isn’t used here.

Example 2 – Math Stuff

Now you have another basket of goodies: all the even numbers (including positives and negatives and zero). We can write this in the fancy math way like this: { x | x is an even integer }

The operation is addition.

If you combine (add) two things from the basket (any two even numbers), what do you get?

8 + 24 = 32

Is it back in the basket, i.e. is the answer an even number? YES!

Try some more to make sure. Technically you should prove it, but usually your intuition is good enough – especially in a high school or undergraduate class.

So we write “the set of even numbers is closed under the operation of addition.” Again, notice that we have BOTH set and operation involved.

Example 3 – More Math Stuff

Here’s your set: all the integers (positive, negative and zero whole numbers).

And the operation: Division.

If you combine (divide) two things from the basket (integers), what do you get?

27 ÷ 3 = 9

Looks good, right? 9 is an integer. But what about  27  2 = 13.5?

13.5 isn’t an integer because it has a decimal piece. Some bits (integers) when combined (divided) turn out to be in the basket (another integer). But not all do. And they ALL have to work in order for it to be called closed.

So this is NOT a closed system. Or we can write, “the set of integers is not closed under the operation of division.”

Example 4 – Even More Math Stuff

Set: {1, 5, 6, 7, 14}

Operation: addition.

Is it closed?

1 + 5 = 6 check!

5 + 6 = 11 oops! 11 isn’t in that set.

This set is not closed under the operation of addition. You might notice that a finite set (a set with only so much stuff) of regular numbers won’t ever be closed under addition – just add the two biggest numbers and you “bust” out of the set.

Going Beyond…

Your sets can be just about anything: real numbers, rationals (fraction-like numbers), whole numbers. They can even be bunches of matrices, polynomials, imaginary numbers, complex numbers or functions! The operations would be one of the normal operations that you would use on things in that set anyway.

Questions about how to teach this? Questions on how to learn it? Leave them in the comments.


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6 Responses to How To Teach Algebraic Closure

  1. This is a good thought-provoking post, Bon. But to be precise, the term “Algebraic Closure” has a specific meaning in abstract algebra, the study of groups, rings, fields, and their properties. It means throw in the roots of all polynomials with coefficients in a certain ring or field. Thus the alg closure of the reals R is the complex nos C, and the alg closure of C is also C, so C is “algebraically closed”. The alg closure of the rationals Q is a subfield Qbar of the complex nos. Clearly not at the level you intended for your audience, but a term is a term, no matter what names you like to make up. ;-}

    • Good point, Dan.

      I am rather sloppily throwing around English, aren’t I? I seemed to have forgotten that a set being closed algebraically under an operation cannot be casually called the concept of “algebraic closure.”

      I studied rings for my masters degree and modules on non-commutative rings for my PhD*. Fields seemed too “nice” for my blood. So there’s a good chance I totally ignored the words “algebraic closure” in undergrad and never hit on them again in grad school.

      I very much appreciate the correction, Dan.

      *or maybe it was non-commutative modules on rings – I didn’t finish my PhD due to a freak accident with marriage, San Diego and the selling of a great pick-up truck. All bad ideas.

  2. Bon,

    what you termed “being closed” is mathematically -and more plainly – is “being defined”.

    An operation is defined on a set if the result of the operation always belongs to the set. So that division is not an operation on the integers, but is an operation on the rationals.

    As Dan said, “being algebraically closed” means that the set contains roots of the polynomials with coefficients from the set.

    • Indeed, Alexander, “algebraically closed” is the incorrect term. But “closure” or “closed” is not incorrect. An algebraic structure is closed under an operation if the result of the operation acting on any two elements is in the set. I have confirmed my recollection here.

      If a set is closed under an operation, then the operation is defined for all pairs of elements. It can be defined for some elements and not for others. Subtraction in the counting numbers is indeed defined, but only for pairs of numbers where the first is greater than the second. And therefore subtraction in the counting numbers isn’t closed.

      Division IS an operation on the integers, it just isn’t defined for all integers or, equivalently, it isn’t closed.

      I did make things quite ambiguous with my “algebraically closed” – simply because I meant closed in an algebraic structure (not the formal definition wrt polynomial roots). But I’m pretty sure that I’m good on the definition of closed.

      Thanks, Alexander!

  3. Hi
    Your use of the term ‘Algebraic Closure’ seems perfectly acceptable here to me, e.g. it is quite clear you are not talking about Topological Closure (say). The other 2 contributors seem to be confusing ‘Closure’ as a property of a set as opposed to ‘Closure’ as a property of an operation on a set.


    John M
    PS ‘Being defined’ and ‘Being Closed’ are absolutely not the same.

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