BASE TWO AND BASE TWO
Verify that a machine is a base-two machine. (That is, explain why is the appropriate value for in the picture below.)
Write the numbers through as given by a machine and as given by a machine.
Does there seem to be an easy way to convert from one representation of a number to the other?
(Explore representations in and machines too?)
Now consider a machine. Here three dots in a box are replaced by two dots: one in the original box and one one place to the left. (Weird!)
Verify that a machine is also a base two machine.
Write the numbers through as given by a machine. Is there an easy way to convert the representation of a number to its representation, and vice versa?
FUN QUESTION: What is the “decimal” representation of the fraction in each of these machines?
How does long division work for these machines?
A DIFFERENT BASE THREE
Here’s a new type of base machine. It is called a machine and operates by converting any two dots in one box into an antidot in that box and a proper dot one place to the left. It also converts two antidots in one box to an antidot/dot pair.
- Show that the number twenty has representation in this machine.
- What number has representation in this machine?
- This machine is a base machine:
Explain why equals .
Thus the machine shows that every number can be written as a combination of powers of three using the coefficients , and .
- A woman has a simple balance scale and five stones of weights and pounds.
I place a rock of weight pounds on one side of the scale. Explain how the women can place some, or all, of her stones on the scale so as to make it balance.
- Suppose instead I place a pound rock on the woman’s scale. Can she make that stone balance too?