Download Properties of Real Numbers

Standards 6, 25
PROPERTIES OF REAL NUMBERS
THE NUMBER LINE
AND
NUMBER SYSTEMS
PROPERTIES OF REAL NUMBERS
EXAMPLES OF PROPERTIES
OF REAL NUMBERS
SIMPLIFYING EXPRESSIONS
APPLYING PROPERTIES
END SHOW1
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ALGEBRA II STANDARDS THIS LESSON AIMS:
Standard 6:
Students add, subtract, multiply, and divide complex numbers.
Estándar 6:
Los estudiantes suman, restan, multiplican, y dividen números
complejos.
Standard 25:
Students use properties from number systems to justify steps in
combining and simplifying functions.
Estándar 25:
Los estudiantes usan propiedades de los sistemas numéricos para
justificar pasos al combinar y simplificar funciones.
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Standards 6, 25
THE NUMBER LINE
INTEGERS
NEGATIVE INTEGERS
-12 -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1
POSITIVE INTEGERS
0
1
3
2
4
5
6
7
8
9 10 11 12
NATURAL NUMBERS
WHOLE NUMBERS
NATURAL NUMBERS:
1, 2, 3, 4, …
WHOLE NUMBERS:
0, 1, 2, 3, 4, …
POSITIVE INTEGERS:
1, 2, 3, 4, …
NEGATIVE INTEGERS:
-1, -2, -3, -4, …
INTEGERS:
…, -3, -2, -1, 0, 1, 2, 3, …
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Standards 6, 25
RATIONAL NUMBERS:
m
n
where:
m and n are
integers.
The following numbers can be expressed as fractions and therefore
they are Rational numbers:
-8
7
5
1
-8
7
=
=
5
25 = =
0.25 =
1
1
1
4
3
0
0.75 =
20
10 = 10
0
=
20
4
=
=
400
1
1
1
IRRATIONAL NUMBERS
3.1416
None can be expressed as a fraction!
2
1.4142
7
2.6457
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Standards 6, 25
REAL NUMBERS
Q
Z
W
I
N
R= reals
Z= integers
I= irrationals
W= Wholes
Q= rationals
N= naturals
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Standards 6, 25
PROPERTIES OF REAL NUMBERS
For any real numbers a, b, and c:
COMMUTATIVE PROPERTY:
•Addition:
a+b=b+a
•Multiplication: a b = b a
5+7 =7+5
1+6=6+1
3.6 + 1.1 = 1.1 + 3.6
9 6 =6 9
4 20 = 20 4
6.4 5.2 = 5.2 6.4
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Standards 6, 25
PROPERTIES OF REAL NUMBERS
For any real numbers a, b, and c:
ASSOCIATIVE PROPERTY:
•Addition:
(a + b) + c = a + (b + c)
(3 + 4) +1 = 3 + (4 + 1)
(2 + 5) + 7 = 2 + (5 + 7)
(6.2 + 4.1) +3.3 = 6.2 + (4.1 + 3.3)
15
4
•Multiplication: a b c= a b c
7
2
3 15
=
5
4
7
2
3
5
34 45 6 = 34 45 6
5.7 7.2 2.3 = 5.7 7.2 2.3
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Standards 6, 25
PROPERTIES OF REAL NUMBERS
For any real numbers a, b, and c:
IDENTITY PROPERTY:
•Addition:
a + 0 = 0 + a=a
•Multiplication: a 1 = 1 a = a
5+0 =0+5 =5
1+0=0+1 =1
3.6 + 0 = 0 + 3.6 = 3.6
9 1 =1 9 =9
4 1 =1 4 =4
6.4 1 = 1 6.4 = 6.4
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Standards 6, 25
PROPERTIES OF REAL NUMBERS
For any real numbers a, b, and c:
INVERSE PROPERTY:
•Addition: a + (-a) = (-a) + a=0
5 + (-5) = (-5) + 5 = 0
3 + (-3) = (-3) + 3 = 0
3.6 + (-3.6) = (-3.6)+ 3.6 = 0
If a=0 then
•Multiplication:
a 1= 1 a = 1
a a
1 1
2 =1
=
2
2
1= 1
5=1
5 5
3= 3 5
=1
3
5 5
2
5
5
3
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Standards 6, 25
PROPERTIES OF REAL NUMBERS
For any real numbers a, b, and c:
DISTRIBUTIVE PROPERTY:
•Distributive:
a(b+c) = ab + ac
and
(b+c)a = ba + ca
3(5+1) = 3(5) + 3(1)
and
(5+1)3 = 5(3) + 1(3)
4(2+6) = 4(2) + 4(6)
and
(2+6)4 = 2(4) + 6(4)
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Standards 6, 25
Name the property shown at each equation:
a) 1 45 = 45
Identity property (X)
b) 56 + 34 = 34 + 56
Commutative property (+)
c) (-3) + 3 = 0
Inverse property (+)
d) 5(9 +2) = 45 + 10
Distributive property
e) (2 + 1) +b= 2 + (1 + b)
Associative property (+)
f) -34(23) = 23(-34)
Commutative property (X)
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Standards 6, 25
Simplify 3(4c -7d) + 5(2c + 9c)
3(4c -7d) + 5(2c + 9d) = 3(4c) – 3(7d) +5(2c) +5(9d)
Use distributive property
=12c – 21d + 10c +45d
Multiply
= 12c + 10c – 21d + 45d
Use commutative property
to group like terms
= 22c +24d
Add like terms
Simplify 1 (12-4x) + 3 (15x-10)
4
5
1 (12-4x) + 3 (15x-10) 1
=( )(12) – ( 1 )(4x) + ( 3 )(15x) – ( 3 )(10)
4
5
5
4
4
5
Use distributive property
= 3 – x + 9x -6 Multiply
= 3 -6 - x + 9x Use commutative property to group like terms
= 8x-3
Add like terms and commutative property
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