Motivated by School Mathematics Project, Book 1, Cambridge, 1965,
.
Bases for Integers
Hindu-Arabic System.
How many different numerals do we use in writing down the numbers
from 1 to 99? (99 consists of the numeral, 9, written twice.) …
the numbers from 1 to 999?
We use base 10, denary numbers or decimal arithmetic,
and place value, so that
Negative Numbers Chapter 12
For each nonnegative number, a, we associate a positive shift number,
+a, and a negative shift number, −a. We think of
these numbers as marching orders, e.g., +3 means 3 paces
forward , and +4 means 4 paces forward . Then we can
talk about +3 + −4 = −1.
Next we map numbers of the for +a, −b to a number
line - the position arrived at starting from 0. Call these new guys
shift numbers also.
"The difference between a shift number a and a shift number b can
be described by the shift needed to move from one to the other."
(p. 200). I think this means "b − a" is defined as: If b has been
ordered what must be ordered to obtain the result of a, i.e., solve
a + ? = b.
The examples say: the position of +5 in relation to the
position of −3 is +8, since a positive shift of 8 is
needed to reach +5 from −3,
−3 + +8
= +5
+5 − −3
≡ +8.
Similarly, since
+5 + −8
= −3
−3 − +5
≡ −8.
Algebraically, since +a + −a = +0 = −0
(returning to the same position!), we could apply associativity and
commutativity to show that the "rule" is:
At the end, subtracting a shift number (from another) is
equivalent to reversing the direction of the shift it represents and
adding.
+7 − −3
= +7 + +3
= +10.
+7 − +3
= +7 + −3
= +4.
+5 − +8
= +5 + −8
= −3.
−5 − −2
= −5 + +2
= −3.
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