1. Way out base: 2i

2. Donald Knuth
author of the seminal multi-volume work, “The Art of Computer Programming”
the father of the analysis of algorithms
creator of the TeX computer typesetting system
and many, many other contributions
see

3. Quater-imaginary base
“The quater-imaginary numeral system was first proposed by Donald Knuth in 1955, in a submission to a high-school science talent search.” – from
base: 2i
digits: { 0, 1, 2, 3 }

4. Quater-imaginary base
Example: What is 1032i in base 10?

5. Quater-imaginary base
k (2i)k i i i
7 -128i
Example: What is 1032i in base 10?
First, let’s build a table for 2i powers.
1 x (2i)2 + 0 x (2i)1 + 3 x (2i)0 -1

6. Quater-imaginary base
Example: i is (-3-8i)10
1 x (2i)3 + 1 x (2i)2 + 0 x (2i)1 + 1 x (2i)0
-8i
-8i - 3

7. Quater-imaginary base
Example: i is -1310
1 x (2i)6 + 0 x (2i)5 + 3 x (2i)4 + 0 x (2i)3 + 0 x (2i)2 + 0 x (2i)1 + 3 x (2i)0
-13

8. Quater-imaginary base
So what is 10.22i in base 10?
1 x (2i)1 + 0 x (2i)0 + 2 x (2i)-1
= 2i x 1 / (2i)
= 2i + 2 / (2i)
= 2i + 1 / i
= 2i – i
= i
Does 1 / i = -i?
Proof:
i x (1/i) = -i x i (multiply both sides by i)
i / i = - (i x i)
1 = - (-1)
1 = 1

9. In researching this presentation, I noticed something very interesting (and somewhat disturbing) . . .

10. Donald Knuth or Dwight from “The Office”?????