What is an optional leading zero in a Turing Machine Number?

To best understand what is meant by optional leading zero, you should first read the FAQ How do I figure out the Turing Machine Number?, if you haven't done so already. Here you can find a step-by-step procedure of how to translate the instructions of a Turing Machine into the Turing Machine Number.

Notice that step 4 of this procedure is an optional step, but it's there to remove any optional leading zeros. Removing the optional leading zeros will give you a smaller Turing Machine Number. If you don't remove the optional leading zeros, then you simply get a larger Turing Machine Number. It's still a valid number, and it still translates back into the same Turing Machine.

In step 4, instead of removing the optional leading zeros, you can insert optional leading zeros before any one or more of the individual binary strings. Then, in step 5, when you join the individual binary strings together, you just get a larger binary number. This is just another Turing Machine Number for the same Turing Machine.

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The easiest way to translate the instructions of a Turing Machine into the Turing Machine Number, especially if there are very many instructions, is to use the Notation Converter, which is the second Excel file that you can download from the main page of this website. When using it, you would be converting from Machine State Notation into other numerical notations.

The other numerical notations are binary (for use by the Universal Turing Machine), expanded binary, unary, and denary. Each of these gives you the same value, and it is the value of the Turing Machine Number. This number, that you get from the Notation Converter, does not include any optional leading zeros, and for that reason it is also the smallest of all the possible numbers for the Turing Machine Number.