Basic Rules of Algebra for real numbers
Assume 𝑎𝑎, 𝑏𝑏, 𝑐𝑐, 𝑑𝑑 are real numbers and that 𝑚𝑚, 𝑛𝑛 are positive integers.
Commutativity
Note:
Associativity
Distributive law
Note:
𝑎𝑎 + 𝑏𝑏 = 𝑏𝑏 + 𝑎𝑎
𝑎𝑎 ∙ 𝑏𝑏 = 𝑏𝑏 ∙ 𝑎𝑎
𝑎𝑎 ∙ 𝑎𝑎 = 𝑎𝑎2
𝑎𝑎 + 𝑎𝑎 = 2𝑎𝑎
𝑎𝑎 + (𝑏𝑏 + 𝑐𝑐) = (𝑎𝑎 + 𝑏𝑏) + 𝑐𝑐
𝑎𝑎 ∙ (𝑏𝑏 ∙ 𝑐𝑐) = (𝑎𝑎 ∙ 𝑏𝑏) ∙ 𝑐𝑐
𝑎𝑎(𝑏𝑏 + 𝑐𝑐) = 𝑎𝑎𝑎𝑎 + 𝑎𝑎𝑎𝑎
(𝑎𝑎 + 𝑏𝑏)(𝑐𝑐 + 𝑑𝑑) = (𝑎𝑎 + 𝑏𝑏)𝑐𝑐 + (𝑎𝑎 + 𝑏𝑏)𝑑𝑑 = 𝑎𝑎𝑎𝑎 + 𝑏𝑏𝑏𝑏 + 𝑎𝑎𝑎𝑎 + 𝑏𝑏𝑏𝑏
Factoring Special Polynomials:
Perfect Squares
𝑎𝑎2 − 2𝑎𝑎𝑎𝑎 + 𝑏𝑏 2 = (𝑎𝑎 − 𝑏𝑏)2
Difference of Squares
𝑎𝑎2 − 𝑏𝑏 2 = (𝑎𝑎 − 𝑏𝑏)(𝑎𝑎 + 𝑏𝑏)
Difference/Sum of Cubes 𝑎𝑎3 − 𝑏𝑏 3 = (𝑎𝑎 − 𝑏𝑏)(𝑎𝑎2 + 𝑎𝑎𝑎𝑎 + 𝑏𝑏 2 )
𝑎𝑎2 + 2𝑎𝑎𝑎𝑎 + 𝑏𝑏 2 = (𝑎𝑎 + 𝑏𝑏)2
𝑎𝑎3 + 𝑏𝑏 3 = (𝑎𝑎 + 𝑏𝑏)(𝑎𝑎2 − 𝑎𝑎𝑎𝑎 + 𝑏𝑏 2 )
Identities and Inverses
𝑛𝑛
𝑛𝑛
𝑛𝑛!
𝑛𝑛
(𝑎𝑎 + 𝑏𝑏)𝑛𝑛 = Σ𝑟𝑟=0
� � 𝑎𝑎𝑛𝑛−𝑟𝑟 𝑏𝑏 𝑛𝑛 where � � = (𝑛𝑛−𝑟𝑟)!𝑟𝑟! and 𝑚𝑚! = 𝑚𝑚(𝑚𝑚 − 1) … 1 also 0! = 1
𝑟𝑟
𝑟𝑟
Fractions
𝑎𝑎 + (−𝑎𝑎) = 0 = (−𝑎𝑎) + 𝑎𝑎
𝑎𝑎 ∙ � � = 1 = � � ∙ 𝑎𝑎, 𝑎𝑎 ≠ 0
𝑎𝑎
𝑏𝑏
𝑎𝑎
𝑏𝑏
Binomial Equation
𝑎𝑎 + 0 = 𝑎𝑎 = 0 + 𝑎𝑎
𝑎𝑎
𝑏𝑏
𝑐𝑐
𝑑𝑑
+ =
Exponents
+
𝑎𝑎𝑎𝑎
𝑐𝑐
+ =
𝑐𝑐
𝑏𝑏
𝑎𝑎+𝑐𝑐
𝑏𝑏
𝑐𝑐
𝑑𝑑
𝑎𝑎
𝑏𝑏
+ =
𝑏𝑏
𝑐𝑐
𝑎𝑎 + =
𝑎𝑎
𝑏𝑏
𝑎𝑎𝑎𝑎
𝑏𝑏𝑏𝑏
− = +
𝑏𝑏𝑏𝑏
𝑏𝑏𝑏𝑏
𝑏𝑏
𝑐𝑐
−𝑐𝑐
𝑑𝑑
𝑎𝑎0 = 1, 𝑎𝑎 ≠ 0
=
=
𝑎𝑎 ∙ 1 = 𝑎𝑎 = 1 ∙ 𝑎𝑎
𝑎𝑎𝑎𝑎+𝑏𝑏𝑏𝑏
𝑏𝑏𝑏𝑏
𝑎𝑎
𝑏𝑏
𝑎𝑎𝑎𝑎+𝑏𝑏
𝑐𝑐
1
𝑎𝑎
𝑐𝑐
𝑑𝑑
∙� �=
𝑎𝑎𝑎𝑎
𝑏𝑏𝑏𝑏
𝑐𝑐
𝑏𝑏
𝑎𝑎𝑎𝑎
𝑏𝑏2
𝑐𝑐
𝑑𝑑
𝑎𝑎 𝑑𝑑
𝑏𝑏 𝑐𝑐
∙� �=
𝑏𝑏
𝑐𝑐
1
𝑎𝑎
𝑎𝑎
1
𝑏𝑏
𝑐𝑐
𝑎𝑎 ∙ � � = ∙ � � =
𝑎𝑎𝑎𝑎−𝑏𝑏𝑏𝑏
𝑏𝑏𝑏𝑏
𝑎𝑎
𝑏𝑏
÷ = ∙ =
0𝑎𝑎 = 0, 𝑎𝑎 ≠ 0
1
𝑎𝑎
��
∙ ���
𝑎𝑎 ∙ …��
∙ 𝑎𝑎 = 𝑎𝑎𝑚𝑚
𝑎𝑎𝑎𝑎
𝑏𝑏𝑏𝑏
𝑎𝑎𝑎𝑎
𝑐𝑐
𝑚𝑚
𝑎𝑎𝑚𝑚 = √𝑎𝑎
𝑚𝑚 times
𝑎𝑎𝑏𝑏 is well defined when 𝑎𝑎 and 𝑏𝑏 are not both 0
(in other words 00 is not well defined for the purposes of 1st and 2nd year math courses)
1
2
𝑎𝑎𝑏𝑏 is a complex number if 𝑎𝑎 < 0 and 𝑏𝑏 = 𝑚𝑚 � 𝑛𝑛 �
(in other words the even root of a negative number is not a real number)
𝑎𝑎𝑏𝑏 ∙ 𝑎𝑎𝑐𝑐 = 𝑎𝑎𝑏𝑏+𝑐𝑐
(𝑎𝑎𝑎𝑎)𝑐𝑐 = 𝑎𝑎𝑐𝑐 ∙ 𝑏𝑏 𝑐𝑐
Logarithms
𝑎𝑎𝑏𝑏 + 𝑎𝑎𝑏𝑏 = 2𝑎𝑎𝑏𝑏
If 𝑎𝑎 = 𝑏𝑏 𝑐𝑐 then 𝑐𝑐 = log 𝑏𝑏 𝑎𝑎 =
log 𝑏𝑏 𝑏𝑏 𝑐𝑐 = 𝑐𝑐
log 𝑏𝑏 (𝑎𝑎𝑎𝑎) = log 𝑏𝑏 𝑎𝑎 + log 𝑏𝑏 𝑐𝑐
log 𝑏𝑏 (𝑎𝑎 + 𝑎𝑎) = log 𝑏𝑏 (2𝑎𝑎)
𝑎𝑎 𝑏𝑏
𝑎𝑎 𝑐𝑐
= 𝑎𝑎𝑏𝑏−𝑐𝑐
𝑎𝑎 𝑐𝑐
𝑏𝑏
� � =
𝑎𝑎 𝑐𝑐
𝑏𝑏𝑐𝑐
𝑐𝑐
�𝑎𝑎𝑏𝑏 � = 𝑎𝑎𝑏𝑏𝑏𝑏
𝑎𝑎 −𝑐𝑐
𝑏𝑏
� �
=
𝑎𝑎 −𝑐𝑐
𝑏𝑏−𝑐𝑐
=
𝑏𝑏𝑐𝑐
𝑎𝑎 𝑐𝑐
𝑏𝑏 𝑐𝑐
𝑎𝑎
=� �
but 𝑎𝑎𝑏𝑏 + 𝑎𝑎𝑐𝑐 and 𝑎𝑎𝑏𝑏 + 𝑐𝑐 𝑏𝑏 cannot be simplified
ln 𝑎𝑎
, 𝑏𝑏
ln 𝑏𝑏
> 0 and 𝑏𝑏 ≠ 1.
log 𝑏𝑏 (𝑎𝑎𝑐𝑐 ) = 𝑐𝑐 log 𝑏𝑏 𝑎𝑎
𝑎𝑎
𝑐𝑐
log 𝑏𝑏 � � = log 𝑏𝑏 𝑎𝑎 − log 𝑏𝑏 𝑐𝑐
but log 𝑏𝑏 (𝑎𝑎 + 𝑐𝑐) cannot be simplified