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MAT081 :: Lecture Note :: Week 04
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Overview
Assignment(s)

Some Arithmetic Properties

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 + a

Commutative 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 ⋅ a

Associative property of addition

   a + (b + c) = (a + b) + c

Associative 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 3/15ths

PurpleMath.com:: Basic Number Properties: Associative, Commutative, and Distributive

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Factors

A factor is a value that when multiplied with another value results in a product.

   given:  a * b

   a  and  b  are  factors

Factors 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 = 9

The 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 number

UTM.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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Multiples, LCM, GCF

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 number
LCM 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 = 280

If 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 = 110

Many calculators have built-in 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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Introduction to Fractions

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 part-of-a-whole 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 denominator

Since 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 fraction

A 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 Divided-By 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 15
Division

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
    -
    7
Equivalent 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   20
Addition/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 _ | _ _ _ _ _ _ _ _ _ _     9/20
Some Older Related BABs
External Hyperlinks

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More About Fractions

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 > 1/4
Equivalent Fractions

Two fractions are equivalent if their cross-products 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 * c

Exercise.

       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 number 1 1/2.

Improper fraction to mixed number is calculated as follows.

  1. divide the denominator into the numerator and record the whole number portion of the quotient
  2. 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
    12
Mixed 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
      ---------
          y

Example.

       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 Division Tidbit

A math book said the fraction 9 / 9 equals 1 because 1 ⋅ 9 equals 9. 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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