Hi Jana. I'll show you how to do this in base 4, and you can use the same idea for base 5.
The number of digits is the same as the base. For base 10, we have 10 digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9.
For base 4, there are 4 digits: 0, 1, 2, 3. You never see any digit larger than a 3 in a base 4 number.
So we start counting:
0
1
2
3
but we are out of digits, so now we have to increase the digit on the left (this would be called the tens digit if we were in base ten, but in base four, this is the fours digit). Then we can reset the ones digit to a zero:
10
"onezero", which looks like a ten, is 4 in base 4. There is a "1" in the fours column and a zero in the ones column. And you can check: (1 x 4) + (0 x 1) = 4 + 0 = 4.
Now we continue increasing the ones:
11
12
13
and now again we have to carry to the left column and reset the ones column:
20
21
22
23
again:
30
31
32
33
Now at "33", we have (3x4) + (3x1) = 23 + 3 = 15. We want to count to 20, so we have to keep going, but if we increase the "fours" column, we have to carry again to a new left column. In base ten, this is like 99 going to 100. In base 4, this is 33 going to 100. This third column on the left is 4x4 which is 16. This is comparable to the 10x10 = 100s column in base ten. So we continue:
100
101
102
103
110
And now we have (1x4x4) + (1x4) + (0x1) = 16 + 4 + 0 = 20. So we are done. To count to 20 in base four we write:
0="0", 1="1", 2="2", 3="3", 4="10", 5="11", 6="12", 7="13", 8="20", 9="21", 10="22", 11="23", 12="30", 13="31", 14="32", 15="33", 16="100", 17="101", 18="102", 19="103", 20="110".
Now you do the same thing with the base 5 system.
Cheers,
Stephen La Rocque.
