Integers and Leftins

Positive Integers

     LZ+ = {aLf | i<0 di=0 and n such that i>n, di=0}

In other words, all the digits to the right of the decimal point are 0 and when looking at the digits to the left you eventually hit a point where all the remaining digits are 0.

     Z+ LZ+

Clearly a positive integer (dn ... d1 d0) is equal to the leftin where di=0 i>n. Addition, subtraction and multiplication behave the same for these leftins.

Negative Integers

It's interesting to see what happens when we try subtracting a member of LZ+ from 0.


Thus 0-1 = ...(9). 0-2 = ...(9)8. 0-3 = ...(9)7. In general, a negative integer starts with an infinite number of repeating digits of 9 base 10, or, in general, k-1 for a given base k. When you try adding or multiplying these numbers together, they behave just like you would expect the negative integers to behave. This is similar to a concept in the computer science world known as 2's complement math for binary (base 2).

     LZ- = {aLf | i<0 di=0 and n such that i>n, di=k-1}

     Z - LZ-

All Integers

     LZ = LZ+ 0 LZ-.

     Z LZ

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© Scott Sherman 1999