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MAT081 :: Lecture Note :: Week 04
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Math.com says, "Believe it or not, the properties of numbers were not invented by evil mathematicians to torture math students!"
Commutative property of addition
a + b = b + aCommutative property of multiplication
a times b = b times a a x b = b x a a * b = b * a a(b) = b(a) ... ab = ba a ⋅ b = b ⋅ aAssociative property of addition
a + (b + c) = (a + b) + cAssociative property of multiplication
(ab)c = a(bc)Distributive property
a(b + c) = a(b) + a(c)What about subtraction and division? Subtraction is neither commutative nor associative; and, division is neither commutative nor associative.
Assume a != b (!= reads "not equal") a  b != b  a (what if b = a?) 5  3 equals 2 3  5 equals 2 a / b != b / a 0 / 7 equals 0 7 / 0 is undefined 15 / 3 equals 5 3 / 15 equals 0 with remainder 3/15thsPurpleMath.com:: Basic Number Properties: Associative, Commutative, and Distributive
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A factor is a value that when multiplied with another value results in a product.
given: a * b a and b are factorsFactors are often used when reducing (or simplifying) fractions.
A positive integer greater than one is either a composite or prime number.
A prime number is has only two factors: 1 and itself (i.e. 1 and the number itself are the only whole numbers to evenly divide into the number). A composite number is a number that is not prime (i.e. it has more than two factors).
3 is a prime number because it only has two factors: 1 and 3. 9 is a composite number because it has the factors 1, 3 and 9.
1 * 3 = 3 1 * 9 = 9 3 * 3 = 9The first fifty prime numbers are:
2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97 101 103 107 109 113 127 131 137 139 149 151 157 163 167 173 179 181 191 193 197 199 211 223 227 229 + 1 is not prime (nor composite) + 2 is the only even prime numberUTM.edu:: First 10,000 Prime Numbers [opens new window]
A composite number can be written as a product of prime factors. This is called prime factorization.
The Fundamental Theorem of Arithmetic states that every positive integer (except the number 1) can be represented in "exactly one way apart from rearrangement as a product of one or more primes."
Prime factors can be found as follows.
Divide the number a by the smallest possible prime number and work up the list of prime numbers until the result is itself a prime number.PurpleMath.com:: Factoring Numbers [opens new window]
MathForum.org:DrMath:: Divisibility Rules
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Every number has an infinite number of multiples.
The multiples for a number 'n' can be found by multiplying the 'n' by 1, 2, 3 and so on.
multiples of 2 are 2, 4, 6, 8, 10, 12, ..., 200, ..., 1000, ... 2 * 1 equals 2 2 * 2 equals 4 2 * 3 equals 6 2 * 4 equals 8 2 * 5 equals 10 2 * 6 equals 12 2 * 7 equals 14 ... 2 * 100 equals 200 ... 2 * 500 equals 1000 ...Definition: Common multiples are multiples that are common to a set (collection) of two or more numbers.
multiples of 3: 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, ... multiples of 4: 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, ... common multiples of 3 and 4 are 12, 24, ... [exercise] No or Yes: 48 is a common multiple of 3 and 4? [answer] Yes because 48 is evenly divisable by both 3 and 4.When adding, subtracting or comparing fractions, they must a common denominator. A common denominator for two fractions can be quickly obtained by multiplying the two denominators together. Sometimes the least (or lowest) common denominator is used by finding the LCM (Least Common Multiple) for the denominators. Note: The LCM is also be called the Lowest Common Multiple.
Definition: The LCM of a set of numbers is the smallest number that evenly divides each number in the set.
common multiples of 3 and 4 are 12, 24, 36... the least common multiple is 12 because 12 < 24 < 36 < ...The LCM for a set of numbers can be found using the following technique (algorithm).
1) Do a prime factorization of each number. 2) Multiply together factors that are gathered as follows: + one of each factors that each number have in common + all factors that are unique to each numberLCM Examples
Find the LCM for 4 and 6. lcm(4, 6) = _____ prime factorization of 4: 2 2 prime factorization of 6: 2 3 lcm factors: 2 2 3 lcm(4,6) = 2 * 2 * 3 = 12  Find the LCM for 27 and 45. lcm(27, 45) = _____ prime factorization of 27: 3 3 3 prime factorization of 45: 3 3 5 lcm factors: 3 3 3 5 lcm(27, 45) = 3 * 3 * 3 * 5 = 135  Find the LCM for 98 and 312. lcm(98,312) = _____ prime factorization of 98: 2 7 7 prime factorization of 312: 2 2 2 3 13 lcm factors: 2 2 2 3 7 7 13 lcm(98, 312) = 2 * 2 * 2 * 3 * 7 * 7 * 13 = 15,288  Find the LCM for 8, 10, 14 and 20. lcm(8, 10, 14, 20) = _____ prime factorization of 8: 2 2 2 prime factorization of 10: 2 5 prime factorization of 14: 2 7 prime factorization of 20: 2 2 5 lcm factors: 2 2 2 5 7 lcm(8, 10, 14, 20) = 2 * 2 * 2 * 5 * 7 = 280If prime factorizations have been calculated on a set of numbers, then their Greatest Common Factor (GCF) can be found. Note: When dealing with fractions, the GCF can be called the GCD (Greatest Common Divisor).
Definition: The GCF of a set of numbers is the greatest (largest) number that evenly divides into each number in the set.
The GCF for a set of numbers can be found using the following technique (algorithm).
1) Do a prime factorization on each number. 2) Multiply together all the factors that each number have in common.GCF Examples
Find the GCF for 4 and 6. gcf(4, 6) = ______ prime factorization of 4: 2 2 prime factorization of 6: 2 3 gcf factors: 2 gcf(4, 6) = 2  Find the GCF for 27 and 45. gcf(27, 45) = _____ prime factorization of 45: 3 3 5 prime factorization of 27: 3 3 3 gcf factors: 3 3 gcf(27, 45) = 3 * 3 = 9  Find the GCF for 98 and 312. gcf(98, 312) = _____ prime factorization of 98: 2 7 7 prime factorization of 312: 2 2 2 3 13 gcf(98, 312) = 2  Find the GCF for 8, 10, 14 and 20. gcf(8, 10, 14, 20) = _____ prime factorization of 8: 2 2 2 prime factorization of 10: 2 5 prime factorization of 14: 2 7 prime factorization of 20: 2 2 5 gcf factors: 2 gcf(8, 10, 14, 20) = 2  Find the GCF for 330 and 770. gcf(330, 770) = _____ prime factorization of 330: 2 3 5 11 prime factorization of 770: 2 5 7 11 gcf factors: 2 5 11 gcf(330, 770) = 2 * 5 * 11 = 110Many calculators have builtin functions to calculate LCMs and GGFs. LCM and GCF tools can also be found on the web, but you have to make sure they work correctly. {Venturaes.com:: LCM and GCF Calculator [opens new window]}
PurpleMath.com:: LCM and GCF [opens new window]
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Instructor to class: How do we define fraction?
Instructor to class: Have you ever defined fraction as follows? A fraction is an unsolved division problem (i.e. it is a quotient of two numbers).
[humor]
Five out of every four people have difficulty understanding fractions.Fractions are used to represent partofawhole of something. Examples: Zelmo ate 1/2 of the whole pie. Edith spent 3/4 of a full hour sending an email message to Zelmo. Truman completed only 2/9 of the nine part exercise.
x x ÷ y is also the fraction  (also written x/y) y 'x' (dividend) is the numerator 'y' (divisor) is the denominatorSince the divisor divides into the dividend, the denominator divides into the numerator.
[terminology exercise: No or Yes: Dividing the denominator into the numerator results in a terminator.]A fraction can never have a denominator of zero because division by zero is undefined.
Ignoring negative numbers... A proper fraction is a fraction having a numerator that is less than the denominator. An improper fraction is a fraction having a numerator greater than or equal to the denominator.
2/3  proper fraction (2 < 3) 7/4  improper fraction (7 > 4) 5/5  improper fraction; equals 1 0/8  proper fraction (0 < 8) 9/0  undefined fraction (denominator is 0)Although improper fractions are called improper, there is nothing improper about them (i.e. they are not "bad" fractions).
A unit fraction is a fraction that has one as its numerator.
Every whole number 'n' can be written as a fraction by using 'n' in the numerator and one in the denominator.
A mixed number is a whole number plus a fraction.
1/8  unit fraction 5/1  the value 5 7 1/4  mixed number; 7 is a whole number; 1/4 is a fraction 7 1/4 is 7 + 1/4; most of the time the fractional part of mixed number is a proper fractionA fraction consisting of two integers is called a rational number. [Observe how the word rational contains the word ratio. Fractions are ratios.]
A fraction is negative if either the numerator is negative or the denominator is negative. If both the numerator and denominator are negative, then the fraction is positive.
GDT::BAB:: About the Fraction Five DividedBy Eight
Multiplication
Multiply the numerators to get the numerator of the product. Multiply the denominators to get the denominator of the product.
2 4 8 2 * 4 is 8  *  =  3 5 15 3 * 5 is 15Division
To divide one fraction by another one, multiply the top fraction by the reciprocal of the bottom fraction.
If dividing mixed numbers, then convert the mixed numbers to improper fractions and then perform the division.
Reciprocal is the multiplicative inverse of a number. For a fraction, it's obtained by "flipping the fraction."
2 1  ÷  5 7 rewritten 2 2 7 14 4   *  =  = 2  5 5 1 5 5  1  7Equivalent Fractions
Given a fraction, both the numerator and denominator can be multiplied or divided by the same number and the result is an equivalent fraction.
3 5 15  *  =  4 5 20Addition/Subtraction
Fractions can be added or subtracted only if they are like fractions (i.e. have common denominators). Typically we want the common denominator to be the smallest common denominator, but to help get started we will obtain a common denominator by using the product of the two denominators. Once a common denominator has been determined and the numerators adjusted (i.e. equivalent fractions calculated), then the numerators are added or subtracted.
Example
1 1  +  5 4 multiply the two denominators to get a common denominator 5 * 4 equals 20 1(n) 1(n)  +  20 20 adjust the numerators finding equivalent fractions given the common denominator 1(4) 1(5) 9  +  =  20 20 20 A + B _ _ _ _ _ _ _ _ _ _  _ _ _ _ _ _ _ _ _ _ 0/20 A A A A B B B B B _  _ _ _ _ _ _ _ _ _ _ 9/20Some Older Related BABs
 Cheap Gas Prices Along Route 66 In Illinois [07 February 2005]
 Learning About Fractions In The Early Days [01 February 2006]
 Mile Markers on Camelback Mountain [19 July 2005]
 BARS:: Incrementing Fractions on Northbound Loop 101 in Scottsdale, Arizona [24 May 2005]
 BARS:: Pima Road and Cave Creek Road Junction in Carefree, Arizona [05 January 2006]
 BARS:: Southbound Loop 101 in Tempe, Arizona [10 February 2006]
 BARS:: Superstition Springs Exit on US Hwy60 in Mesa, Arizona [12 February 2007]
 BARS:: Fractions on Yavapai County Road 10 Near Prescott, Arizona [24 September 2006]
 BARS:: Heading East on I40 To Winslow, Arizona [18 March 2006]
External Hyperlinks
 Math.com:: Homework Help Hot Subject: Fractions
 Wikipedia.org:: Fraction (mathematics)
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Comparing Fractions
In order to determine the relationship between fractions, it is necessary for the fractions to have a common denominator (similar to adding and subtracting fractions). If two fractions have a common denominator, then you compare the values of the numerators to determine the relationship between the fractions. [The product of the two denominators gives a common denominator.]
Which is smaller, 1/4 or 1/3? 1) determine a common denominator 2) determine equivalent fractions using the common denominator 3) compare the numerators 1/4 and 1/3 have a common denominator of 12 (4 x 3) 1/4 is equivalent to 3/12 1/3 is equivalent to 4/12 4 > 3; therefore, 1/3 > 1/4Equivalent Fractions
Two fractions are equivalent if their crossproducts are equal. In other words, two fractions A and B are equivalent if the numerator of A times the denominator of B equals the denominator of A times the numerator of B.
a c  is equivalent to  b d if a * d equals b * cExercise.
38 19 Is  equivalent to  ? 96 48 38 * 48 equals 1824 96 * 19 equals 1824 1824 equals 1824; therefore, the fractions are equivalent and the answer is Yes.FunBrain.com:: Fresh Baked Fractions
Improper Fraction to Mixed Number
Many times improper fractions are expressed as mixed numbers. For example, the improper fraction
3/2
is equal to the mixed number1 1/2
.Improper fraction to mixed number is calculated as follows.
 divide the denominator into the numerator and record the whole number portion of the quotient
 the remainder is the numerator of the answer's fractional part and the divisor is its denominator
Example.
15   is 15 ÷ 7 which is 7) 15 7 7 goes into 15 twice with a remainder of one; therefore, the mixed number result is 2 1/7.Exercise.
Write 55/12 as a mixed number. 55   = 55 ÷ 12 = 12) 55 12Mixed Number to Fraction
A mixed number can be converted into fraction using the following technique.
Given the mixed number x a  y 1) multiply the denominator of the fractional part (y) with the whole number (a) 2) take the product obtained from step 1) and add the numerator of the fractional part (x) 3) the sum obtained from step 2) becomes the numerator of the result with the denominator of the fractional part being the denominator of the result a ⋅ y + x  yExample.
2 10 * 7 + 2 70 + 2 72 10  =  =  =  7 7 7 7 [pemdas]Most of the time mixed numbers will result in improper fractions.
Doing Arithmetic with Mixed Numbers
When performing arithmetic with mixed numbers it is best to convert the mixed numbers to fractions and then do arithmetic with the fractions. In most cases you will want to convert your answers back to mixed numbers.
Example.
3 1 5  * 3  5 3 28 10 280 10 2  *  =  = 18  = 18  5 3 15 15 3
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A math book said the fraction
9 / 9
equals1
because1 ⋅ 9
equals9
. What does this mean?When you do a division problem, you can check your answer by multiplying the quotient with the divisor and adding any remainder. The result, if correct, should equal the dividend.
Example.
10 / 4 = 2r2 10 is the dividend and 4 is the divisor quotient is 2 with remainder 2 2 x 4 + 2 = 10 divisor times quotient plus reminder gives us the dividend
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