KAY174 MATHEMATICS I Prof. Dr. Doğan Nadi Leblebici

ALGEBRA REFRESHMENT AND EQUATIONS PURPOSE: TO GIVE A BRIEF REVIEW OF SOME TERMS AND METHODS OF MANIPULATIVE MATHEMATICS.

SETS AND REAL NUMBERS A SET is a collection of objects. Example: The set A of even numbers between 5 and 11. The Set A={6, 8, 10} An OBJECT in a set is called a member or element of that set. The Object in the Set A is 6 or 8 or 10. Positive integers are natural numbers. Example = {0, 1, 2, 3, ….} Rational numbers are numbers such as ½ and 5/3, which can be written as ratio (quotient) of two integers.

SETS AND REAL NUMBERS A rational number is one that can be written as p/q where p and q are integers and q≠0. Because we can not divide by 0. All integers are rational. All rational numbers can be represented by decimal numbers that terminate or by nonterminating repeating decimals. Terminating Decimal: 3/4=.75 Nonterminating Repeating Decimal: 2/3=.6666…. Nonterminating Nonrepeating Decimals are called irrational numbers. Example: ∏ (Pi) and √2 are irrational.

SETS AND REAL NUMBERS Together, rational numbers and irreational numbers form the set of real numbers.

SETS AND REAL NUMBERS Real Number Venn Diagram

SOME PROPERTIES OF REAL NUMBERS 1. The Transitive Property If a = b and b = c, then a = c. 2. The Commutative Property x + y = y + x or x × y = y × x We can add or multiply two real number in any order. 3. The Associative Property a + (b + c) = (a + b) + c or a(bc) = (ab)c

SOME PROPERTIES OF REAL NUMBERS 4. The Inverse Property Additive Inverse: a + (-a) = 0 or Multiplicative Inverse a.a -1 = 1 5. The Distiributive Property a(b + c) = ab + ac and (b + c)a = ba + ca

OPERATIONS WITH SIGNED NUMBERS PROPERTYEXAMPLE a – b = a + (-b)2 – 7 = 2 + (-7) = -5 a – (-b) = a + b2 – (-7) = = 9 -a = (-1)(a)-7 = (-1)(7) a(b + c) = ab + ac6(7 + 2) = = 54 a(b - c) = ab - ac6(7 - 2) = = 30 -(a + b) = -a - b-(7 + 2) = -7 – 2 = -9 -(a - b) = -a + b-(7 - 2) = = -5 -(-a) = a-(-2) = 2 a(0) = (-a)(0) = 02(0) = (-2)(0) = 0

OPERATIONS WITH SIGNED NUMBERS PROPERTYEXAMPLE (-a)(b) = -(ab) = a(-b)(-2)(7) = -(2.7) = 2(-7) = -14 (-a)(-b) = ab(-2)(-7) = 2.7 = 14 a/1 = a7/1 = 7 or -2/1 = -1 a/b = a(1/b)2/7 = 2(1/7) 1/-a = -1/a =1/-4 = -1/4 = a/-b = -a/b =2/-7 = -2/7 = -a/-b = a/b-2/-7 = 2/7 0/a = 00/7 = 0 a/a = 12/2 = 1

OPERATIONS WITH SIGNED NUMBERS PROPERTYEXAMPLE a(b/a) = b2(7/2) = 7 a(1/a) = 12(1/2) = 1

OPERATIONS WITH SIGNED NUMBERS PROPERTYEXAMPLE

EXPONENTS AND RADICALS The product “x.x.x” is abbreviated “x 3 ”. In general, for n a positive integer, x n is the abbrevation for the product of n x’s. The letter n in x n is called the exponent and x is called the base. LAWEXAMPLE İf x≠0

EXPONENTS AND RADICALS LAWEXAMPLE İf x≠0

EXPONENTS AND RADICALS LAWEXAMPLE

EXPONENTS AND RADICALS LAWEXAMPLE

OPERATIONS WITH ALGEBRAIC EXPRESSION If numbers, represented by symbols, are combined by the operations of addition, substraction, multiplication, division, or extraction of roots, then the resulting expression is called an algebraic expression. For example: is an algebraic expression in the variable x.

OPERATIONS WITH ALGEBRAIC EXPRESSION In the fallowing expression, a and b are constants, 5 is numerical coefficient of ax 3 and 5a is coefficient of x 3. Algebraic expression with one term is called monominals, two terms is binominals, three terms is trinominals, more than one term is also called multinominal.

OPERATIONS WITH ALGEBRAIC EXPRESSION A polynominal in x is an algebraic expression of the form Where n is a positive integer and c 0, c 1, …..c n are real numbers with c n ≠0. We call n the degree of polynominal. Thus, 4x 3 – 5x 2 + x – 2 is a polynominal in x of degree 3. A nonzero constant such as 5 is a polynominal of degree zero.

OPERATIONS WITH ALGEBRAIC EXPRESSION Below is a list of special products that can be obtained from distributive property and are useful in multiplying algebraic expressions.

OPERATIONS WITH ALGEBRAIC EXPRESSION Below is a list of special products that can be obtained from distributive property and are useful in multiplying algebraic expressions.

FACTORING If two or more expressions are multiplied together, the expressions are called factors of the product. Thus if c=ab, then a and b are both factors of the product c. The process by which an expression is written as a product of its factors is called factoring. Listed below are factorization rules.

FACTORING Listed below are factorization rules.

EQUATIONS – LINEAR EQUATIONS An equation is a statement that two expressions are equal. The two expressions that make up an equations are equal. The two expressions that make up an equation are called its sides or members. They are seperated by the equality sign “=“. Examples:

EQUATIONS – LINEAR EQUATIONS Each equation contains at least one variable. A variable is a symbol that can be replaced by any one of a set of different numbers. When only one variable is involved, a solution is also called a root. The set of all solutions is called the solution set of the equation.

EQUATIONS – QUADRATIC EQUATIONS A quadratic equation in the variable x is an equation that can be written in the form ax 2 + bx + c = 0 Where a, b, and c are constants and a ≠ 0

EQUATIONS – QUADRATIC EQUATIONS Quadratic Formula