
Base five numeration system
The main difference between base five numeration system and our familiar base 10 numeration system is that
grouping is done in groups of 5 instead of 10.
For instance, to represent 24 in base 10 using sticks, you could use two groups of ten and 4 as shown below
However, to represent the same number in base 5, you make groups of 5
Notice that you have 4 groups of five and 4. You write 44_{five} or 44_{5}
Thus, 24_{10} = 44_{5}
Be very careful when you read 44_{five}! You don't read it as fourty four because this is how you read
four tens and four in base 10. We are not in base 10 anymore
44_{5} is read four four base 5
In like manner, if you group 17 in groups of five, you can get three groups of 5 and 2
You write 17_{10} = 32_{5} and 32_{5} is read three two base 5
Look at the following pattern:
18_{10} = 34_{5}
18_{10} = 33_{5}
17_{10} = 32_{5}
16_{10} = 31_{5}
15_{10} = ?
What do you think 15 is in base 5?
Yes, 15_{10} = 30_{5} = ?
This is important. It is not 3 just because we have three fives in 15
In fact, what it means is three fives in the second place value and zero number in the first place value
The first ten base 5 numerals are shown below:
1_{5 } 2_{5 }
3_{5 } 4_{5 }
10_{5 } 11_{5}
12_{5 } 13_{5 }
14_{5 } 20_{5 }
Notice that the first four numbers have the same representation in base 10 and in base 5
The second difference between base five and base ten is that in base ten, we use the 10 digits 0, 1, 2, 3 , 4 , 5, 6, 7, 8, and 9
In base 10, when you have 10 tens, you replace it by one hundred and starting from the right one hundred is in the third position
In general, place values are power of 10: 10^{0} = 1, 10^{1} = 10, 10^{2} = 100, 10^{3} = 1000, etc. representing first, second, third place etc repectively.
On the other hand,
In base 5, we use only 5 digits 0, 1, 2, 3 , and 4
In base five, when you have 5 fives, you replace it by one twenty five and starting from the right twenty five is in the third position
In this case, place values are power of 5: 5^{0} = 1, 5^{1} = 5, 5^{2} = 25, 5^{3} = 125, etc representing first, second, third place etc.respectively..
Having said that, what is 57 in base 5
Group 57 in groups of 25 and 5
57 has 2 groups of 25, 1 group of 5 and 2, so 57_{10} = 212_{5}

