Exploring the Reals

     The Real numbers include the integers, the rational, and the irrational numbers and are represented as points on the number line. Every real number has its own position on this number line, and this one to one correspondence with no gaps makes these numbers a continuum.

   To understand the Real numbers, we  need to understand the concept of repeating and non-repeating decimals.   As it turns out, every rational number can be represented in one of these ways.  

For example:     1/4  = 0.25           non-repeating    (terminating)

                       1/3  = 0.3333...     repeating          (non-terminating)

                       1/10 = 0.1            non-repeating    (terminating)

                       1/7  =  0.142857 142857 ....   repeating  (non-terminating)

Even more surprising, a repeating decimal can be expressed as a terminating decimal (ignoring the repeating 0's)!

Example:          0.3999999...  =  0.4

and here's why,

 (*)                  0.300... + 0.099999... =  0.300...  + x   (i.e., let x = 0.0999...)

     multiply by 10

                       3.000...  +0.9999...       = 3.00...  + 10x

     subtract 3                   0.9999...        =  10x

     multiply by 10             9.999...          =  100x

     subtract 0.999             9.0000...        =  100x - 0.999...

     note that 0.999... = 10x  and substitute     here  ^

                                      9.000...          =  100x  - 10x

                                      9.000...           = 90x

      divide by 90               1/10                = x

      going back to the equation above (*)

      we have   0.3999...  = 3/10 + x = 3/10 + + 1/10 = 4/10 = .4000... = 0.4

If we include the repeating 0's  (as in 0.40000...) then every rational number can be represented as a non-terminating decimal.  If the repeating sequence begins right after the decimal point, then the decimal number is "periodic".  If the decimal number contains digits after the decimal not part of the repeating sequence then the decimal number is called "mixed".

Example:

     11/26 = 0.4230769230769...   230769  is the repeating sequence,  a bar is usually drawn above these digits to indicate the sequence. like so

when the repeating digits are understood, then an ellipsis is sufficient.

Conversions between decimals and fractions.

1.  To convert a fraction to a decimal number carry out the division until no remainders remain or a repeating sequence occurs.

2.  To convert a decimal number to a fraction, place the digits of the period in the numerator and in the denominator place one less than the power of 10 of the digits.

Examples:

                   0.33333....   repeats in the first position after the decimal so

                                     the denominator is  10 -1 = 9  and we have  3/9 = 1/3

                   0.2727...      27 repeating so 100 -1 = 99 and 27/99 = 9/11

                   2.35858...    2.3  + 0.0585858 = 2.3 + 1/10 (0.585858...)

                                    = 2.3 + 1/10( 58/99 ) = 2.3 + 58/990

                                    =  2 + 3/10 + 58/990 = 2 + (3*99 + 58)/990

                                    = 2 + 355/990 = 2 + 71/198 = 2  71/198  = 467/198

Notice in this last example 0.058585... required 1/10 to be factored out to move the first 5 next to the decimal point.

So, in review,  the Real Numbers are all non-terminating, repeating decimal numbers that represent all Rational Numbers and can approximate all Irrational Numbers. 

All Real Numbers have a unique position on the number line (where irrational numbers can be handled with nested intervals, to be covered at a later time.)

Operations with real numbers. 

Although a calculator is normally used to perform operations with real numbers, it is well worth the effort to make sure your student understands how real numbers are comprised of repeating and non repeating decimals.  So performing operations by hand will reinforce this concept using a calculator to perform multiplication or  division along the way is acceptable.

Addition/Subtraction

     To add or subtract real numbers, line up the decimal points, place implied zeroes to get an equal number of decimals, then carry out the operation as you would with integers.

Example:  13.254 + 2.1

Example:    0.0016 + 23 + 4001.1    

                          notice in this example leading zeroes are not shown since they are implied.

Example:    adding repeated decimals. 

     1) first convert to fractions as above

     2) combine fractions

     3) convert back to decimal

Multiplication

    Ignore the decimal points, multiply the numbers, then place the decimal point counting from the right of the result to the left the total number of decimals in the  numbers multiplied.

 

Example: 

                      

 

    Repeated decimals must first be converted to fractions, then multiply the

fractions, then convert back to decimal.

Example:  with repeated decimals

	Explanation for 2nd step

                        

 Division:

     Move the decimal in the dividend to the right the number of decimals in the divisor.  Carry out the division as usual.

Example:   In this example the second step shows moving the decimal to the right two places, the number of decimals in the divisor. Also, the division stops at the second decimal since the least number of significant digits is 1 (in the dividend) and the second decimal would be used to round up or round down the first decimal.  In this case the answer is simply 109.2.

                

                                                                                                   

Example:  repeating decimals, convert to fractions, invert then multiply.  Finally, convert back to decimal.