Math 105 Fraction Notes Name Page one Giovanini

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1 Math 0 Fraction Notes Name Page one Giovanini REDUCING FRACTIONS. Cancel the biggest common factor to both numerator and denominator 2. Fractions must always be reduced. ADDING OR SUBTRACTING OF LIKE FRACTIONS. Add or subtract numerators, keep the denominator the same. 2. Check for reducing. ADDING OR SUBTRACTING OF UNLIKE FRACTIONS. Find the LCD, write all denominators using LCD as denominator. 2. Follow above rules for like fractions. 3. Check for reducing. MULTIPLYING FRACTIONS. Check for any canceling or related factors on both top and bottom. 2. Cancel, then take top top bottom bottom DIVIDING FRACTIONS. Invert (flipflop) the divisor, (the fraction or number after the divide sign.) (Reciprocal), 2. If the fractions are mied numbers, you must rewrite the fractions as improper fractions before you invert and multiply. 3. Cancel related factors if you can. 4. Multiply the resulting fraction, same steps as above.. Division may be written horizontally or vertically. SIMPLIFLYING COMPLEX FRACTIONS Comple fractions are fractions in which numerator or denominator or both contain fractions. Method One. Perform add or subtract in denominator and/or numerator. May first have to find LCD. 2. Do indicated division from this result. 3. Check for reducing. Method Two. Multiply numerator and denominator of the comple fraction by the LCD OF ALL THE FRACTIONS APPEARING IN THE COMPLEX FRACTION. 2. Then simplify these results. 3. Check for reducing.

2 When a fraction is negative the minus sign can go with the numerator, denominator, or out in front of the fraction. Page 2 y y y Addition of Signed Numbers. If the signs are the same ADD, Keep the sign. 2. If the signs are different SUBTRACT, answer takes the sign of the larger number. Subtraction of Signed Numbers. Change the subtract sign to an add, then change the following sign to its opposite. 2. Then follow the rules for addition of signed numbers. Multiplication and Division of Two Signed Numbers. If the signs are the same, answer is POSITIVE. 2. If the signs are different, answer is NEGATIVE. When multiplying an EVEN amount of negative numbers, the answer is POSITIVE. When multiplying an ODD amount of negative numbers, the answer is NEGATIVE. The Sets of Real Numbers The real numbers are any number on the number line. They are made up of the rational numbers and the irrational numbers. The real numbers represent all positive and negative numbers, all fractions and all terminating, repeating and non-repeating decimals. The rational numbers are any number that can be written as a fraction or terminating or repeating decimal. A terminating decimal is a decimal that has a set number of terms. A repeating decimal is a decimal that has one or more digits that repeat in a pattern forever. We would put a bar over the group of numbers that are repeated. In order to find the equivalent terminating decimal divide the denominator into the numerator, add the decimal, and some zeros. The irrational numbers are any number that can t be written as a fraction or terminating or repeating decimal. These are the non-repeating decimals. They never form a pattern and never terminate. The following are eamples of irrational numbers. 2, 3,, 7 and π

3 The Sets of Real Numbers Page 3 The Real Numbers are any number on the number line. The Rational numbers are any number that can be written as a fraction or terminating or repeating decimal. The irrational numbers are any number that can t be written as a fraction or terminating or repeating decimal. Integers..., 4, 3, 2,, 0,, 2, 3, 4,... Whole Numbers 0,, 2, 3, 4,... Natural Numbers, 2, 3, 4,... Real numbers / \ Rational numbers Irrational numbers / Integers / Whole numbers / Natural numbers Sometimes a number needs to be written in a more simplified form to decide which set of numbers it belongs to Given the set of numbers 4, 4π, 2, 0,, 0,, 7,.6, Which of the above numbers are real numbers? 2. Which of the above numbers are rational numbers? 3. Which of the above numbers are irrational numbers? 4. Which of the above numbers are integers?. Which of the above numbers are whole numbers? 6. Which of the above numbers are natural numbers?

4 Properties of Real Numbers Page 4 a. ASSOCIATIVE PROPERTY Associative property of addition a b c a b c Associative property of multiplication a b c a b c b. COMMUNITIVE PROPERTY Commutative property of addition a b b a Commutative property of multiplication a b b a c. DISTRIBUTIVE PROPERTY ab c ab ac y 30 2y 3a9b 4 27ab 2a d. ADDITIVE IDENTITY a 0 a and 0 a a 0 and and e. ADDITIVE INVERSE a a 0 and a a and and f. MULTIPLICATIVE IDENTIY a a and a a 6 6 and and 2 2 g. MULTIPLICATIVE INVERSE a and a a a 7 and 7 and 7 7 h. MULTIPLICATION PROPERTY OF ZERO a 0 0 and 0 a and and 0 3 0

5 Notes on Empty set, Order of Operations, and Absolute Value Page Empty set or null set is a set that has no elements. The symbol for empty set is. This is another symbol for empty set, but use. Variables are letters that can take on different values depending on the problem. Multiplication can be shown with a dot, or parentheses around one or both of the numbers. Write the dot between the two numbers so it doesn t look like the decimal. Please don t use the times sign, as it looks like the letter. Order of Operations. Copy the problem. 2. Substitution, if there are any variables in the problem substitute the given values into the problem and simplify. 3. Parentheses, Brackets, the Division Bar, the Absolute Value Bars, all act as grouping symbols. Work inside the innermost one first and simplify. 4. Simplify all roots and radicals in order from left to right.. Multiply and divide in order from left to right 6. Add or Subtract in order from left to right. PBMDAS PLEASE BURY MY DEAD AUNT SALLY PARENTHESES, POWERS& ROOTS, BRACKETS, MULTIPLICATION, DIVISION, ADDITION, AND SUBTRACTION PEMDAS PLEASE EXCUSE MY DEAD AUNT SALLY ABSOLUTE VALUE PARENTHESES, BRACKETS, EXPONENTS & ROOTS, MULTIPLICATION, DIVISION, ADDITION, AND SUBTRACTION Absolute value is the distance between zero and the given number on the number line. Since absolute value is a distance it is never negative. However there may be a minus sign out in front of the absolute value bars. If there is, it is understood to be a negative one in front of the absolute value bars. Then the final answer would be negative. Absolute value has two vertical bars as the symbol. The absolute value of zero is zero, since we haven t moved either left or right on the number line. These bars act as one of the grouping symbols. Simplify inside the bars first, before taking the absolute value of what is inside the bars.

6 Math0 Inequality Notes Name Page 6 Giovanini An inequality is a statement that shows two quantities are not equal. The symbols for inequality are,,,. This is the less than sign. This is the less than or equal to sign. This is the greater than sign. This is the greater than or equal to sign. Set Builder notation is a method of writing a set, using a special notation of set braces and a straight line. has some property 3 For eample Some books use parentheses and brackets while some books use open or closed dots to determine if the endpoints are included in the graph of an inequality. Some books also use interval notation as a shortcut way of writing the solution set. An interval is a section or piece of a number line. Interval notation is a special notation that uses symbols to describe the interval or section of the number line. When writing a solution set in interval notation, parentheses are always used net to the infinity symbol., Parentheses or open dots indicate the endpoints are not included in the graph. Also unshaded portions of a number line indicate points not included in graph. Brackets or closed dots indicate the endpoints are included in the graph. Also shaded portions of a number line are used to indicate points included in the graph. Describe in words how to graph each of following inequalities, make sure to tell which endpoints to use. Also write the solution set in interval notation. 9 Line with an arrow going to the right, parentheses at 9. Sol. Set 9, Line with an arrow going to the left, bracket at. Sol. Set, 7 Line with an arrow going to the right, bracket at 7. Sol. Set 7,

7 3 Line with an arrow going to the left, parentheses at 3. Page 7 Sol. Set, 3 Describe in words how to graph each of following inequalities, make sure to tell which endpoints to use. Also write the solution set in interval notation. 4 Parentheses at and bracket at 4, shade between the two numbers Sol. Set,4 2 7 Bracket at 2 and parentheses at 7, shade between the two numbers Sol. Set 2,7 Parentheses at, parentheses at, shade between the two numbers Sol. Set, 2 9 Bracket at 2 and bracket at 9, shade between the two numbers Sol. Set 2,9 Describe in words how to graph each of following inequalities, make sure to tell which endpoints to use. Also write the solution set in interval notation