a. Write a real number with ten (10) digits to the right of the
decimal point that is a rational number between 0 and 2 with a 2 digit
repeating pattern.
b. Write a real number with ten digits to the right of the decimal
point that is an irrational number between 2 and 4.
c. If x-10=35, which property of equality for real numbers justifies that x=45.
Show the use of this property.

Hi,

Thank you for your question. I've left a few spaces between each
explanation and the answer, so that you can attempt the answers after
reading the pointers.

a. A rational number can be written as x/y where both x and y are
natural numbers. In certain cirumstances there will be an infinite
number of decimal points. You might want to try looking for a rational
number where these occur using your calculator.




















For example: 15/11 = 1.3636363636... 

The ... indicate that the pattern is repeated indefinitely. There is a
2 digit repeating pattern.


b. An irrational number is a number that cannot be written as x/y with
both x and y natural. sqrt(x) indicates the "square root of x".




















sqrt(5) = 2.2360679774...

... indicates that there is an infinite sequence of digits after the decimal point.

Of course 2<sqrt(5)<4

Here is a proof that shows that sqrt(5) is irrational:

Proof that for no rational number r = p/q, (p/q)2 = 5
http://www.cut-the-knot.org/do_you_know/numbers.shtml#rational


c. The property of equality for real numbers that is mentionned is as follows:
Let x and y be 2 equal real numbers, such that we have x = y. Let c be
a real number, we have the following equality:
x + c = y + c




















In your case the property is used as follows:
x - 10 = 35
we use c = 10
therefore:
x - 10 + 10 = 35 + 10
which gives us
x = 35 + 10
x = 45


I hope this answers your question, if you need any more clarifications
please do not hesitate to ask.

Thanks.
endo

Answer to a) and b)

Every real number with (only) ten digits to the right of the decimal
point is a rational number.  It can always be represented by a
fraction of whole numbers.  The numberator is the product of the real
number and 10 billion (10^10).  The denominator is 10 billion.  Both
are whole numbers, thus the quotient is rational.

Thus, a) has an infinite number of answers; it is up to you to choose
one, and b) has no answers.

Hi,

I'm pretty sure that the indication that specifies 10 digits after the
decimal point is only for representation purposes. It does not imply
that the numbers actually only contain 10 digits after the decimal
point.

Thanks.
endo

Assuming that you want to get the representation of x.yzyzyzyzyz... as
a rational number of the form p/q, you can use this procedure ->

a = x.yzyzyzyzyz...	(1)
100*a = xyz.yzyzyzyz... (2)
---------------------------
subtract (1) from (2) ->
  99*a = xyz.yzyzyz... - x.yzyzyz...
=> a = (xyz.00 - x.00)/99

a is the rational number (p/q) representation you want for x.yzyzyz
... You can cancel the common factors from the numerator and
denominator to reduce it to the form where p and q are co-prime, i.e.,
they have no common factors.

e.g.

    a = 3.545454...
100*a = 354.5454...
--------------------
=> 99*a = 351
=> a = 351/99 = 39/11