How to Convert to Base5
Most human civilizations settled on a base10 or base20 numbering system because human hands have 10 fingers, and our hands and feet have 20 digits. Considering we have five fingers on one hand, it's not such a stretch to imagine a base5 numbering system. In fact, the system of Roman numerals is in some ways a hybrid between a base5 and base10 system, though it is not a true "base" system since it doesn't use 0 as a place holder digit.
Base5, also called quinary and pental, uses only the digits 0, 1, 2, 3, and 4. Instead of powers of 10, it's based on powers of 5. Thus, five is written as "10," twentyfive as "100," one hundred twentyfive as "1000," etc. As in base10 (decimal) the digit 0 is used as a place holder when moving up to the next power.
Base10 to Base5 Conversion: Remainder Method
The remainder method is the easiest way to convert from one integer base to another integer base. With the remainder method, first divide your number by 5 and take note of the quotient and remainder. Next, divide the quotient by 5 and note the new quotient and new remainder. Keep dividing the new quotients by 5 until you get a quotient of zero. Next, string together your list of remainders from last to first. This string is your base5 number.
As an example, lets convert the base10 number 2874 to a quinary number. Applying the process of division with remainders we have
2874 ÷ 5 = 556 remainder 4
556 ÷ 5 = 111 remainder 1
111 ÷ 5 = 22 remainder 1
22 ÷ 5 = 4 remainder 2
4 ÷ 5 = 0 remainder 4
Reading the list of remainders from last to first and concatenating them gives us 42114 as the base5 representation of 2874.
As another example, let's convert the decimal number 503 to base5. Repeating the same method gives us
503 ÷ 5 = 100 remainder 3
100 ÷ 5 = 20 remainder 0
20 ÷ 5 = 4 remainder 0
4 ÷ 5 = 0 remainder 4
Therefore, the base5 representation is 4003.
An Alternative Method
An other method to convert from decimal to quinary is to break up the number into a sum of multiples of powers of five. This is analogous to how we represent base10 numbers, for instance, 2784 in base10 is
2*ten^3 + 7*ten^2 + 8*ten^1 + 4*ten^0
= 2*1000 + 7*100 + 8*10 + 4*1
To convert the decimal number 2784 to a base5 integer using this method, we note that the highest power of 5 that divides evenly into it is 5^4 = 625 (to us the familiar base10 notation. If the coefficients are restricted to the set {0, 1, 2, 3, 4} then we get the unique representation
4*five^4 + 2*five^3 + 1*five^2 + 1*five^1 + 4*five^0
Reading the coefficients from left to right gives us 42114 as the base5 representation of 2784, which matches our result from the previous section. We can check that this is correct by computing
4*625 + 2*125 + 1*25 + 1*5 + 4*1
= 2500 + 250 + 25 + 5 + 4
= 2784.
Converting from Base5 to Base10
The reverse conversion, from base5 to base10, is done by expressiong the base5 string as the sum of multiples of powers of 5, and then computing the sum in base10. For example, consider the base5 integer 1042233. The highest power of five present is one less than the number of digits. Because this number has seven digits, the highest power of five is 5^6 = 15625. Thus we can convert this to
1*5^6 + 0*5^5 + 4*5^4 + 2*5^3 + 2*5^2 + 3*5^1 + 3*5^0
= 15625 + 0 + 2500 + 250 + 50 + 15 + 3
= 18443 in base10
Uses of Base5
Common systems of tallying are built on units of five, as shown in the images above. Among natural human languages, only the aboriginal Dhuwal languages of Australia's Northern Territory use a true base5 counting system. Words for numbers are built on clusterings of 5, 25, etc. The word for 625 is "dambumirri dambumirri dambumirri rulu."
In terms of number length, base5 is more efficient than base2, but less efficient than base10. One might think that because base5 uses half as many digts as base10, quinary numbers are twice as long as their decimal equivalents. In fact, quinary numbers are only about 1.43 times as long as their decimal equivalents. This is because
LOG_{10}(5) = 1.430676558.
More Base Conversions
Fractions in Base5
Because 5 is a prime number, the only fractions that yield terminating (nonrepeating decimals) in base5 are those whose denominator is a power of 5. That means common fractions like one half, one third, one fourth, and even one tenth are all repeating decimals in base5. This is one reason why base5 is not commonly used. The more distinct prime factors the base has, the more fractions that can be represented as terminating decimals.
The table below shows representations of selected integers and fractions in base5. In the last column, the part of the decimal in {braces} is the repeating part.
Base10 Integer
 Base5 Equivalent
 Base10 Fraction
 Base5 Equivalent
 

6
 11
 
 1/2
 0.{2}

7
 12
 
 1/3
 0.{13}

8
 13
 
 1/4
 0.{1}

9
 14
 
 1/5
 0.1

10
 20
 
 1/6
 0.{04}

20
 40
 
 1/7
 0.{032412}

21
 41
 
 1/8
 0.{03}

22
 42
 
 1/9
 0.{023421}

23
 43
 
 1/10
 0.0{2}

24
 44
 
 1/11
 0.{02114}

25
 100
 
 1/12
 0.{02}

50
 200
 
 1/20
 0.0{1}

100
 400
 
 1/25
 0.01

200
 1300
 
 1/50
 0.00{2}

500
 4000
 
 1/100
 0.00{1}

1000
 13000
 
 1/125
 0.001

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