All Your Base Are Belong to Us: Counting in Other Bases (Nashville PHP February 2020)

01000001, 2102, 1001, 101, and 41 are all ways to represent the decimal number 65 in bases other than the standard base-10 numeral system we're most familiar with. As programmers, we sometimes encounter these other bases, learn just enough to get by, and move along to the next task. But what if we could think in these other bases? What if we could count in them?

In this talk, we'll learn how to count in other bases, and we'll learn how to convert a number in any base to its base-10 equivalent, without code or a special calculator. We'll explore the most common bases we encounter in our profession (binary, octal, hexadecimal), and we'll play around with some uncommon ones (unary, ternary, duodecimal, sexagesimal, etc.). By the end, all the bases will belong to you!

Transcript

1. All Your Base Counting in Other Bases Ben Ramsey Nashville PHP 11 February 2020 Are belong to us
2. Base !!
3. Base-10 ▪ Otherwise known as “the decimal system” ▪ Greek déka; Latin decem ▪ decade, decagon, decameter, December ▪ A numeral system having 10 symbols to represent numbers
5. 1 24 432 1 489 12 794 676
6. 5 6 7 8 9 0 1 2 3 4
7. 12 794 676
8. 12 794 676 ones
9. 12 794 676 tens
10. 12 794 676 hundreds
11. 12 794 676 thousands
12. 12 794 676 ten thousands
13. 12 794 676 hundred thousands
14. 12 794 676 millions
15. 12 794 676 ten millions
16. Each place (or position) in the number represents a value.
17. positional Notation !!
18. Place-value Notation !!
19. 4676 10
20. 4676 10 digits
21. 4676 10 position
22. 4676 10 0 1 2 3 index
23. 4676 10 radix
24. 4 6 7 6 ⨉ 103 ⨉ 102 ⨉ 101 ⨉ 100 ( ) ( ) ( ) ( ) + + +
25. 4 6 7 6 ⨉ 103 ⨉ 102 ⨉ 101 ⨉ 100 ( ) ( ) ( ) ( ) + + + 4000 600 70 6
26. 4 6 7 6 ⨉ 103 ⨉ 102 ⨉ 101 ⨉ 100 ( ) ( ) ( ) ( ) + + + 4000 600 70 6 = 4676
27. What you say !!
28. Base-2 ▪ Binary ▪ A numeral system having 2 symbols to represent numbers ▪ 0 and 1
29. 1101 0111 2
30. 1101 0111 2 0 1 2 3 4 5 6 7
31. 0 1 1 1 ⨉ 23 ⨉ 22 ⨉ 21 ⨉ 20 ( ) ( ) ( ) ( ) + + + 1 1 0 1 ⨉ 27 ⨉ 26 ⨉ 25 ⨉ 24 ( ) ( ) ( ) ( ) + + + +
32. 0 1 1 1 ⨉ 23 ⨉ 22 ⨉ 21 ⨉ 20 ( ) ( ) ( ) ( ) + + + 1 1 0 1 ⨉ 27 ⨉ 26 ⨉ 25 ⨉ 24 ( ) ( ) ( ) ( ) + + + + 0 4 2 1 128 64 0 16 = 215
33. Base-16 ▪ Hexadecimal ▪ A numeral system having 16 symbols to represent numbers ▪ 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 and commonly A, B, C, D, E, and F for numbers 10-15
34. fab1 16
35. fab1 16 0 1 2 3
36. F A B 1 ⨉ 163 ⨉ 162 ⨉ 161 ⨉ 160 ( ) ( ) ( ) ( ) + + +
37. F A B 1 ⨉ 163 ⨉ 162 ⨉ 161 ⨉ 160 ( ) ( ) ( ) ( ) + + + 61440 2560 176 1
38. F A B 1 ⨉ 163 ⨉ 162 ⨉ 161 ⨉ 160 ( ) ( ) ( ) ( ) + + + 61440 2560 176 1 = 64177