To learn how to count in other bases, recall what the positional notation in base ten means. Each digit stands for a different power of ten, with zero sometimes necessary as a "placeholder." Think of a dial for each digit (kind of like the old electricity meter dials, but all the dials go clockwise) with each needle pointing up to 0 to start. Then, to count, let the rightmost needle move around the dial one number at a time. When a dial gets around to 9 the next step is to click back to 0 with the next left dial advancing one.
Practice counting in base ten by writing the first couple dozen or so numbers in a column (starting with zero).
Now that you have refreshed your memory of base ten, pretend you visit a planet where the inhabitants have seven fingers. Repeat the electric meter dial exercise, but remember that this time (in base seven) the numerals go from 0 to 6. By the way, 10(seven) is not read as "ten," nor even as "seven," but as "one-zero." So it is in other bases.
Practice counting in base seven by writing the first couple dozen or so numbers in a column parallel to the column you had earlier for base 10.
Now draw some of your own electric meter dials for a planet where the inhabitants have only three fingers (i.e., base three). What are the only three numerals that should appear on these dials? To help you count, imagine the dials going around, and after each full revolution of one dial the one next left clicks one number. Practice counting by adding another parallel column for base three to your previous columns for bases ten and seven; count up to about 33(ten) in base three. Notice it takes more digits to indicate a given number when you use a smaller base.
To do arithmetic in base higher than ten requires some new numerals beyond the familiar 0-9 of the decimal system, so we use capital letters for those other numbers. That is, the dials on the electric meters would start with 0, go around to 9, and then continue around with A, B, C..., as needed, before the needle could click back up to 0. For example, on a planet where the inhabitants had 13 fingers the numerals would be 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C.
Draw some electric meter dials for base thirteen and then count in a parallel column for the first couple dozen numbers or so. You should end up with a table like the one below (but it is better to create your own first before checking below), although yours will be a bit longer.
Of course seven, three, and thirteen aren't particularly important bases in science and computer science. Instead bases that are powers of two are important: two (binary), four (quaternary), eight (octal), and sixteen (hexadecimal), with the most important of these being binary and hexadecimal (hex), because the switches that make digital computers can be either on or off (two states).
In fact, the word "bit" comes from BInary digIT, and a bit is the smallest chunk (quantum) of information. When Sagan plays the game of 20 questions to whittle the entire universe down to a dandelion seed, each question adds one bit (yes or no question) of information. Switches in computers can be on or off which can represent yes or no; that is how information is stored in our electronic digital computers. Eight bits make a byte. 2^10 bytes is a kilobyte (kB). But you may only know about MB, GB, or soon even TB (terabytes). These are merely huge conglomerations of switches that can be on or off. We know how to make such collections of switches add and remember, and from there we get the full range of everything comptuters can do.
In the assignment called Bits and Bases (next file) you are asked to count in some important bases, and to convert numbers from one base to another. Turn in your table of counting in other bases, and turn in the converted numbers.
You might practice here first by converting the base ten number 25 to base three; and then convert the base seven number 36 into base ten.
There are some base converters on the web. If you are of a mind to cheat on the assignment, you can easily find them yourself; so, assuming that you are honest, I'll give you some links to save you the trouble--but you should only use them to learn how to do this, not just to mindlessly plug in and get the answers for the assignment. Do the conversions by hand (or in your head) on the assignment, but use these links to learn how.
TCP/IP guide
Dark
Logic
Base Eight
Purplemath
Base Conversion
Base
Converter
Clock Arithmetic
This is the most mathy this course gets.
| Ten | Seven | Three | Thirteen |
| 0 | 0 | 0 | 0 |
| 1 | 1 | 1 | 1 |
| 2 | 2 | 2 | 2 |
| 3 | 3 | 10 | 3 |
| 4 | 4 | 11 | 4 |
| 5 | 5 | 12 | 5 |
| 6 | 6 | 20 | 6 |
| 7 | 10 | 21 | 7 |
| 8 | 11 | 22 | 8 |
| 9 | 12 | 100 | 9 |
| 10 | 13 | 101 | A |
| 11 | 14 | 102 | B |
| 12 | 15 | 110 | C |
| 13 | 16 | 111 | 10 |
| 14 | 20 | 112 | 11 |