Station  13

Division in any base 

Let’s now do the same division problem in another base. But the only tricky part is that I am not going to tell you which machine we are in!

We could be in a $1 \leftarrow 10$ machine again, I am just not going to say. Maybe it will be in a $1 \leftarrow 2$, or a $1 \leftarrow 4$ machine or a $1 \leftarrow 13$ machine.

Since I don’t seem to be telling which machine we’re in let’s call it an $1 \leftarrow x$ machine. (People always seem to use the letter $x$ to represent a quantity whose value they do not know.)

In a $1 \leftarrow 10$ machine the place-value of boxes were the powers of ten: $1, 10, 100, 1000, ...$.
In a $1 \leftarrow 2$ machine the place-value of boxes were the powers of two: $1, 2, 4, 8, 16, ...$.
In a $1 \leftarrow x$ machine the place-values of boxes must be the powers of $x$: $1,x,x{2},x{3}, ...$.

As a check:
If I tell you $x$ is ten in my mind, then we really are getting $1, 10, 100, 1000, ...$ and if I tell you instead $x$ really is two, then we are getting $1, 2, 4, 8, 16, ...$, and so on. So the $1 \leftarrow x$ machine is indeed representing ALL machines all at once.

This high school algebra problem is identical to a grade school arithmetic problem!

What’s going on?

Suppose I told you that $x$ really was $10$ in my head all along. Then $2x^{2} + 7x + 6$ is the number $2 \times 100 + 7 \times 10 + 6$, which is $276$. And $x + 2$ is the number $10 + 2$, that is, $12$. And so we computed $276 \div 12$. We got the answer $2x + 3$, which is $2 \times 10 + 3 = 23$, if I am indeed now telling you that $x$ is $10$.

So we did just repeat a grade-school arithmetic problem!

Aside: By the way, if I tell you that $x$ was instead $2$, then

$\left( 2x^{2}+7x+6\right) = 2 \times 4 + 7 \times 2 + 6$, which is $28$,
$x + 2 = 2 + 2$, which is $4$,
and,
$2x + 3 = 2 x 2 + 3$, which is $7$

Ansd also we just computed $28 \div 4 = 7$, which is correct.

Doing division in an $1 \leftarrow x$ machine is really doing an infinite number of division problems all in one hit.
Whoa!

Use the following questions to master the $1 \leftarrow x$ machine.