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MAT081 :: Lecture Note :: Week 05
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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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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 * 112*2* 3 cancel like factors: 2*2* 11 3 final answer:  11Another 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 11Another 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 41An observation about cancelling like terms...
a(c) a c a  =  *  =  c dividedby c is 1 b(c) b c b
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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 < 10A decimal fraction is a fraction where the denominator is 10 raised to a positive integer exponent.
10^{4} equals 10,000 10^{3} equals 1,000 10^{2} equals 100 10^{1} equals 10 10^{0} 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 0.000000001To 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 = 0.55Trailing 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 fiftynine hundreths 0.459 is four hundred fiftynine thousandths 5000.29 is five thousand and twentynine hundreths 233.056 is two hundred thirtythree and fiftysix thousandthsA NanoMoment
Note: nano is a prefix meaning onebillionth (or
10^{9}
or1/1,000,000,000
or0.000000001
).1 nano...  = 0.000000001 1,000,000,000In everydayworld, 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)."
 GDT::Blog:: Nanotech SmallBlog
 GDT::BAB:: A Mathematical Singularity
 GDT::Computing::Bit:: NanosecondGrace Hopper to ASU
 GDT::BAB:: Popular PhraseNano Giga
External Hyperlink(s)
 Math.com:: Decimals
 ScienceMadeSimple.net:: Fractions to Decimal Conversions Chart
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If necessary, rewrite 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  260.10173Subtraction 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
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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
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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. 4.5Recall 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 rightmost digit. 15.75
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