**Before we Get to Balanced Ternary**

Before we get to balanced ternary notation, a brief introduction to bases. We normally use decimal notation. So the number 123 means 1*100 + 2*10 + 3*1, or, written with exponents 1*103 + 2*102 + 3*100. Most bases are like that. Probably the most familiar one other than decimal is binary, or base 2, which uses only the digits 0 and 1. So 11012 = 1*23 + 1*22 + 0*21 + 1*20 = 810 + 410 + 010 + 110 = 1310

Ternary notation isn’t much used, but it follows the same idea. It uses 0, 1 and 2, and so

1213 = 1*32 + 2*31 + 1*30 = 910 + 610 + 110 = 1610

**So, what about balanced ternary?**

In balanced ternary, there are again 3 digits, but they are now -1, 0 and 1. I will use a bold **1** to mean -1. This is strange! But fun. Let’s try

**Counting in balanced ternary**

1, 1**1**, 10, 11, 1**11**, 1**1**0……

1 = 1 1**1** = 3 – 1 = 2 10 = 3 + 0 = 3 11 = 3 + 1 = 4 1**11** = 9 – 3 – 1 = 5 1**1**0 = 9 – 3 + 0 = 6

**What about negative numbers in balanced ternary notation?**

Negative numbers are where we start to see how cool balanced ternary notation can be. We don’t need a minus sign! In fact, if we just change the 1 to -1 , we change positive to negative. For instance

1**11**t = 510 (see above). And what about **1**11? Well, that -1*9 + 1*3 +1*1 = -9+3+1 = -5!

Neat!

**Adding and subtracting in balanced ternary**

We don’t just want to count, we need to add and subtract. So, the addition table:

1 + 1 = **1**1 1 + 0 = 1 A1 + **1 =** 0

That’s the whole thing! If we want to add two numbers, we can. We just need space. Let’s try 510 + 410.

1 **1 1**

+ 1 1

1 0 0

That was easy. No carrying. How about

1 1 1

+ 1 1

1 1**1** 1**1** = 100 + 1**1**0 + 1**1.** Now, 1**1**0 + 1**1** = 10**1.** Then 100 + 10**1 =** 1**1**0**1t =** 1*27 – 1*9 + 0*3 -1*1 = 1710. Voila! And subtraction is just addition…. Simply turn the second number negative (turn 11 to **11**) and add.

Next there’s multiplication, and the table there is just as simple as that for addition:

1*1 = 1

1*0 = 0 = 0*1

1***1 = 1 = 1***1

0***1 =** 0 = **1***0

But fractions are where things really get cool.

The easy fractions are multiples and powers of 1/3. So 1/3 = .1t. 1/9 = .01t. 1/27 = .001t and so on. How about 2/3? That’s interesting, because it’s 1.**1t** that is, 1 – 1/3. Then there’s ½. It’s a repeating decimal and it can be represented in two ways: 0.111111…t OR 1.**111111…t.**

Source Wolfram Mathworld http://demonstrations.wolfram.com/BalancedTernaryNotation/,

Wikipedia