### Integers and Leftins

#### Positive Integers

*L*_{Z+} =
{a*L*_{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} ^{...} d_{1} d_{0})
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-} =
{a*L*_{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}

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© Scott Sherman 1999