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.
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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This is another FREE ALGEBRA PRINTABLE presented to you from the
Algebra section of
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