MTH 060 Elementary Algebra I Section 1.7 Properties of Real Numbers “There is a valid reason for everything you do in algebra.”

Elements of a Logical System  Objects  Things believed to be true about the objects accepted without proof (faith) as few as possible  Things known to be true about the objects provable based on other believed or known facts  Must be consistent no two facts can contradict each other

Logical Systems - Algebra  Objects numbers/values  constants  variables operators  arithmetic (+ * - / ^) notations for multiplication: notations for division:  relational (=    )

Logical Systems - Algebra  Objects – values & operators  Believed definitions (no ambiguities) properties example from Economics: “Goods & services can only be paid for with goods & services.” (Albert Jay Nock)

Logical Systems - Algebra  Objects – values & operators  Believed – definitions & properties  Known theorems  most common form: p  q  common proof logic: p  f1  f2  …  q

Properties of Algebra  Arithmetic operations additive (i.e. addition) multiplicative (i.e. multiplication)  Relational operations equality (another day) inequality (another day)

Properties of Algebra  Identity x + 0 = x  0 is unique and is called the “additive identity” x * 1 = x (if x  0)  1 is unique and is called the “multiplicative identity” Arithmetic Operations

Properties of Algebra  Inverse For every number x, there is a unique number -x where x + (-x) = 0  x & -x are called “opposites”  subtraction means “add the opposite of” For every number x, other than 0, there is a unique number x -1 where x * x -1 = 1  x -1 is often written 1/x  x & x -1 are called “reciprocals”  division means “multiply by the reciprocal of” Arithmetic Operations

Properties of Algebra  Commutative x + y = y + x  i.e. order in addition is not important x * y = y * x  i.e. order in multiplication is not important  standards: 3x [not x3 … prefer the number first] ab [not ba … prefer variables in alphabetical order] x(a + b) [not (a+b)x … prefer multiplier before parentheses] Arithmetic Operations NOTE: Order is important in subtraction and division!

Properties of Algebra  Associative (x + y) + z = x + (y + z)  i.e. grouping in addition is not important  therefore, x + y + z is not ambiguous (x * y) * z = x * (y * z)  i.e. grouping in multiplication is not important  therefore, xyz is not ambiguous Arithmetic Operations Careful! Don’t mix operators. (x + y) * z  x + (y * z) Also, grouping is important with subtraction and division!

Properties of Algebra  Distributive x(y + z) = xy + xz  only property that involves both operators  “distributive property of multiplication over addition” Arithmetic Operations Careful, addition over multiplication does not work! x +(yz)  (x + y)(x + z) = (2 + 3) 4(2) + 4(3) =

The number 0  Multiplying by 0 – a definition x * 0 = 0 If ab = 0, then a = 0 and/or b = 0.  Dividing into 0 – a theorem (provable) 0 / x = 0 (provided x  0) Why? 0 / x = 0 * x -1 = 0  0 -1 does not exist – a theorem (provable) If 0 -1 exists, then 0 -1 * 0 = 1 (multiplicative inverse property). But, from the above definition, 0 -1 * 0 = 0. This contradiction implies that 0 -1 can’t exist.  You cannot divide by 0 – a theorem (provable) Why? x / 0 = x * 0 -1 which does not exist!

Algebraic Properties Final Note: “In algebra, if you violate any of these properties, you will (may) get incorrect results!” Identities (0 & 1) Inverses (opposites & reciprocals) Commutative (order) Associative (grouping) Distributive (mixed operations) Properties of 0