# How to Convert to Base-5

TR Smith is a product designer and former teacher who uses math in her work every day.

Most human civilizations settled on a base-10 or base-20 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 base-5 numbering system. In fact, the system of Roman numerals is in some ways a hybrid between a base-5 and base-10 system, though it is not a true "base" system since it doesn't use 0 as a place holder digit.

Base-5, 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," twenty-five as "100," one hundred twenty-five as "1000," etc. As in base-10 (decimal) the digit 0 is used as a place holder when moving up to the next power.

## Base-10 to Base-5 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 base-5 number.

As an example, lets convert the base-10 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 base-5 representation of 2874.

As another example, let's convert the decimal number 503 to base-5. 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 base-5 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 base-10 numbers, for instance, 2784 in base-10 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 base-5 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 base-10 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 base-5 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 Base-5 to Base-10

The reverse conversion, from base-5 to base-10, is done by expressiong the base-5 string as the sum of multiples of powers of 5, and then computing the sum in base-10. For example, consider the base-5 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 base-10

Tally mark style used in China and other countries that use Chinese characters.
Tally mark style used in France, Spain, and former French and Spanish colonies.
Tally mark style used in most of Europe and English-speaking countries.

## Uses of Base-5

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 base-5 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, base-5 is more efficient than base-2, but less efficient than base-10. One might think that because base-5 uses half as many digts as base-10, 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

LOG10(5) = 1.430676558.

## Fractions in Base-5

Because 5 is a prime number, the only fractions that yield terminating (non-repeating decimals) in base-5 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 base-5. This is one reason why base-5 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 base-5. In the last column, the part of the decimal in {braces} is the repeating part.

Base-10 Integer
Base-5 Equivalent

Base-10 Fraction
Base-5 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

0

5

8

1

2

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