Lecture 4: Using conditioning
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Problem of the day
When 10 pounds of cucumbers were bought at a market,
they contained 99\% water. After several days they dried
out to 98\% water. How many pounds of water do they contain
now?
Assignment
By today you should have read through the entire first chapter
and worked problems up to page 59. The assignment for Wednesday
January 29 the problems on page 70 shown on the assignment page
Brief review of the lecture
Continuation of absolute value problems. Consider the following
problem. How many real numbers x satisfy
|||x-10|-12|-15|=12? Again we saw conditioning at work.
The equation has 6 solutions, -29, -5, 1, 19, 25, and 49.
We were also concerned with the problem of
representing real numbers. There are many different ways to do this,
but one of the best is the decimal representation. Not only is it
available for EVERY real number, the form of the representation
says a lot about the number. For our purposes, we talk about three
types of decimal representations:
a. those which end in zeros from some point on. We don't write the zeros,
but instead just the nonzero digits of the representation. 4.205 is an example.
The convention is that no zeros are written after which all the digits
are zeros. So 4.205 is the same as 4.2050000... with zeros extending forever.
b. those which repeat in blocks from some point on. For example,
2.99999.... which we usually write as 2.9 with a bar over the 9.
c. those which don't repeat. For example, 1.01001000100001....
>BR>
The first two are the types we get representing RATIONAL numbers, and
the third is the type IRRATIONAL numbers have.
We are concerned today with the problem of converting
those of type b. (all such numbers are rational, remember) to
the form a/b where a and b are integers.
The solution is to give the representation a name, say x. Multiply
it by ten raised to the power equal to the size of the repeating block,
and subtract the former from the latter. The `tails' of the two numbers
disappear and the difference is a decimal of the type a. above. Some easy
arithmetic will finish the job.
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