Properties of Integers:
The set of Integers obey certain properties that
are used heavily in algebraic manipulations. We'll investigate each now.
A. Well Ordered Set
Each integer has a unique predecessor that is 1
less and a unique successor that is 1 more. (This property allows for the use
of mathematical induction in mathematical proofs.)
Example: consider the integer 3.
2 is its predecessor and 4 is its successor.
Example: consider the integer - 5.
-6 is its predecessor and -4 is its successor.
So, we can order them like so:
{ . . ., -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5,
6, . . . }
A number line can and is often used to
reinforce this ordering and to visualize operations involving addition and
subtraction.
... -6
-5 -4 -3
-2 -1 0
1 2
3
4 5 6 ...
Visually, addition can be done by starting with the
first number, and if adding a positive number you move right that number of
bars and if adding a negative number, you move left that number of bars.
Example: 2 + 2 Start at 2. Move right 2 bars and
you land on 4, the answer.
Example: 3 + - 4 Start at 3. Move left 4 bars and
you land on -1, the answer.
Example: - 1 + -3 Start at -1. Move left 3 bars and
you land on -4, the answer.
Example: - 4 + 5 Start at -4. Move right 5 bars and
you land on 1, the answer.
Note, this is all well and good, but breaks down
with something like:
-3 - -5
Remember subtraction is adding a negative number so
we have -3 + - -5, not any better until you recall that - 5 means the opposite
of 5. The opposite of – 5 is 5, and this is what - -5 means, the opposite of
-5.
-3 + - -5 becomes -3 + 5, and now this can be done
the usual way arriving at 2.
All numbers obey the following relation:
Equality is the equivalence relation for numbers. This relation has three properties:
1) Reflexive
2) Symmetric
3) Transitive
If a, b, and c are numbers then
a = a
(reflexive, every number is always equal to itself)
a = b implies
that b = a (symmetric, switching the order
of equality is valid)
a = b, b = c,
then a = c (transitive, equality
holds in sequence from the first to
the last number in the sequence)
B. Additive Identity Exists
As noted earlier, the number '0', is the additive
identity, in other words, adding 0 to any number does not change that number.
Examples: 1 + 0 =
1 -5 + 0 =
-5 0 + 0 = 0
C. Additive Inverse Exists
Any number added to its opposite equals the
additive identity, 0.
Examples: 5 + -5 =
0 - 2 + 2 = 0
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