Math 20 - Lesson 41


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Transcript:
Math 20 - Lesson 41 A Portland community college mathematics telecourse.
A course in arithmetic review. Produced at Portland community college. Can you see that
just by the title alone that this lesson is going to be simply a subset of the previous
lesson? After all, fractions are implied divisions and we already had that. So the last lesson
which was a bit long will make this lesson to be that much shorter. So we simply read
this as 3 divided by 8 and write it accordingly and then we write the decimal points on the
end of the whole number. Since I'm dividing by a whole number the decimal point goes straight
up. I can append as many zeros as I wish and then simply divide. To save you time I will
do it separately. We find after three divisions it comes out evenly hence we have a terminating
decimal representative of the decimal. Again, all fractions are in disguise a division.
Another one, four divided by eleven so setting it up in its division form we get this and
doing the division we begin to get an unending quotient. However we can see the repetition,
the repetition is at 36 so we could write this as .36 with the bar over it or, whoever
is using this might make a statement like this: "Round to the nearest thousandth." So
in a case like that we go out to the nearest thousandth look only at the digit next to
it and that's 5 or more so we add one to the thousandth position which is four. Here we
have a rounded approximation of 4/11 accurate to the nearest thousandth. Some people will
show it with a symbol like this and others with a symbol like that. Both of these are
symbols which mean approximately equal to. Again, a fraction is really a division in
disuse. Another one- two and five sixteenths 2 5/16, that’s one way of saying it. Another
way of saying it is two and five divided by 16. (2. 5÷ 16) Decimal point is on the end,
shoot it straight up, add as many zeros as you think are in necessary and begin to divide.
In this case we get a terminating zero. So we can say that 2 5/16 is 2.3125 and in the
case, exactly. Here is an interesting fact regarding repeating or terminating decimals.
The fact is this- a fraction whose denominator has only 2's and or 5's as prime factors (that
is dividers) will result always in a terminating decimals. All others will be unending, repeating
decimals. Interesting fact. Let's look at some examples. If you divide 7 by 32 (7 ÷ 32)
it will come out to be .21875. It goes out quite a ways but at this point it fits; it
terminates. Notice that its dominator prime factors to 25 which means that two is the
only prime number that divides 32. The same thing with eleven five hundredths 11/500.
If you were to divide that out it would come out to .022 and it quits. If you were to prime
factor it you would get two to the second power times five to the third power (22 · 53
) hence the only prime numbers that will divide 500 is 2 and 5 and that's what our special
note said. If 2 and 5 are the only divisors (prime number wise) of the dominator it will
always terminate. Sometimes it will terminate rather soon, sometimes it will go quite a
ways before it terminates but it will, always, terminate. Here's a rather cute shortcut for
some fractions. Let's say we wished to change this one into a decimal. But first note if
I were to multiply top and bottom by 5- the top of course would be 35 and the bottom becomes
100; a power of 10. Then in that case we recall back a couple of lessons, we can simply make
the top smaller by two decimal places or .35 and sure enough if you divide 7 by 20 you
get .35, so sometimes you can sort of watch the denominators and ask if there is any number
you can multiply it by to make it a power of 10 because if I can, all I need to do is
move the decimal point instead of dividing and that is quite easy. That kind of fraction
occurs quite frequently in business fractions. Can you see that here? If I were to multiply
500 by 2 it becomes 1000 and 2 times the top is easy to do mentally. 10 carry the one,
three, six. But if I am multiplying by 1000 that is the same thing as making the top smaller
by one, two, three places which is .63 because in fact I don't need that end zero now. So
in your work, in other classes, particularly bookkeeping and some of those, watch for that
kind of a denominator and frequently you will be in a position to do this with a calculator.
Here they want it rounded to the nearest ten thousandth. So that's six and here's six and
thirty-seven divided by 89. So on your calculator 37 ÷ 89 = and you get of course a very long
decimal answer. But according to our instructions we don't need all that, we need it rounded
to the nearest ten thousandth so let's see: tenth, hundredth, thousandth, ten thousandths,
look only at the one to its right. It's four or less so I can drop it so there I have converted
that mixed number to an approximate decimal, accurate to the nearest ten thousandth. Let's
play with a couple of problems where this kind of skill is really very necessary. Consider
this one: A shop micrometer reads to the nearest thousandth of an inch. A bolt is supposed
to be 27/32 of an inch in diameter. What would that be on the micrometer? Well I don't have
a micrometer here but perhaps if you have seen one you can get the idea. Here's a bolt
and a micrometer is basically a tool like this C-clamp which you would tighten up on
the bolt and as you were doing that you would have a scale here that reads these turn around
in terms of the distance that it is going down. Because of that you can get this to
read very very accurate to the nearest thousandth of an inch rather routinely and in some specialized
cases to the nearest ten thousandth of an inch. The point is many standardized bolts
come marked with a diameter of 32/64 but most standard micrometers will be calibrated to
read in the decimal system. So you are trying to check the bolt to see if it really is this
diameter. Then what would you expect to see on the micrometer? So our problem is basically
this stated arithmetically and that is to simply convert 27/32 to a decimal accurate
to the nearest thousandth. Now to do that all we have to do is remember that this symbol
means "divided by." Let's also note this, 32 is divisible only by the prime number 2.
That means if I were to divide 27 by 32 it will terminate at some point. Well this read
27 divided by 32. Decimal point is on the end, go straight up, I want it to the nearest
thousandth so here is tenth, hundredth, thousandth, however in round off situations you always
go one digit further to see if I round up or back. In doing this division you would
get these four digits. Actually there is one more there but since we want it only to the
nearest thousandth all we need is one more digit and since that's five or more we would
add one to this. So that means if you were trying to read this and if this were actually
the diameter of that bolt then on the micrometer the best we could see is .844. It would be
closer to 4 than 3 because 7 is closer to 4 than it is down to 3. So you see by this-
Whenever, in a text book, you see a problem that looks like this, it seems to be rather
sterile and out of this world, they really do come from real world situations. Another
one. There are decimal rulers which read to the nearest hundredth of an inch then in such
a ruler what would 7/16 read as? Because most standard school rulers will read to the sixteenth
or 32nd but you might look around a shop the next time you are near one and see if you
can find one of these decimal rulers; they are fairly common. We simply again, remember
that this means divided by so that reads seven divided by sixteen 7 ÷ 16 and what the problem
is stating is that the ruler cannot read anything beyond 100 so whatever is beyond here really
is irrelevant to the ruler. But we do need one more digit here to find out if the ruler
will read to the next highest or the one down below so performing this division either by
hand or by calculator you would get .437 and so on but again we only need one more digit
to tell this is more than 5 therefore we round this up. So we can now claim that 7/16 is
approximately equal to the decimal number .44. That is closer to .44 than
it is to .43 so on our decimal ruler this is actually what we would see. We would see
that edge of our measuring item almost against the next four in the hundredths place; just
a squidge under it. Here's an interesting problem. A science book, most of them, gives
the value of this symbol which we call pi as 3.1416. We use this number a whole lot
when working with spheres and circles. Whereas another book says to use 22/7. Can one really
do this? As we are really asking is this - is 22/7 really equal to 3.1416?
Assuming this is correct, that is accurate to the nearest ten thousandths. Well let's
check this. You might recall many many elementary schools use this as a value of pi before you
got into decimals in the later grades. Let's divide 22 by 7 (22/7) and see if it comes
out to be this. So, 22 divided by 7, decimal point is here and goes straight up and this
actually divided and if I do so I get this quotient. Let's compare them and you will
actually get a feel of what to round off really means. We see they agree at the whole number.
They agree at the tenth. They agree at the hundredth but at the thousandth they disagree
however if I were to round this decimal value of Pi to the nearest thousandth this would
round to 2 of course this one would round to three doesn't it? So what we can really
say is that 22/7 is approximately equal to Pi and we can say more than that
we can tell you exactly how approximate it is. So the correct statement is that 22/7
is not equal to Pi however it is correct to say that 22/7 is approximately equal to Pi
to the nearest hundredth. So if this amount of accuracy is good enough you could use this
fraction as an approximator. If you need more accuracy than that then this would be not
good enough. By the way while we are here let me show you a cute device to remember
Pi to as many places as you need within normal usage. If you remember the phrase "May I have
a large container of coffee?” (a phrase many people use), and count the letters - three, point, one,
four, one, five, nine, two, six. (3.145926) out to that many places in fact that is what
Pi is equal to. Whereas 22/7 if you were to work that with more accuracy you would find
that it is approximately equal to that decimal (3.1428571) Again you can see that they agree
to the nearest hundredth. Beyond that, Pi and 22/7 are entirely different numbers. So
if you can remember this phrase, (an easy one to remember), it will give you Pi to quite
a few places. Let's see if we can verbalize this. Tenth, hundredth, thousandth, ten thousandths,
hundred thousandths, millionths, ten millionths so this phrase gives you Pi to the nearest
ten millionths. More accuracy then you would even use in most sciences. By the way a question
frequently asked of me in class at this point is "Where does that Pi, that number, come
from? How does that fit into mathematics?" Well if you don't know it's very simple and
very interesting. Well if you were to take any circle and start on the rim and measure its distance
around which we call its circumference; let's call that capital C. Then measure the distance
across the circle that goes through the center which we call the diameter and call that D
and divide these two (C/D). Depending on how accurate the circle is drawn you would come
out close to that value of Pi 3.1416 well actually 159 etc. . . . Of course that rounds
to six at the nearest ten thousandths. The more accurate that circle was the closer it
would be to Pi. The size of the circle had nothing to do with it. If you were to take
a smaller circle and measure its circumference (c) and its diameter which would also be smaller
(d) and divide these two, the ratio would again be that number Pi. So the
definition Pi is nothing more than the ratio or the number you get when you divide the
length of the circumference by the diameter and using very advanced math we can prove
that this number which goes on and on forever, is approximately equal to that number. And
interestingly enough that number, 3.1415926 etc. . . . is not a repeating decimal and
is not a terminating decimal. It's a very interesting number and advanced mathematics
spends a lot of time studying and using it. But never the less, here we have a decimal
number that goes on forever and very frequently we need that number and we of course cannot
go on forever so we must specify where we are going to quit and that is nothing more
than this and the last lesson is about and that is rounding decimal numbers. Therefore
it is necessary that by the completion of this lesson that you realize to ask a simple
question like changing a fraction (two of them here) into a decimal is it not enough?
Because if you were to do that by performing the division, in this case 7/19 (seven divided
by nineteen) or in this case 3/40 (three divided by forty) we now need to realize that as I
do either one of these that I realize that I will get either a decimal that goes on forever,
(with a repeating cycle of course somewhere in it), or a decimal that might go on a very
long way. For instance in this particular case since 40 (the denominator) has only 2's
and 5's as prime divisors, this will be a terminating decimal somewhere. It might go
out a long way and this one will never terminate. It will keep on giving me digits forever and
ever. So whenever I am converting a fraction to a decimal and I intend to use that answer
I have to quit somewhere. So at this point you realize that it is not enough to say simply
to change to a decimal. You must specify where I am to quit, that is where am I to round?
How much accuracy is needed? And that has to be specified to you external to the problem,
there are no math rules telling you where to round off; it depends what you want to
use the decimal equivalent for. So in this particular case let's assume that somehow
someone has determined it needs to be accurate to the nearest ten thousandths. Therefore
as we perform the division we will go out to the nearest tenth, hundredth, thousandth,
ten thousandth and one extra digit (we don't need this now) to find out if I round up or
down. In this case, if I were to follow through with that I would get this and the next digit
is next than four or less so I round down so 7/19 to the nearest ten thousandths is
the decimal point .3684 and it's very important to realize that this is not exactly equal
to that. It's closer to this than .3683 or .3685 that it is the closest approximation
to that place. Then doing the same thing to this fraction that we wish to have as a decimal
accurate to the nearest ten thousandth we would get a situation where it actually terminated
before I got to the nearest ten thousandths however as I wrote this more likely I would
not terminate it here but actually to insert a zero because when I say that it terminates,
what I am really meaning is that I get zeros as my repeating digit from that point on.
So in a sense a terminating decimal is one in which zero is the reporting portion. If
you were to give an answer to somebody that means you are going to give them this without
this so they don't know that it came from a fraction that just happened to terminate
at the nearest thousandths. If you needed ten thousandths you need to tell the person
that you are going to give this to that hey, that ten thousandths digit is a legitimate
and accurate zero. So sometimes even though a zero does not contribute to a value of a
number it is necessary to keep it in order to convey how much accuracy you have to the
person to whom you are going to have this number. And do remember that if whole numbers,
mixed numbers are involved that the whole number part doesn't enter into the computation.
It's simply the whole number of the decimal. Then to find the decimal portion we work only
with the proper fraction portion so 3/11 three divided by eleven. In this case let's see
if a pattern comes out. We haven't followed through too many to find that pattern but
if we wanted the nearest hundredth we would go only to the hundredth and then one extra
to see where the round off it. So let's see eleven into thirty is twice, twenty-two, eight,
eleven into eighty is seven, seventy- seven, three, eleven into 30 is twice, then at this
point you can see I've got 30 here, then 80, 30 here, then 80, so in fact it's the 27 that
is going to repeat but in fact we want it only to the nearest hundredth. The digit next
to it is 4 or less so I drop them so this is approximately equal to 2.27 (? 2.27). If,
however, I were to say this, 2.27 with a bar over it, for now we know that the bar means
to go on with this pattern forever then we can say more than being approximately. We
can even say it's exactly equal. So that's a convenient notation isn't it? So this lesson
is a very simple one, one of the simpler you have had. I give you lots of practice of division
and of course rounding off. But it's one that's used a whole lot in shops, in industry and
to be surely in science. In the next lesson we'll show you yet another way of showing
a fraction as a decimal in an exact way; a method you perhaps haven't seen before but
is used frequently in stock markets. So, until then this is your host Bob Finel- Good Luck.