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MAT081 :: Lecture Note :: Week 05
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Overview
Assignment(s)

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
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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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Reducing Factions

Fractions are often expressed in their lowest form (i.e. simplified or reduced).

The following technique can be used to reduce fractions.

   1) Do a prime factorization of the numerator.
   2) Do a prime factorization of the denominator.
   3) Cancel (eliminate) "like" factors.

Example.

               12
   reduce:   ------
               44

                          2 * 2 * 3
   prime factorizations: ------------
                          2 * 2 * 11


                          2 * 2 * 3
   cancel like factors:  ------------
                          2 * 2 * 11

   
                  3
   final answer: ----
                  11

Another reduction techique is to use the greatest common factor of the numerator and denominator.

   x                 x ÷ gcf(x,y)
   -   reduces to    ------------
   y                 y ÷ gcf(x,y)

Example.

               12
   reduce:   ------
               44

   gcf(12, 44) equals 4

   12 ÷ 4       3
   ------  =  ----
   44 ÷ 4      11

Another fraction reducing strategy is as follows.

   1) Determine if the numerator or the denominator is the
      simplist prime factorization and do a prime factorization.
   2) Cancel the prime factors that are also factors of the
      fraction part that was not prime factorized.

Example.

              25
   reduce:  ------ 
              205

   do a prime factorization on the numerator

     25     5 * 5 
   ------ = -----
     205     205 

   5 evenly divides 205 41 times
   5 does not evenly divide 41

     25       5
   ------ = -----
     205      41

An observation about cancelling like terms...

	a(c)  a   c   a
	--- = - * - = -    c divided-by c is 1
	b(c)  b   c   b

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Introduction to Decimal Numbers

A decimal number is written with the whole number followed by a dot (decimal point) followed by the fractional part. A decimal number falls between two integers that differ by one in value.

   2 < 2.43 < 3
   8 < 8.0001 < 9
   99 < 99.9999 < 10

A decimal fraction is a fraction where the denominator is 10 raised to a positive integer exponent.

   104 equals 10,000
   103 equals  1,000
   102 equals    100
   101 equals     10
   100 equals      1
   10-1 equals     1/10      equals   0.1
   10-2 equals     1/100     equals   0.01
   10-3 equals     1/1000    equals   0.001
   10-4 equals     1/10,000  equals   0.0001
   ...
   10-9 equals     0 1/1,000,000,000  equals   0.000000001

To the left of the decimal point are the ones, tens, hundreds, thousands, and so on. On the fractional side of decimal number are the tenths, hundreths, thousandths, and so on. There are no oneths.

Decimal numbers that lie between zero and one (and zero and a negative one) are often prefixed with a zero.

 
   .1 = 0.1       .375 = 0.375      -.55 = -0.55

Trailing zeroes after the decimal point are not necessary; however, in science, engineering, statistics and other fields, trailing zeros are retained to show a level of confidence in the accuracy of the number.

When writing a decimal number in English, use the word and to represent the decimal point.

      7.59  is  seven and fifty-nine hundreths
     0.459  is  four hundred fifty-nine thousandths
   5000.29  is  five thousand and twenty-nine hundreths
   233.056  is  two hundred thirty-three and fifty-six thousandths
A Nano-Moment

Note: nano is a prefix meaning one-billionth (or 10-9 or 1/1,000,000,000 or 0.000000001).

                   1
   nano...   -------------  =  0.000000001
             1,000,000,000

In everyday-world, GDT replaces nano with "very, very, very small." For example, a nanofoo is a very, very, very small foo. It doesn't matter what foo is; whatever it is, it is very, very, very small.

Let's get smaller (i.e. closer to zero)...

A nanosecond is a very, very, very short second (i.e. a billionth of a second).

   From Fall 2004:
   "Optical 'rulers' are lasers that emit pulses of light 
    lasting just 10 femtoseconds (10 quadrillionths of a 
    second, or 10 millionths of a billionth of a second)."
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Adding and Subtracting Decimals

If necessary, re-write the problem vertically and lineup the decimal points. It is okay to pad decimal numbers with zeros.

   example:
      4.451 + 13.3

         4.451
      + 13.3
      --------
        17.751

      or

         4.451
      + 13.300
      --------
        17.751

   example:
      5.00073 + 255.101

          5.00073
      + 255.10100
      -----------
        260.10173

Subtraction works the same way; i.e., lineup the decimal points prior to doing the subtraction.

   5.55 - 4.02

     5.55
   - 4.02
   ------
     1.53

[top]


Multiplying Decimals

Execute the multiply as if dealing with whole numbers (i.e. it is not necessary to lineup the decimal points).

Insert the decimal point in the product by starting at the right and moving a number of places equal to the sum of the decimal places in both numbers multiplied.

   5.50 * 2.1

       5.50
   *    2.1
   --------
        550
   +  1100
   --------
      11550

   5.50 has 2 digits to the right of the decimal point.
   2.1 has 1 digit to the right of the decimal point.

   2+1 is 3; therefore, the decimal point goes left of 
   the 3rd digit from the right

   11.550  or 11.55

[top]


Dividing Decimals

If the divisor has a decimal point, then make it a whole number by moving the decimal point to the right.

Move the decimal point in the dividend to the right by the number of moves made in the divisor.

Execute a whole number division ignoring any decimal point in the dividend.

Insert a decimal point in the quotient directly above the decimal point in the dividend.


       _________
   3.5 ) 15.75

           45
      _________
   35 )  157.5
        -140
         ---
          175
         -175
          ---
            0

   Insert the decimal point into the quotient.

   4.5

Recall that answers to division problems can be checked by multiplying the quotient (result) by the divisor. The product should equal the dividend.

     4.5
   * 3.5
   -----
     225
   +135
   -----
    1575

   Both 4.5 and 3.5 have 1 digit to the right of the decimal.
   Insert decimal point into the product 2 digits left of the
   right-most digit.

   15.75

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