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

1004^4

… (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

1/2

… (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.