Base five numeration systemThe 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. 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 Look at the following pattern: 19_{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} 




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