## Notes on Embedding

“Embedding” is one of those mathematics ideas that is really simple but we manage to make it seem really complicated.  Typically, the first time somebody studying mathematics hears or uses the term is well into their university studies.  And yet the idea of embedding is there, in the background, throughout elementary school and on.

I’ll come back in later posts to issues such as introducing fractions, introducing negative numbers, introducing non-terminating non-repeating decimal numbers, introducing arrays and matrices and so on, all of which are examples of the underlying idea of embedding.

In this post, I’ll merely introduce the idea of embedding by looking at even numbers, and contrasting the even numbers with the odd numbers.

At quite a young age, kids typically notice that when you add even numbers together, you end up again with an even number.  Later, they notice that the same thing is true when you multiply: when you multiply even numbers together, you end up again with an even number.  This allows for an interesting thought experiment: imagine that suddenly all the odd numbers ceased to exist, or became illegal, or could only be used after paying a fee to Halliburton.

evens

Though life without the odd numbers may be strange and awkward, at least you wouldn’t have to re-learn your arithmetic with even numbers.  You can still add the even numbers together, or subtract, or multiply, just like you always did.  The fancy way to say this is that even numbers are embedded in the whole numbers (with respect to addition and multiplication).

It’s only when you try to divide that you have to be careful.  When you divide 6 / 2, you cannot do this inside of the system of even numbers.  So you either stop, or you pay the steep price to Halliburton.  But then, you may remember that before, there were whole numbers that couldn’t be divided inside of the system of whole numbers either.  In the old days, 1 / 2 was strange because it wasn’t a whole number.  The trick you learned then is to write it as a fraction: 1 / 2 = $\frac{1}{2}$.   That trick can work in the even world just as well:  6 / 2 = $\frac{6}{2}$.  Done!

Now imagine that many decades have gone by, and you are one of the few remaining people that remembered the days in which there were odd numbers.  Kids in school learn to count 2, 4, 6, 8,… and they learn to skip count 6, 12, 18, 24, …  They learn to add, they learn to multiply.  They may never miss the odd numbers!  In fact, to avoid accidentally stumbling onto odd numbers, they aren’t even given cubes anymore as manipulatives, but instead use these:

And here you are, one of the few people in the world who know that the Even System isn’t the only system, isn’t the end-all system, but can be embedded smoothly into a larger system, and that in that larger system there are many more divisions by 2 that work without needing fractions.  And yet, if you tried to show this to some kids, they might have a lot of resistance to the idea.  “Why do we need to learn this?”  “These new numbers you’re showing us sure seem odd!”  “Are you sure these new numbers are really numbers?  I can’t picture them with my dumbbells.”  “How can $1 \times 1 = 1$?  We all know that when you multiply two numbers, the result is bigger!”

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