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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