Lesson 1 Using properties of real numbers

A set is a collection of objects  If all the members of one set are also members of a second set, then the first set is a subset of the second.

Real numbers are a set of numbers consisting of several subsets of numbers  Real numbers consist of rational numbers (can be written as a quotient of integers) and irrational numbers  Rational numbers include the  natural numbers-1,2,3,4,…,  the whole numbers- 0,1,2,3,….. and  the integers-….-3,-2,-1,0,1,2,3…

Identifying subsets of numbers  Identify the subsets of which each number is a member:  -16  32  3/4  Pi  Square root of 2  -(square root of 20)

Properties of addition and multiplication  If a,b,c are real numbers  Property addition multiplication  1)Closure a +b is real # ab is real #  2)commutative a+b=b+a ab = ba  3)associative (a+b)+c=a+(b+c) (ab)c=a(bc)  4)distributive a(b+c)= ab +ac

Identifying properties of real numbers  Which property is demonstrated:  23+10=10+23  3x8=8x3  -3(7(-5))=(-3(7))(-5)  5(7+11)=5(7)+5(11)  (12+30)+20= 12+(30+20)

Simplify and identify each property used    (12+18) + (4+ 56)   90

Identify the property used in each step  5(23)  5(20+3)  (5x20) +(5x3)   115

More properties of addition and multiplication  Property addition multiplication  Identity a+0=a,0+a=a a x1=a,1 x a=a  Inverse a+(-a)= 0 a x 1/a = 1,  a cannot be 0

Finding inverses  Find the additive inverse of 3a  Find the multiplicative inverse of  2/y-5  Find the additive inverse of 3-d  Find the multiplicative inverse of 5n/12p

Lesson 2 evaluating expressions and combining like terms  An algebraic expression can contain variables (letters that represent unspecified numbers)  When you replace the variables in an expression with selected numbers and simplify using the order of operations, you have evaluated the expression

Order of operations  1. parentheses and grouping symbols  2. exponents  3. Multiply and divide from left to right  4. add and subtract from left to right  PEMDAS

Evaluating expressions  If a= -2 and b=4  Find a(-b-a)-ab  If a = 5and b= -3  Find ab 2 -(a-b) and a 2 b +3  -2

Practice with calculator (be careful- use parentheses with negative numbers)  Evaluate 3mp +2p 2 if m = 6 and p =-3  Evaluate 5p 2 (6m+p) if m = 4 and p=-2  Evaluate m 2 p 2 -(m-p) if m= 4 and p=-2

terms  The terms of an algebraic expression are separated by addition and subtraction symbols  LIKE TERMS have the same variables raised to the same power. (the order of the variables does not matter)  Constant terms are always like terms.  Add like terms by adding the coefficients of the terms

Simplifying expressions  3xy - 2x yx + 3x  7mp m + 5pm -3

Questions for understanding  Explain the difference between (-x) 2 and -x 2  What is wrong with this and what should the answer be? 3x 2 + x + 4x = 8x 4  What is the difference between a set and a subset?  Can a number be both rational and irrational?  Explain.