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| Natural Numbers and Integers |
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The Basic Operations, Rules and Properties |
Addition |
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Subtraction |
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The use of parenthesis |
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Multiplication |
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The
commutative property |
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The associative
property |
|
The distributive property of
multiplication over addition and subtraction |
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Division |
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| Addition |
|
a + b
= c, a
and b
are terms, and c
is the sum. |
| To
add integers having the same sign, keep the same sign and
add the absolute value of each number. |
|
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| To add integers with different
signs, keep the sign of the number with larger absolute value and subtract smaller absolute
value from the larger. |
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| The
additive inverse or opposite of a is
- a,
so that |
a
+ (-
a) = 0, and
a
+ 0
= a. |
| The
commutative property of addition, |
a
+ b
= b
+ a. |
| The
associative property of addition, |
(a
+ b)
+ c
= a
+ ( b
+ c). |
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|
| Subtraction |
|
a - b
= c, a
- minuend, b
- subtrahend, and c
- difference |
| Subtract an integer by adding its
opposite, so
that |
-
(+ a)
= -
a,
-
(- a)
=
+ a |
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| Examples: |
|
5 -
7
=
5
+ (-
7)
= -
2 |
|
-
1
-
8
=
-
1
- (
+ 8)
= - 1+
(-
8)
= -
9 |
| |
|
4 -
( -
3)
=
4
+ (
+
3)
= 7 |
|
-
5
- (-
9)
=
-
5
+
(+ 9)
= -
5+
9
= 4 |
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| The
use of parenthesis: |
|
(a
+ b)
+ c
= a
+ b
+ c |
|
(-
5
+ 3)
+ 6
= -
5
+ 3
+ 6 = 4 |
|
a
+ (b
+ c)
= a
+ b
+ c |
|
-
7
+ (-
3
+ 8)
= -
7
-
3
+ 8 = -
2 |
|
a -
( b
+ c)
= a -
b -
c |
|
6 -
( 5 -
12)
= 6
-
5
+ 12 = 13 |
|
- (a
+ b) + c
= -
a -
b
+ c |
|
- (-
2
+
7)
+ 3
= 2
-
7
+ 3 = -
2 |
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| Multiplication |
|
a · b
= c, a
and b
are the factors c
is the product |
| The
properties and rules: |
| The commutative property: |
|
a
· b
= b
· a |
|
3 · (-
7) = -
7 ·
3 = -
21 |
| The associative property: |
|
(
a
· b )
· c
= a · (
b
· c ) |
|
( 2 · 5
) · 7
=
2 ·
( 5 ·
7 ) = 70 |
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| a
· 0
= 0 |
7
· 0 =
0 |
|
a
= 1
· a
and -
a
= (-
1)
· a |
-
18 = (-
1)
· 18 |
| (-
a )
· (-
b )
= a · b |
(-
3)
· (-
4) = 12 |
|
(-
a )
· b
= a · (-
b )
= -
a · b |
3
· (-
4) = -12 |
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|
| The
distributive property of multiplication over addition and subtraction: |
|
(a
+ b)
· c
= a · c
+ b · c |
|
(-
3
+ 5)
· 6
= (-
3)
· 6
+
5 · 6
= -
18
+ 30
= 12 |
|
(a -
b)
· c
= a · c -
b · c |
|
( 7
-
4)
· (-
3)
= 7 ·
(-
3)
-
4 · (-
3)
= -
21+
12 =
-
9 |
| a · c
+ b · c
=
(a
+ b)
· c |
|
(-
5
)
· 3
+ 2
· 3
= (-
5
+ 2)
· 3
= (-
3) ·
3 =
-
9 |
| a · c
-
b · c
=
(a -
b)
· c |
|
9 · (-
6)
- 2
· (-
6)
= (9 -
2)
· (-
6)
= 7 ·
(-
6) =
-
42 |
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| Examples: |
(-
7
+ 4)
· (-
2)
= (-
7)
· (-
2)
+ 4
· (-
2)
= 14 -
8
= 6
or (-
3) ·
(-
2)
= 6 |
|
| -
9 · a
+ 5
· a
= (-
9
+
5)
· a
= -
4a |
|
3
·
(1 -
a)
+ 5
= 3
-
3
· a
+ 5
= 8
-
3a |
|
|
| Division |
|
a ¸ b
= c, a
is the dividend, b
is the divisor and c
is the quotient |
|
| Basic
identities: |
|
0
¸ a = 0
and a
¸
1
= a |
|
0
¸ 5
= 0, -
3
¸
1
= -
3 |
|
| If both the dividend and divisor signs are the same the quotient will
be positive, if they are different, the quotient will be negative. |
| (-
a )
¸
(-
b )
= a ¸ b |
|
-
28
¸
(-
4) =
28 ¸ 4
= 7 |
| -
a ¸ b
= (-
a )
¸ b = -
(a ¸ b) |
|
45
¸
(-
15) =
-
45 ¸ 15
= -
3 |
|
| Division is neither commutative nor
associative, thus |
| a
¸ b is
not the same as b
¸ a |
|
(36 ¸ 6)
¸2
= 6
¸ 2
= 3, while 36 ¸(6
¸ 2)=
36 ¸ 3
= 12 |
| (a ¸ b)
¸ c is
not the same as a
¸(b ¸ c) |
|
18 ¸
6 ¸
3 =
3 ¸ 3
= 1, while 18
¸(6
¸ 3)
=
18 ¸ 2
= 9 |
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| Beginning
Algebra Contents A |
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| Copyright
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