MAT082 :: Lecture Note :: Week 05
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Overview

• BAB::Fractions on Route 66 in Illinois
• BAB::Fractions at the Smithsonion Institute
• BAB::Fractions on the Carefree Highway
• Introduction to decimal numbers.

Assignments

• Order of Operations Practice
• To Be Completed

Quote of the Week {Furman.edu:: Mathematical Quotations Server}
I have learned, that if one advances confidently in the direction
of his dreams, and endeavors to live the life he has imagined, he
will meet with a success unexpected in common hours.
-- Henry David Thoreau (01817-01862) {American author; more...} [life/dream] [log]
BARS of the Week {NASA.gov:: Astronomy Picture of the Day}
Casa Grande AZ | I-55 in MO | Hwy-79 in MO [log]

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)." GDT::Blog:: Nanotech SmallBlog GDT::BAB:: A Mathematical Singularity GDT::Computing::Bit:: Nanosecond--Grace Hopper to ASU GDT::BAB:: Popular Phrase--Nano Giga

External Hyperlink(s)

• Math.com:: Decimals
• ScienceMadeSimple.net:: Fractions to Decimal Conversions Chart

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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]

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]

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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