Elgebra Math 10
Real number
Irrational numbers π , √��
Rational numbers Integers Whole
Natural
3 5 1 2 4 2 2 3
Rational Like:
Integers {…, -3, -2, -1, 0, 1, 2, 3…….} Whole {0, 1, 2, 3…} Natural {1, 2, 3…}
, , ,
Properties of real numbers 1234-
Reflexive property a=a Symmetric property a = b then b = a Transitive property a = b and b = c then a = c Principle of substitution if a = b then we can substitute b for a in any expirations
Commutative properties a+b=b+a , a.b=b.a Associative properties a+(b+c)=(a+b)+c=a+b+c a.(b.c)=(a.b).c=a.b.c Distributive properties a.(b+c)=a.b+a.c (a+b).c=a.c+b.c Identity Properties 0+a=a+0=a a.1=a.a=a additive inverse Properties a + (- a ) = - a +a = 0 Multiplicative inverse properties a. =
1 �� 1 ��
Multiplication by zero a.0=0
. �� = 1 if b ≠ 0
0 ��
Division properties =0
�� ��
= 1 if a ≠ 0
�� −�� −�� �� �� �� −�� −�� �� ��
Rules of signs a(-b ) = - (ab) Exponents ��n = a.a.a…….a , (-a)b = - (ab) , ( -a ) ( -b ) = ab , - ( -a ) = a , ��0 = 1 if a ≠ 0 , ��−n =
1 �� n
=
=-
,
=
n factors ,
,
if a ≠ 0
1
Laws of exponents
��n ��m = ��m+n , (��m )n = ��mn Square roots √��2 =|��|
(����)n = ��n �� n ,
�� �� �� ��
= ����−�� =
�� ��−��
���� �� ≠ 0
,
( )�� =
�� ��
�� �� ���� ,
���� �� ≠ 0
Geometry Review
�� 2 = ��2 + �� 2
Pythagorean Theorem
Geometry Formulas
1
Area = LW Perimeter = 2L + 2W
Area = 2bh
Circumference = 2πr = πd
Area = π�� 2
Volume = LWH Surface area= 2LW+ 2LH+2WH
Volume= π�� 2 ℎ =π�� 2 ℎ + 2πrℎ Surface area=
Volume= 3 ���� 3
4
Surface area=4π�� 2
Polynomials
Special Products Difference of two squares
( �� + �� )2 = �� 2 + 2���� + ��2 ( �� − �� )2 = �� 2 − 2���� + ��2
( x – a )( x + a ) = �� 2 − ��2
Squares of binomials or perfect squares
( �� + �� )3 = �� 3 + 3���� 3 + 3��2 �� + ��3 ( �� − �� )3 = �� 3 − 3���� 3 + 3��2 �� + ��3 Differences of two cubes (�� − ��)(�� 2 + ���� + ��2 ) = �� 3
Irrational numbers π , √��
Rational numbers Integers Whole
Natural
3 5 1 2 4 2 2 3
Rational Like:
Integers {…, -3, -2, -1, 0, 1, 2, 3…….} Whole {0, 1, 2, 3…} Natural {1, 2, 3…}
, , ,
Properties of real numbers 1234-
Reflexive property a=a Symmetric property a = b then b = a Transitive property a = b and b = c then a = c Principle of substitution if a = b then we can substitute b for a in any expirations
Commutative properties a+b=b+a , a.b=b.a Associative properties a+(b+c)=(a+b)+c=a+b+c a.(b.c)=(a.b).c=a.b.c Distributive properties a.(b+c)=a.b+a.c (a+b).c=a.c+b.c Identity Properties 0+a=a+0=a a.1=a.a=a additive inverse Properties a + (- a ) = - a +a = 0 Multiplicative inverse properties a. =
1 �� 1 ��
Multiplication by zero a.0=0
. �� = 1 if b ≠ 0
0 ��
Division properties =0
�� ��
= 1 if a ≠ 0
�� −�� −�� �� �� �� −�� −�� �� ��
Rules of signs a(-b ) = - (ab) Exponents ��n = a.a.a…….a , (-a)b = - (ab) , ( -a ) ( -b ) = ab , - ( -a ) = a , ��0 = 1 if a ≠ 0 , ��−n =
1 �� n
=
=-
,
=
n factors ,
,
if a ≠ 0
1
Laws of exponents
��n ��m = ��m+n , (��m )n = ��mn Square roots √��2 =|��|
(����)n = ��n �� n ,
�� �� �� ��
= ����−�� =
�� ��−��
���� �� ≠ 0
,
( )�� =
�� ��
�� �� ���� ,
���� �� ≠ 0
Geometry Review
�� 2 = ��2 + �� 2
Pythagorean Theorem
Geometry Formulas
1
Area = LW Perimeter = 2L + 2W
Area = 2bh
Circumference = 2πr = πd
Area = π�� 2
Volume = LWH Surface area= 2LW+ 2LH+2WH
Volume= π�� 2 ℎ =π�� 2 ℎ + 2πrℎ Surface area=
Volume= 3 ���� 3
4
Surface area=4π�� 2
Polynomials
Special Products Difference of two squares
( �� + �� )2 = �� 2 + 2���� + ��2 ( �� − �� )2 = �� 2 − 2���� + ��2
( x – a )( x + a ) = �� 2 − ��2
Squares of binomials or perfect squares
( �� + �� )3 = �� 3 + 3���� 3 + 3��2 �� + ��3 ( �� − �� )3 = �� 3 − 3���� 3 + 3��2 �� + ��3 Differences of two cubes (�� − ��)(�� 2 + ���� + ��2 ) = �� 3