In base four, each digit in a number represents the number of copies of that power of four. That is, the first digit tells you how many ones you have; the second tells you how many fours you have; the third tells you how many sixteens (that is, how many four-times-fours) you have; the fourth tells you how many sixty-fours (that is, how many four-times-four-times-fours) you have; and so on.
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The methodology for conversion between decimal and base-four numbers is just like that for converting between decimals and binaries, except that binary digits can be only "0" or "1", while the digits for base-four numbers can be "0", "1", "2", or "3".
(As you might expect, there is no single solitary digit in base-four math that represents the quantity "four".)
I will do the same division that I did before for binaries, keeping track of the remainders. (You may want to use scratch paper for this.)
Naturally, since this is base-4, I'll be dividing by 4s.
Then 357_{10} converts to 11211_{4}.
I'll divide again by 4s.
Note: Once I got to that "3" on top, I had to stop, because four cannot divide into 3.
Reading the numbers off the division, I see that 807_{10} converts to 30213_{4}.
I will list out the digits, and then number them from the RIGHT, starting at zero:
digits: |
3 |
0 |
2 |
1 |
3 |
numbering: |
4 |
3 |
2 |
1 |
0 |
Each digit stands for the number of copies I need for that power of four:
3×4^{4} + 0×4^{3} + 2×4^{2} + 1×4^{1} + 3×4^{0}
= 3×256 + 0×64 + 2×16 + 1×4 + 3×1
= 768 + 32 + 4 + 3
= 807
As expected, 30213_{4} converts to 807_{10}.
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I can't think of any particular use for base-seven numbers, but they will serve us by providing some more practice with conversions.
I do the division, this time by 7s:
Then 357_{10} = 1020_{7}.
When I got to that "5" on top, I had to stop, because 7 can't divide into 5.
Then 13346_{10} = 53624_{7}.
I will list the digits, and count them off from the RIGHT, starting at zero:
digits: |
5 |
3 |
6 |
2 |
4 |
numbering: |
4 |
3 |
2 |
1 |
0 |
Then I'll do the multiplication and addition:
5×7^{4} + 3×7^{3} + 6×7^{2} + 2×7^{1} + 4×7^{0}
= 5×2401 + 3×343 + 6×49 + 2×7 + 4×1
= 12005 + 1029 + 294 + 14 + 4
= 13346
Then 53624_{7} = 13346_{10}.
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