Finger Counting Balanced Ternary

Upon finding out the awesome benefits of base 3, having not only the most efficient integer radix economy, but also being psychologically more easily interpretable, I scrambled to find out what is the best way for me to adapt to using this base.

We usually gravite to base 10 naturally, as we learn to count on ten fingers. In order to adapt to base 3, the adoption should be at a fundamental level - finger counting. How can we adapt finger counting so it was obvious to a kindergartener that 10 goes after 2?

I was looking around online trying to figure out what is some better way to understand the concept. Then I found about Balanced Ternary. Balanced ternary is interesting. Instead of the numbers 0, 1, 2; you have T, 0, 1. The T, or an upsidedown 1 "1", represents -1.

Consider, a number system that doesn't require the negative sign to show something's negative! And uses 3 of the most fundamental numbers in mathematics! Awesome!

I started playing around with the idea of representing each balanced ternary number on fingers. It ended up easier than expected.

You have 1:

A negative one, or T, is just an opposing 1, or:

To represent a zero? Well, 1 + T = 0, so..:

Awesome now there is a way to represent a zero without just metaphorically showing an O or a knuckle for nothing.

Background in Bases

Before I go further, I'm going to give a little background on how number base work.

Let's say in the normal decimal system, I have the number "123". How much exactly is 123 equal to?

Well, it's 100 + 20 + 3. That is derived from the following:

123 => 1 * 102 + 2 * 101 + 3 * 100 = 100 + 20 + 3 = 123

Okay, now that I explained how bases work, lets get back to the balanced tenary, which has a base of 3.

How do we represent the next number, 2? As per how balanced ternary was designed, you build up numbers by both adding and subtracting. Since we are in base 3, moving over the decimal by one digit over, would equal 3:

10 => 1 * 31 + 0 * 30 = 3 + 0 = 3

Okay, we have 1 which equals 1, and we have 10 which equals 3, but how do you get 2? That's where T comes in:

1T => 1 * 31 + T * 30 = 3 + -1 = 2

How to represent 1T with fingers? Well, we know how to represent 1, and we know how to represent a T, so it seems to make sense to put one next to the other:

Awesome! Now how do we represent 3, or 10? Similar to how we build 0 above:

Representing 11 is easy:

What is the next balanced ternary number? 1TT:

1TT => 1 * 32 + T * 31 + T * 30 = 9 + -3 + -1 = 5

The numbers go on in the same simple pattern..

1T0:

1T1:

10T:

100:

101:

11T:

110:

111:

If you continue and use 4 fingers, you can get to 1111 which is equivilent to 40. How efficient! Much better than just the usual maximum of 10.

After playing around with this balanced ternary finger counting a bit, I realized an interesting benefit in certain situations.

What is 111 + -11?

+ =

It's awesome when you can visually see the subtraction going on.

What is 100 + -11?

+ =

I believe this method of looking at numbers has a lot of potential, and may help people subtract numbers as affectively as they add them.

Did you find another interesting side effect of using this method? Please let me know.

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