1.3 FRACTIONS

1. 1.3 FRACTIONS REVIEW Variables-letters that represent numbers examples: x, y, z, a, b, c Multiplication-can be shown many ways examples: ab a b a(b) (a)b Factors-numbers (or variables) multiplied together to get a product

2. Fractions a/b Where a is the numerator (top) and b is the denominator (bottom) The fraction bar means to divide A fraction is said to be reduced or simplified or in lowest terms when the numerator and denominator have no common factors except one. To simplify a fraction, find the GCF and divide both numerator and denominator by that number GCF is the biggest number that goes into the top and bottom (see appendix B for more info on GCF)

3. a nice trick to remember Note: you can only cross cancel across a multiplication sign-never do this across an add, subtract, or division sign. Below you can cross cancel the two’s and then multiply. (like reducing before you multiply)

4. Forms of fractions Proper fractions: ½ or a/b when a<b the numerator is smaller than the denominator Improper fractions: when a>b; the numerator is larger than the denominator Mixed numbers: 3 ½ or A b/c have two parts – a whole number part and a fraction part

5. Changing forms (circle trick) A nice trick for changing from a mixed number to an improper fraction is: -multiply the denominator and the whole number -add this to the numerator

6. Changing forms The method for changing from a improper fraction to a mixed number is to divide. Remember the fraction bar is a divide sign. Can you see the 3 ½ ?

7. Multiplying fractions To multiply fractions: Mutliply the numerators, multiply the denominators and reduce In other words, take top times top; bottom times bottom and reduce

8. Dividing fractions To divide fractions: By definition, division is multipying by the reciprocal. So . . . -leave the first fraction as is -flip the second fraction -multiply (take top times top; bottom times bottom) -reduce

9. Adding and subtracting fractions -find a common denominator (LCD) -rewrite each fraction with new denominator -add or subtract numerators as indicated -keep denominator the same -reduce See appendix b for more info on LCD

10. 1.4 REAL NUMBERS In this section we will be working with set notation. A set is a collection of elements listed within braces Example {a,b,c,d,e} – this set has 5 elements { } 0 -- this set has no elements. It is called the empty set or null set. {1,2,3 . . } – this set has an infinite number of elements

11. Natural Numbers {1,2,3, . . .} This set includes the positive numbers-no decimals or fractions. Also referred to as the counting numbers in some books.

12. Whole Numbers {0,1,2,3, . . .} This set includes the natural numbers and zero; still no decimals or fractions

13. Integers { . . . -2,-1,0,1,2, . . .} This set includes the positive and negative “whole” numbers; again, no decimals or fractions

14. Rational Numbers There are a lot of numbers in this set. This set includes any number that can be written as a fraction. Fractions; Repeating and Terminating decimals as well; (1/3 = 0.333333…. Or ½= 0.5) And all the whole numbers. (put a 1 under them 5 = 5/1)

15. Irrational Numbers We don’t work with these a lot. Common examples are and certain square roots. This set includes any number that can not be written as a fraction. These are non-repeating, non-terminating decimals.

16. Real Numbers This where we spend most of our time. This set includes natural numbers, whole numbers, integers, rational numbers, irrational numbers. Everything we have talked about so far. Real numbers are any number that can be represented on a number line.

17. Real Number System • Venn diagram Natural Whole Integers Rationals Reals Irrationals

18. You can use a number line to compare numbers. To the left things get smaller. To the right things get bigger. 1.5 INEQUALITIES < less than less than or = > greater than greater than or = = equal to = not equal to

19. Absolute Value By definition, absolute value is the distance away from zero on a number line. Denoted by straight lines or bars on either side of a number or an expression -3 = 3 3 = 3 0 = 0

20. 1.6 Addition of Real Numbers Addition is combining When adding two numbers with the same sign, add the absolute values of the numbers and keep the sign the same. When adding two numbers with different signs, find the difference of the absolute values of the numbers and take the sign of the number with the larger absolute value. Note: additive inverse means opposite

21. Addition examples: 3 + 2 = 5 the numbers have the same sign, add the numbers and keep the sign the same -3 + -2 the numbers have the same sign, add the numbers and keep the sign the same -3 + 2 the numbers have different signs, find the difference of the numbers and take the sign of the number with the larger absolute value 3 + -2 the numbers have different signs, find the difference of the numbers and take the sign of the number with the larger absolute value

22. 1.7 Subtraction By definition, subtraction means to add the opposite You will rewrite every subtraction problem into an addition problem. Then use the rules for addition that we went over in 1.6 a – b = a + -b 3 – 2 = 3 + -2 -3 – 2 = -3 + -2 -3 - -2 = -3 + 2 3 - -2 = 3 + 2

23. 1.8 Multiplication/Division Because multiplication and division are so closely related, the chart below works for both operations When multiplying or dividing two numbers: If the signs are the same, your answer is positive. If the signs are different, your answer is negative.

24. In other words + + = + - - = + + - = - - + = -

25. 1.9 Exponents, Parenthesis, and Order of Operations Exponents- An exponent tells the number of times the base appears as a factor. 2 is the exponent or power 4 is the base is read 4 to the 2nd power or 4 squared

26. means take 4 x 4 means take 4 x 4 x 4If no exponent appears, we assume the exponent is one – not zero.

27. Order of Operations Order of Operations exists because when there is more than one operation involved, if we do not have an agreed upon order to do things, we will not all come up with the same answer. The order of operations ensures that a problem has only one correct answer.

28. Order of Operations Parenthesis (or grouping symbols) Exponents Multiplication or Division from Left to Right Addition or Subtraction from Left to Right PEMDAS

29. In the parenthesis step, you may encounter nested parenthesis. Below you will see the same problem written two ways: once with nested parenthesis and the other with a variety of grouping symbols (including brackets, braces, and parenthesis). (( 5 x ( 2 + 3 )) + 7 ) – 2 OR {[ 5 x ( 2 + 3 )] + 7 } - 2

30. 1.10 Properties of Real Numbers In general, these properties are things that you already know to be true. This just puts a name to the idea that you already understand. You will need to memorize these (or think of tricks to remember the names of them).

31. Commutative Commutative Property says the order does not matter when you are adding or multiplying. In other words, you can add or multiply in any order, it does not affect the answer. Commutative Property of Addition A + B = B + A Commutative Property of Multiplication A x B = B x A

32. Associative The Associative Property says when you are adding or multiplying three or more numbers, grouping symbols can be placed around any two adjacent numbers without changing the result. Associate Property of Addition ( a + b ) + c = a + ( b + c ) Associative Property of Multiplication ( a x b ) x c = a x ( b x c )

33. The Commutative Property and the Associative Property do not apply to Division or Subtraction. Distributive Property of Multiplication over Addition a ( b + c ) = ab + ac Take something that is out front of the parenthesis and distribute it through everything in the parenthesis

34. Identity and Inverse Identity Property In the Identity Property, whatever you start with, you end with the same thing. The additive identity is zero a + 0 = a 0 + a = a The multiplicative identity is one a x 1 = a 1 x a = a

35. Identity and Inverse Inverse Property With the Inverse Property, you end up with the IDENTITY as the answer. The multiplicative inverse means reciprocal a x = 1 x a = 1 The additive inverse means opposite a + -a = 0 -a + a = 0