How do we write our numbers? When ten units get together, they make one new thing. We keep this new thing in a separate box to keep track of its status. This is called the 1←10 rule. Some people think we use the 1←10 rule because we have ten fingers. But what if we had a different number of fingers? Or a different type of rule altogether?

In this discussion, we took on the following tasks.

Part One

Task 1: Convert numbers to base 3 or base 5 using the 1←3 or 1←5 rule. Can every number be written in base 3? Can every number be written in base 5?

Task 2: Convert numbers from base 3 or 5 to our regular system using the rules in reverse

Task 3: Add and subtract numbers in base 3 or 5 (without converting to decimal)
(hint: carrying and borrowing require using the rule in different ways)

Task 4: Write the powers of 2 in base 5 (hint: double each answer to get the next)

Task 5: Complete the following multiplication problems in base 5
12034 x 3
30442 x 120
... (make up your own)

Task 6: Try doing some division problem in base 5 (hint: use the rule to help you find the divisor)
12304 / 21
23222 / 14
... (make up your own)

In base 5, five things explode together to become just one. But what if some things explode to become something other than one?

Part two

We can also find new bases if we let the number on the left side of the arrow be something other than one.

Task 7: Try converting numbers to base 3/2 using the 2←3 rule. Can every whole number be written in base 3/2?

Task 8: Try converting numbers from base 3/2 back to base 10. Try doing 21221. Now try doing 21212. What difficulties do you encounter?

Which numbers in base 3/2 can be successfully converted back to base 10?

Which fractions in base 10 can be successfully written in base 3/2? Can 15/4 be written (without using fractions) in base 3/2? Can 5/4?

Task 9: Try converting numbers to base -4 using the -1←4 rule. Can every number be written in base -4? (First try allowing negative digits, an then try avoiding negative digits.)

Task 10: Try doing some addition and multiplication problems in base -4. Use the traditional “carrying” algorithm.