Slide 1:ALGEBRA Engr. Lizette Ivy G. Catadman Instructor / Lecturer LECTURE NOTES & PROBLEM SETS: PART 1


SETS OF NUMBERS :SETS OF NUMBERS


RATIONAL NUMBERS :RATIONAL NUMBERS A rational number is a number that can be expressed as a fraction or ratio (rational).  The numerator and the denominator of the fraction are both integers.


RATIONAL NUMBERS :RATIONAL NUMBERS When the fraction is divided out, it becomes a terminating or repeating decimal. (The repeating decimal portion may be one number or a billion numbers.)


RATIONAL NUMBERS :RATIONAL NUMBERS


IRRATIONAL NUMBERS :IRRATIONAL NUMBERS An irrational number cannot be expressed as a fraction. Irrational numbers cannot be represented as terminating or repeating decimals.  Irrational numbers are non-terminating, non-repeating decimals.


REAL NUMBERS :REAL NUMBERS Any number found on the number line. It is a number whose name will be the "address" of a point on the number line. Its absolute value is the distance of that point from 0.


REAL NUMBERS :REAL NUMBERS The real numbers are the numbers needed for measuring. Rational and irrational numbers. An actual measurement can result only in a rational number. An irrational number can result only from a theoretical  calculation.


SIGNED NUMBERS :SIGNED NUMBERS NUMBERS HAVING A PLUS (+) SIGN OR A MINUS (-) SIGN. IF NO SIGN PRECEDES A NUMBER THE (+) SIGN IS TO BE UNDERSTOOD. (+) : POSITIVE NUMBER OR ADDITION (-) : NEGATIVE NUMBER OR SUBTRACTION THE NEGATIVE 5 IS TO BE ADDED TO POSITIVE 6. THE 5 IS SUBTRACTED FROM 6.


ABSOLUTE VALUE :ABSOLUTE VALUE FOR POSITIVE NUMBERS OR ZERO: THE NUMBER ITSELF. FOR NEGATIVE NUMBER: CHANGE THE SIGN OF THE NUMBER, OR TAKE THE POSITIVE EQUIVALENT OF THE NEGATIVE NUMBER. SYMBOL ( ? ? )


Slide 11:CONSTANT A SYMBOL WHICH, THROUGHOUT A DISCUSSION, DOES NOT CHANGE; OR A SYMBOL HAVING A FIXED VALUE. 3, ?, -1/3, 100 VARIABLE A SYMBOL WHICH MAY CHANGE, THROUGHOUT A DISCUSSION, IN VALUE; OR A SYMBOL WITHOUT A FIXED VALUE. A, B, C,…, a, ß, d, ?


FUNDAMENTAL OPERATIONS :FUNDAMENTAL OPERATIONS ADDITION COMMUTATIVE LAW: THE SUM OF TWO NUMBERS IS THE SAME IN WHATEVER ORDER THEY ARE ADDED. A + B = B + A (NOTE: IF NO NUMBER PRECEDES A VARIABLE, IT IS UNDERSTOOD TO BE “1”.)


FUNDAMENTAL OPERATIONS :FUNDAMENTAL OPERATIONS ADDITION ASSOCIATIVE LAW: THE SUM OF THREE OR MORE NUMBERS IS THE SAME IN WHATEVER WAY THE NUMBERS ARE GROUPED. A + B + C = ( A + B ) + C = A + ( B + C ) = ( A + C ) + B


FUNDAMENTAL OPERATIONS :FUNDAMENTAL OPERATIONS MULTIPLICATION COMMUTATIVE LAW: THE PRODUCT OF TWO NUMBERS IS THE SAME IN WHATEVER ORDER THEY ARE MULTIPLIED. AB = BA


FUNDAMENTAL OPERATIONS :FUNDAMENTAL OPERATIONS MULTIPLICATION ASSOCIATIVE LAW: THE PRODUCT OF THREE OR MORE NUMBERS IS THE SAME IN WHATEVER WAY THE NUMBERS ARE GROUPED. ABC = ( AB ) C = A ( BC ) = ( AC ) B


FUNDAMENTAL OPERATIONS :FUNDAMENTAL OPERATIONS MULTIPLICATION DISTRIBUTIVE LAW: THE PRODUCT OF A NUMBER AND THE SUM OF OTHER NUMBERS ARE THE SAME AS THE SUM OF THE PRODUCTS OBTAINED BY MULTIPLYING EACH OF THE OTHER NUMBERS BY THE FIRST NUMBER. A ( B + C ) = AB + AC


FUNDAMENTAL OPERATIONS :FUNDAMENTAL OPERATIONS MULTIPLICATION QUANTITIES EQUAL TO THE SAME QUANTITIES OR TO EQUAL QUANTITIES ARE EQUAL TO EACH OTHER. IF EQUAL QUANTITIES ARE ADDED TO EQUAL QUANTITIES, THE SUMS ARE EQUAL.


FUNDAMENTAL OPERATIONS :FUNDAMENTAL OPERATIONS MULTIPLICATION IF EQUAL QUANTITIES ARE SUBTRACTED FROM EQUAL QUANTITIES, THE REMAINDERS ARE EQUAL.


FUNDAMENTAL OPERATIONS :FUNDAMENTAL OPERATIONS MULTIPLICATION IF EQUAL QUANTITIES ARE MULTIPLIED/DIVIDED BY EQUAL QUANTITIES (EXCEPT ZERO), THE PRODUCTS/QUOTIENTS ARE EQUAL. A QUANTITY MAY BE SUBSTITUTED FOR ITS EQUAL IN ANY ALGEBRAIC EXPRESSION.


FUNDAMENTAL OPERATIONS :FUNDAMENTAL OPERATIONS DIVISION OF POLYNOMIALS LONG DIVISION METHOD TO CHECK ANSWER OBTAINED: THE DIVISOR TIMES THE QUOTIENT PLUS THE REMAINDER SHOULD EQUAL TO THE DIVIDEND. SYNTHETIC DIVISION POLYNOMIALS WHEREIN REAL ROOTS CAN BE OBTAINED. USED IN FACTORING POLYNOMIALS


OTHER BASIC OR FUNDAMENTAL PROPERTIES OF REAL NUMBERS :OTHER BASIC OR FUNDAMENTAL PROPERTIES OF REAL NUMBERS IDENTITY PROPERTY ADDITIVE IDENTITY: A + 0 = A MULTIPLICATIVE IDENTITY: A x 1 = A INVERSE PROPERTY ADDITIVE INVERSE: A + ( -A ) = 0 OR ( - A ) + A = 0 MULTIPLICATIVE INVERSE: A x ( 1 / A ) = 1 OR ( 1 / A ) x A = 1


OTHER BASIC OR FUNDAMENTAL PROPERTIES OF REAL NUMBERS :OTHER BASIC OR FUNDAMENTAL PROPERTIES OF REAL NUMBERS PROPERTIES OF ZERO A x 0 = 0 IF ( A / B ) = 0, THEN A = 0 IF ( AB ) = 0, THEN A = 0 OR B = 0 PROPERTIES OF NEGATIVES - ( -A ) = A ( -A ) B = - ( AB ) = A ( -B) = -AB ( -A ) ( -B ) = AB -A = -1 ( A )


OTHER BASIC OR FUNDAMENTAL PROPERTIES OF REAL NUMBERS :OTHER BASIC OR FUNDAMENTAL PROPERTIES OF REAL NUMBERS PROPERTIES OF EQUALITY REFLEXIVE PROPERTY: A = A SYMMETRIC PROPERTY: IF A = B, THEN B = A TRANSITIVE PROPERTY: IF A = B & B = C, THEN A = C


OTHER BASIC OR FUNDAMENTAL PROPERTIES OF REAL NUMBERS :OTHER BASIC OR FUNDAMENTAL PROPERTIES OF REAL NUMBERS PROPERTIES OF EQUALITY ADDITION PROPERTY: IF A = B, THEN A + C = B + C MULTIPLICATIVE PROPERTY: IF A = B, THEN AC = BC SUBSTITUTION PROPERTY: IF A = B – C; B = 3 & C = 2; THEN, A = 3 – 2 = 1


ALGEBRAIC EXPRESSIONS :ALGEBRAIC EXPRESSIONS FORMED BY USING CONSTANTS AND VARIABLES AND THE ALGEBRAIC OPERATIONS OF:


Slide 26:ALGEBRAIC EXPRESSION VARIABLE x TERMS FACTORS 2: NUMERICAL COEFFICIENT x: LITERAL COEFFICIENT


Slide 28:ALGEBRAIC EXPRESSION WITH...


Slide 29:The degree is 2. The degree is 6. The degree is 3.


HIERARCHY OF OPERATIONS :HIERARCHY OF OPERATIONS EVOLUTION (POWER) AND/OR INVOLUTION (ROOT) MULTIPLICATION AND/OR DIVISION ADDITION AND/OR SUBTRACTION


SYMBOLS OF GROUPING :SYMBOLS OF GROUPING SYMBOLS USED TO ENCLOSE TERMS WHOSE SUM IS TO ACT AS A SINGLE NUMBER.


SYMBOLS OF GROUPING RULES :SYMBOLS OF GROUPING RULES Removing a symbol of grouping preceded by a minus sign changes the sign of all terms previously enclosed by that symbol. If the sign preceding a symbol of grouping is the plus sign, removal of that symbol of grouping does not affect the signs of the terms previously enclosed by it.


SYMBOLS OF GROUPING RULES :SYMBOLS OF GROUPING RULES Enclosing several terms by a symbol of grouping preceding a minus sign changes the sign of all the terms it is made to enclose. Enclosing several terms by a symbol of grouping preceded by a plus sign does not affect the signs of the terms enclosed.


SYMBOLS OF GROUPING RULES :SYMBOLS OF GROUPING RULES


SYMBOLS OF GROUPING RULES :SYMBOLS OF GROUPING RULES


FUNCTIONS :FUNCTIONS A VARIABLE IS SAID TO BE A FUNCTION OF ANOTHER IF EACH OF A SET OF VALUES OF ONE, THERE CORRESPONDS ONE OR MORE VALUES OF THE OTHER. y is a function of x. y is the dependent variable, x is the independent variable.


Slide 37:FUNCTIONS


Slide 38:LAWS OF EXPONENTS


Slide 39:LAWS OF RADICALS n is the index (root) or order of the radical. a is the radicand. m is the exponent (power). If no index is indicated, it is understood to be “2”.