Welcome to our exploration of ternary systems using Boolean algebra. While traditional Boolean algebra works with two states, zero and one, ternary systems extend this concept to three states: zero, one, and two. This creates a three-valued logic system that can represent more complex relationships and handle uncertain or unknown states. Let's examine how these systems compare and their practical applications.
Now let's explore the fundamental ternary logic operations. The ternary AND operation returns the minimum of its inputs, so AND of zero and two equals zero. The ternary OR operation returns the maximum of its inputs, making OR of one and two equal to two. The ternary NOT operation follows a specific pattern: NOT zero equals two, NOT one equals one, and NOT two equals zero. These truth tables show all possible combinations and demonstrate how ternary logic extends traditional Boolean operations to handle three-valued systems.
Ternary number representation uses base-3 positional notation, where each position represents a power of 3. The rightmost digit represents 3 to the power 0, which equals 1. The next position is 3 to the power 1, which equals 3, and so on. To convert decimal 19 to ternary, we repeatedly divide by 3 and collect remainders: 19 divided by 3 gives 6 remainder 1, then 6 divided by 3 gives 2 remainder 0, and finally 2 divided by 3 gives 0 remainder 2. Reading the remainders from bottom to top gives us 201 in base 3. Converting back, 122 in base 3 equals 1 times 9 plus 2 times 3 plus 2 times 1, which equals 17 in decimal.