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Station  13
Division in any base 

Here’s the division problem 276÷12276 \div 12 we did earlier in a 1101 \leftarrow 10 machine. We see the answer 2323.
Stare at this picture for a moment – it will soon sneak back up on us.


Question 1

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 1101 \leftarrow 10 machine again, I am just not going to say. Maybe it will be in a 121 \leftarrow 2, or a 141 \leftarrow 4 machine or a 1131 \leftarrow 13 machine.

Since I don’t seem to be telling which machine we’re in let’s call it an 1x1 \leftarrow x machine. (People always seem to use the letter xx to represent a quantity whose value they do not know.)

Try putting some dots in the rightmost box and make them explode.
You’ll see that it is quite annoying that I am not telling you which machine we’re in!


Question 2

In a 1101 \leftarrow 10 machine the place-value of boxes were the powers of ten: 1,10,100,1000,...1, 10, 100, 1000, ....
In a 121 \leftarrow 2 machine the place-value of boxes were the powers of two: 1,2,4,8,16,...1, 2, 4, 8, 16, ....
In a 1x1 \leftarrow x machine the place-values of boxes must be the powers of xx: 1,x,x2,x3,...1,x,x{2},x{3}, ... .

As a check:
If I tell you xx is ten in my mind, then we really are getting 1,10,100,1000,...1, 10, 100, 1000, ... and if I tell you instead xx really is two, then we are getting 1,2,4,8,16,...1, 2, 4, 8, 16, ..., and so on. So the 1x1 \leftarrow x machine is indeed representing ALL machines all at once.

Okay … without any warning, here’s a high-school algebra problem:
Compute (2x2+7x+6)÷(x+2)\left( 2x^{2}+7x+6\right) \div \left( x+2\right).


Question 3

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


What’s going on?

Suppose I told you that xx really was 1010 in my head all along. Then 2x2+7x+62x^{2} + 7x + 6 is the number 2×100+7×10+62 \times 100 + 7 \times 10 + 6, which is 276276. And x+2x + 2 is the number 10+210 + 2, that is, 1212. And so we computed 276÷12276 \div 12. We got the answer 2x+32x + 3, which is 2×10+3=232 \times 10 + 3 = 23, if I am indeed now telling you that xx is 1010.

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

Aside: By the way, if I tell you that xx was instead 22, then

(2x2+7x+6)=2×4+7×2+6\left( 2x^{2}+7x+6\right) = 2 \times 4 + 7 \times 2 + 6, which is 2828,
x+2=2+2x + 2 = 2 + 2, which is 44,
2x+3=2x2+32x + 3 = 2 x 2 + 3, which is 77

Ansd also we just computed 28÷4=728 \div 4 = 7, which is correct.

Doing division in an 1x1 \leftarrow x machine is really doing an infinite number of division problems all in one hit.

Compute (2x3+5x2+5x+6)÷(x+2)\left( 2x{3}+5x{2}+5x+6\right) \div \left( x+2\right) in the 1x1 \leftarrow x machine.


Question 4

Use the following questions to master the 1x1 \leftarrow x machine.

Compute (2x4+3x3+5x2+4x+1)÷(2x+1)\left( 2x{4}+3x{3}+5x^{2}+4x+1\right) \div \left( 2x+1\right).


Question 5

Compute (x4+3x3+6x2+5x+3)÷(x2+x+1)\left( x{4}+3x{3}+6x^{2}+5x+3\right) \div \left( x^{2}+ x+1 \right)


Question 6

Here’s a polynomial division problem written in fraction notation. Can you do it?

x4+2x3+4x2+6x+3x2+3\dfrac {x{4}+2x{3}+4x^{2}+6x+3} {x^{2}+3}


Question 7

Show that (x4+4x3+6x2+4x+1)÷(x+1)\left(x{4}+4x{3}+6x^{2}+4x+1\right) \div \left(x+1\right) equals x3+3x2+3x+1x{3}+3x{2}+3x+1.


Let's go to the next station!

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