How to Convert a Number to Base3
The base3 numbering system is known as ternary, or less frequently trinary. In the ternary number system every number is represented as a string of 0s, 1s, and 2s. For example, the regular base10 (decimal) number ninetyseven (97) is written as 10121 in ternary. Base3 numbering is not as frequently used as binary, octal, or hexadecimal, however it has some unique uses and some interesting properties. To convert a base10 number into base3, you divide the number and successively divide the resulting quotients by 3 and then read the string of remainders from end to beginning.
Some examples of computing base3 representations are worked out below along with some interesting properties of ternary numbers.
Converting to Base3 with the Remainder Method
Let's convert the decimal number 4,517 as an example. The first step is to divide it by three and note the remainder. This gives us a quotient of 1,505 remainder 2.
Next, divide 1,505 by 3. This gives us a quotient of 501 remainder 2. Continuing, we get
501/3 = 167 remainder 0.
167/3 = 55 remainder 2.
55/3 = 18 remainder 1.
18/3 = 6 remainder 0.
6/3 = 2 remainder 0.
2/3 = 0 remainder 2.
Now if we read the list of remainders from bottom to top we get 20012022 as the base3 representation of 4,517.
Second Example
Let's convert the base10 number 507 to base3.
 507/3 = 169 remainder 0
 169/3 = 56 remainder 1
 56/3 = 18 remainder 2
 18/3 = 6 remainder 0
 6/3 = 2 remainder 0
 2/3 = 0 remainder 2
Reading from bottom to top the base3 representation is 200210.
Third Example
Let's convert the base10 number 25,000 to ternary. Working out the successive remainders gives us
 25000/3 = 8333 remainder 1
 8333/3 = 2777 remainder 2
 2777/3 = 925 remainder 2
 925/3 = 308 remainder 1
 308/3 = 102 remainder 2
 102/3 = 34 remainder 0
 34/3 = 11 remainder 1
 11/3 = 3 remainder 2
 3/3 = 1 remainder 0
 1/3 = 0 remainder 1
Reading the list of remainders from bottom to top gives you 1021021221 as the base3 representation of the decimal number twentyfive thousand.
Converting Base3 Back to Base10
To check that 1021021221 is the correct base3 representation of 25,000, you convert the base3 representation to the base10 form and see if the answers match. To do this, take the digits of the base3 form and multiply by the power of three that corresponds to its place, starting with 3^0 = 1 in the units place, 3^1 = 3 in the "tens" place, 3^2 = 9 in the "hundreds" place, 3^3 = 27 in the thousands place, etc. Then add them all up to obtain the base10 form. With our example of 25,000 = 1021021221, this gives us
1*(3^9) + 0*(3^8) + 2*(3^7) + 1*(3^6) + 0*(3^5) + 2*(3^4) + 1*(3^3) + 2*(3^2) + 2*(3^1) + 1*(3^0) = 25,000.
Related Base Conversions
Properties of Ternary Numbers
In base10 you can tell whether or not a number is even by looking at the last digit  it will be one of 0, 2, 4, 6, or 8. But this method doesn't work in base3 since even numbers may end in either 0, 1, or 2. Instead, you count the number of 1's in the string. The ternary representation of an even number has an even number of 1s. That is, it will contain either zero, two, four, six,...etc. instances of the digit 1. Likewise, an odd number has an odd number of 1s.
Here are some other properties of base3:
 Every multiple of 3 ends in the digit 0.
 Every multiple of 9 ends in the digits 00.
 Every square number that is not a multiple of 3 ends in the digit 1. By extension, every even power ends in 1 unless the power is a multiple of 3.
 It is conjectured (but not yet proven) that for every integer n greater than 15, the base3 expansion of 2^n contains at least one of each digit.
 The base3 expansion of a number is approximately 2.0959 times longer, on average, than the base10 expansion. In comparison to binary, the base2 expansion is about 1.58496 times longer than the base3 expansion, on average.
 Every number of the form 3^n is represented by a 1 followed by n 0s.
 Every number of the form 3^n  1 is represented by a string of n 2s.
 Every number of the form (3^n  1)/2 is represented by a string of n 1s.
 The first fifteen Fibonacci numbers in base3 are 1, 1, 2, 10, 12, 22, 111, 210, 1021, 2001, 10022, 12100, 22122, 111222, and 211121.
Base10 to Base3 Conversions
Decimal
 Ternary
 Decimal
 Ternary
 Decimal
 Ternary
 

1
 1
 13
 111
 25
 221
 
2
 2
 14
 112
 26
 222
 
3
 10
 15
 120
 27
 1000
 
4
 11
 16
 121
 28
 1001
 
5
 12
 17
 122
 29
 1002
 
6
 20
 18
 200
 30
 1010
 
7
 21
 19
 201
 31
 1011
 
8
 22
 20
 202
 32
 1012
 
9
 100
 21
 210
 33
 1020
 
10
 101
 22
 211
 34
 1021
 
11
 102
 23
 212
 35
 1022
 
12
 110
 24
 220
 36
 1100

So far this tutorial has only covered how to convert positive integers from decimal to ternary, but just as fractions and irrational numbers can be represented by base10 decimal numbers, so can they be represented by base3 "decimals." Here is a table of some common fractions and irrational numbers in their ternary forms. The part of the decimal in braces { } is repeated, while ellipses... indicate a nonrepeating nonterminating string (aka, irrational number).
number
 decimal
 

1/2, one half
 0.5
 0.{1}

1/3, one third
 0.{3}
 0.1

2/3, two thirds
 0.{6}
 0.2

1/4, one fourth
 0.25
 0.{02}

3/4, three fouths
 0.75
 0.{20}

1/5, one fifth
 0.2
 0.{0121}

2/5, two fifths
 0.4
 0.{1012}

1/6, one sixth
 0.1{6}
 0.0{1}

pi
 3.141592653...
 10.010211012...

e
 2.718281828...
 2.201101121...

square root of 2
 1.414213562...
 1.102011221...

square root of 3
 1.732050807...
 1.201202122...

phi, the golden ration
 1.618033988...
 1.121200112...

Irrational numbers have a nonrepeating, nonterminating decimal expansion. If a number is irrational in base10 it is irrational in any integer base. Rational numbers either terminate or repeat. Rational numbers in base10 are still rational in base3, but whether the expansion terminates or repeats may be different. In base10, the decimal form of a rational number only terminates if the denominator is of the form (2^m)*(5^n) for some nonnegative integers m and n, otherwise it repeats. In base3, the decimal expansion only terminates if the denominator is of the form 3^n for some nonnegative n, otherwise it repeats.
Practical Uses of Ternary
Most computing systems are twostate systems based on binary, with a bit being the smallest unit. A bit represents either an on or off state, corresponding to 1 and 0. In threestate systems, the smallest unit is a "trit" which takes on three values. The extra state is called a highimpedance state.
The ternary number system has the best average radix economy of any integerbase number system. Radix economy is defined as the number of digits needed to express an integer multiplied by the total number of digits available. The lower the number the more economical the system. The average number of digits needed to express a number in base3 is log_3(K), where K is the number in question and log_3() is the logarithm base3. Since ternary has three available digits, the radix economy is 3*log_3(K).
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