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.
1/4 = 0.25
1/3 = 0.3333...
1/10 = 0.1
1/7 = 0.142857 142857 ....
Even more surprising, a
repeating decimal can be expressed as a terminating
decimal (ignoring the repeating 0's)!
0.3999999... = 0.4
and here's why,
0.300... + 0.099999...
= 0.300... + x
multiply by 10
= 3.00... + 10x
Pause here for a moment, the 3
on each side of the equation has been removed.
multiply by 10
= 100x - 0.999...
that 0.999... = 10x and substitute
= 100x - 10x
divide by 90
going back to the equation above (*)
0.300... + x
we have 0.3999... =
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".
= 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
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.
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 =
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
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.
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 +
0.0016 + 23 + 4001.1
notice in this example leading
zeroes are not shown since they are implied.
adding repeated decimals.
1) first convert to fractions as above
2) combine fractions
3) convert back to decimal
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
must first be converted to fractions, then multiply
fractions, then convert back to
Example: with repeated
Explanation for 2nd step
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.
decimals, convert to fractions, invert then
multiply. Finally, convert back to decimal.