Integers and Leftins

Positive Integers

L_Z⁺ = {aL_f | i<0 d_i=0 and n such that i>n, d_i=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⁺ L_Z⁺

Clearly a positive integer (d_n ^... d₁ d₀) is equal to the leftin where d_i=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 L_Z⁺ from 0.

 ...00000
-...00001
 ________
 ...99999

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).

L_Z^- = {aL_f | i<0 d_i=0 and n such that i>n, d_i=k-1}

Z ^- L_Z^-

All Integers

L_Z = L_Z⁺ 0 L_Z^-.

Z L_Z

Back to Index