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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 |
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The distributive property of
multiplication over addition and subtraction |
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Division |
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Order of operations -
parenthesis as grouping symbols |
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The
divisibility of integers |
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Prime
and composite numbers |
 The
integer prime factorization |
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The
greatest common divisor/factor (GCD/GCF) |
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The
least common multiple (LCM) |
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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 |
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|
| 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, so |
| 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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|
| Order
of operations -
parenthesis as grouping symbols |
| When expressions have more than one operation, we have to follow
rules for the order of operations. First do |
| all multiplication and
division operations working from left to right. Next do all addition
and subtraction. |
|
| Example: |
-
8 -
3 ·
(-
4)
+ 15
¸
(-
5) =
-
8 -
(-
12)
+
(-
3) =
-
8
+ 12
-
3
= 1 |
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| We often use grouping symbols, like parentheses, to help us
organize complicated expressions into |
| simpler ones. Do operations
in parentheses and other grouping symbols first. If there are grouping
symbols |
| within other grouping symbols do the innermost first. |
|
| Example: |
-
3
+ 2
· [-
2 -
3 ·
(-
4 ¸
2
+ 1)]
=
-
3 + 2
· [-
2 -
3 ·
(-
2 +
1)] =
-
3
+ 2
· [-
2 -
3 ·
(-
1)]
=
|
| |
=
-
3
+ 2
· [-
2 +
3]
=
-
3
+ 2
=
-
1 |
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| The
divisibility of integers |
| A
divisor of an integer n, also called a
factor of n, is an integer which
evenly divides n
without leaving a |
| remainder. Numbers divisible by 2 are called even and those that are not are called
odd. |
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| Example: |
7 is
divisor of 21 because 21 ¸ 7
=
3. We also say 21 is
divisible by 7 or 21 is a multiple of 7. |
| |
The divisors
of 21 are { 1, 3, 7, 21 }. |
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| The
divisibility rules |
| |
A number is divisible by
2 if the last digit is divisible by 2. |
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| |
A number is divisible by
3 if the sum of its digits is divisible by 3. |
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| |
A number is divisible by
4 if the number given by the last two digits is divisible by 4. |
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| |
A number is divisible by
5 if the last digit is 0 or 5. |
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| |
A number is divisible by
6 if it is divisible by 2 and by 3. |
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| |
A number is divisible by
7 if the result of subtracting twice the last digit from the number (with the last digit removed) is divisible by 7 (e.g. 581 is divisible by 7 since
58 -
2 ·
1 =
56 is divisible by 7). |
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| |
A number is divisible by
8 if the number given by the last three digits is divisible by 8. |
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| |
A number is divisible by
9 if the sum of its digits is divisible by 9. |
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| |
A number is divisible by
10 if the last digit is 0. |
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| Prime
and composite numbers |
| A
prime number is a natural number greater than one whose only
positive divisors are one and itself. |
| A
natural number that is greater
than one and is not a prime is called a composite number. The
sequence |
| of prime numbers begins 2, 3, 5, 7, 11, 13, 17, 19, 23, ... |
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| The
integer prime-factorization |
| Every positive integer greater than 1 can be factored as
a product of prime numbers.
|
| Given a number
n divide by the first prime from the list of primes, if it does not divide
cleanly, |
| divide
n by the next prime, and so on.
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| |
| Example: |
Thus, the prime factors of
350
= 2
· 5
· 5
· 7.
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|
| 350
| 2 |
| 175
| 5 |
| 35
| 5 |
|
7 | 7 |
| 1
| |
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| The
greatest common divisor (GCD) |
| The greatest common divisor of two integers is the largest integer
that divides both numbers. |
| Greatest common divisors can be computed by determining the prime
factors of the two numbers and |
| comparing the factors, as in the following example: |
|
| Example: |
72
= 2
· 2
· 2
· 3
· 3,
90 = 2
· 3
· 3
· 5,
=>
GCD(72, 90) = 2
· 3
· 3
= 18. |
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|
| Two numbers are called
relatively prime if their greatest common divisor equals 1.
Thus, for example |
| 12
= 2
· 2
· 3,
25 = 5
· 5
=>
GCD(12, 25) = 1. |
|
| The
least common multiple (LCM) |
| A
common multiple is a number that is a multiple of two or more
numbers. |
| The least common multiple of two integers a and
b
is the smallest positive integer that is a multiple of |
|
both a and
b. To
find the least common multiple of two numbers: - first list the prime
factors of each number, |
| - then multiply each factor the greatest
number of times it occurs in either number. If the same factor occurs |
|
more than once in both numbers, multiply the factor the greatest
number of times it occurs. |
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| Example: |
Let's find the
LCM of 24, 54 and 60, |
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|
the prime factors: 24 = 2
· 2
· 2
· 3,
54 = 2
· 3
· 3
· 3,
60 = 2
· 2
· 3
· 5, |
|
==>
LCM(24, 54, 60) = 2
· 2
· 2
· 3
· 3
· 3
· 5 = 1080. |
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| Beginning
Algebra Contents |
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| Copyright
© 2004 - 2012, Nabla Ltd. All rights reserved. |
