In our current arithmetic notation, we write complex numbers as sums of a real part and an imaginary part. We write the imaginary part as the coefficient of a term in 𝑖, which we define as √-1. Note that -1 has two square roots, either one of which you may consider +𝑖; the other one is always -𝑖. So for example, 12+34𝑖 is a typical complex number, actually a complex integer. On this page, I'll consider only integers, although the principles could be extended to the entire complex plane, including 3.14159+2.71828𝑖.

This notation is adequate, but awkward. We can imagine that at some period in the past, we wrote mixed fractions with a plus sign: spoken "one and a half" became 1+½. Then we learn to omit the plus sign - 1½ - and finally, we began to use a decimal point to make it a single number: 1.5.

Let's start by trying to do the same thing with complex numbers: we want to omit the + or -. Well, we still have to separate the real from the imaginary, and we still have to indicate the sign of both the real and imaginary parts, so a straightforward replacement only makes things worse: -1|-𝑖.

But a balanced notation gets rid of the signs. Let's start with balanced ternary, which uses the digits 1, 0, and -1. In Diego's original presentation (on which this page is based), he used a turned 1 to indicate -1. That's a good idea, but turned 1 isn't available in Unicode. I'm going to use the Musa digits   .

Here are the numbers from 0 to 13 in balanced ternary:

													

And here are the numbers from 0 to -13:

													

All the negative numbers start with . To negate a number, just change  to  and vice versa.

In normal Musa notation, we use  to separate the real and imaginary parts of a complex number. We also use its inverted form  to reverse the sign of the imaginary part, since this notation applies to non-balanced bases, too. So we could write 1-𝑖 as .

But we can do something cleverer: we can write the imaginary part below the real part. Since Musa letters are composed of two shapes, we can choose the letter whose top matches the real part, and whose bottom matches the imaginary part. Since we want to read them both as numerals - shapes - we'll choose a font that doesn't connect them with a stem, and doesn't flip or truncate shapes. Wolfgang Musa Partiture is such a font, and in addition, it stretches the two shapes along different axes to help distinguish them.

In Wolfgang, 1-𝑖 is , while -3+5𝑖 is .

We could use this stacking trick to create a complex Janus notation, with 169 digits! But let's stay with ternary, to keep things manageable.

Diego's original system went a step further: his symbols for -1 0 +1 (remember, -1 is turned 1) and his symbols for -𝑖 0𝑖 +𝑖 could be combined to form single symbols for ±1±𝑖: he simply adds a dot above for +𝑖 and a dot below for -𝑖. Again, those symbols aren't available in Unicode - a shame, since they're so elegant. That said, the vertical lines with dots and flags aren't easy to read. Here's his original system:

But he didn't have Musa! I do - and I'm going to take advantage to replace those symbols using a simple trick: I'm going to use the Musa digits for 3 and -3 to represent 𝑖 and -𝑖. Since those digits aren't used in ternary, there's no ambiguity. And since 3-1 = 2 and 3+1 = 4, I'm going to use the Musa digits for 2 and 4 to represent 𝑖-1 and 𝑖+1. The negative-𝑖 side is a little trickier: since -3-1 = -4 and -3+1 = 2, we'll use the Musa digits for -4 and -2 to represent -𝑖-1 and -𝑖+1. That results in the following grid of symbols:

	-1+𝑖		+𝑖		1+𝑖
	-1		0		1
	-1-𝑖		-𝑖		1-𝑖

One important thing to notice about this grid is that to convert any digit into its complement - its additive inverse, its negative - all you need to do is rotate it. And the complement of a full number is just spelled with the complement of each of its digits. For example, the complement of    is   .

On this page, I'll use Musa numeric order as the 1D order for this 2D grid. I also use the Musa names for the shapes to read complex ternary numbers aloud.

								
Ka	Ta	Sa	Wa	Pi	Wi	Si	Ti	Ki

And I'll use  as the symbol for this complex balanced ternary base.

Now let's expand this grid to cover all the two-digit numbers:

	-4+4𝑖		-3+4𝑖		-2+4𝑖		-1+4𝑖		+4𝑖		1+4𝑖		2+4𝑖		3+4𝑖		4+4𝑖
	-4+3𝑖		-3+3𝑖		-2+3𝑖		-1+3𝑖		+3𝑖		1+3𝑖		2+3𝑖		3+3𝑖		4+3𝑖
	-4+2𝑖		-3+2𝑖		-2+2𝑖		-1+2𝑖		+2𝑖		1+2𝑖		2+2𝑖		3+2𝑖		4+2𝑖
	-4+𝑖		-3+𝑖		-2+𝑖		-1+𝑖		+𝑖		1+𝑖		2+𝑖		3+𝑖		4+𝑖
	-4		-3		-2		-1		0		1		2		3		4
	-4-𝑖		-3-𝑖		-2-𝑖		-1-𝑖		-𝑖		1-𝑖		2-𝑖		3-𝑖		4-𝑖
	-4-2𝑖		-3-2𝑖		-2-2𝑖		-1-2𝑖		-2𝑖		1-2𝑖		2-2𝑖		3-2𝑖		4-2𝑖
	-4-3𝑖		-3-3𝑖		-2-3𝑖		-1-3𝑖		-3𝑖		1-3𝑖		2-3𝑖		3-3𝑖		4-3𝑖
	-4-4𝑖		-3-4𝑖		-2-4𝑖		-1-4𝑖		-4𝑖		1-4𝑖		2-4𝑖		3-4𝑖		4-4𝑖

Here's the addition table. It should really be an addition tesseract, but I can't display that here. To add two numbers, start at the right and work your way left. If the sum of two digits of the addends is itself a two-digit number, write the righthand digit and carry the lefthand digit to the next column, just as you do in any base.

									
									
									
							;	;	
									
									
									
									
									
									

There is no subtraction table. To subtract a from b, (b-a), just take the complement of a and add it to b, (b+(-a)).

Here's a multiplication table.

									
									
									
									
									
									
									
									
									
									

Diego shows a geometric algorithm for division, but it's too complicated for this page.