Unlike standard numeral systems, which use an integer (such as 10 in decimal, or 2 in binary) as their bases, it uses the imaginary number
It is able to (almost) uniquely represent every complex number using only the digits 0, 1, 2, and 3.
[1] Numbers less than zero, which are ordinarily represented with a minus sign, are representable as digit strings in quater-imaginary; for example, the number −1 is represented as "103" in quater-imaginary notation.
Adding these two series together then gives the total value of the number.
Because of the separation of these two series, the real and imaginary parts of complex numbers are readily expressed in base −4 as
in base b can be converted to a decimal number using the formula For the quater-imaginary system,
to a decimal number, fill in the formula above: Another, longer example:
To convert an arbitrary complex number to quater-imaginary, it is sufficient to split the number into its real and imaginary components, convert each of those separately, and then add the results by interleaving the digits.
To find the quater-imaginary representation of the imaginary component, it suffices to multiply that component by 2i, which gives a real number; then find the quater-imaginary representation of that real number, and finally shift the representation by one place to the right (thus dividing by 2i).
For example, the quater-imaginary representation of 6i is calculated by multiplying 6i × 2i = −12, which is expressed as 3002i, and then shifting by one place to the right, yielding: 6i = 302i.
Finding the quater-imaginary representation of an arbitrary real integer number can be done manually by solving a system of simultaneous equations, as shown below, but there are faster methods for both real and imaginary integers, as shown in the negative base article.
To find the representation, first write out the general formula, and group terms: Since 7 is a real number, it is allowed to conclude that d1, d3 and d5 should be zero.
This configuration gives the resulting digit string for 710.
For example, to find the representation of 6i, it is possible to use the general formula.
However, for 6i it is easily seen by looking at the formula that if d1 = 3 and all other coefficients are zero, we get the desired string for 6i.
That is: For real numbers the quater-imaginary representation is the same as negative quaternary (base −4).
If both x and y are finite binary fractions we can use the following algorithm using repeated Euclidean division: For example: 35+23i=121003.22i A radix point in the decimal system is the usual .
the radix point marks the separation between non-negative and negative powers of b.
Using the radix point the general formula becomes: or If the quater-imaginary representation of the complex unit i has to be found, the formula without radix point will not suffice.
For the imaginary part, if d5 = d3 = d−3 = 0 and when d1 = 1 and d−1 = 2 the digit string can be found.
Using the above coefficients in the digit string the result is: It is possible to add and subtract numbers in the quater-imaginary system.
In doing this, there are two basic rules that have to be kept in mind: Or for short: "If you add four, carry +1.
This is the opposite of normal long addition, in which a "carry" in the current column requires adding 1 to the next column to the left, and a "borrow" requires subtracting.
In quater-imaginary arithmetic, a "carry" subtracts from the next-but-one column, and a "borrow" adds.
The second digit from the right would become −1, so add 4 to give 3 and then carry +1 two places to the left.
For long multiplication in the quater-imaginary system, the two rules stated above are used as well.
Then the resulting partial products are added where each is shifted to the left by one digit.
Below is a table of some decimal and complex numbers and their quater-imaginary counterparts.
gives rise to an injective mapping with some suitable
is a Cantor set which allows to linearly order