Math 20 - Lesson 34

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Math 20 Lesson 34 A Portland Community College mathematics telecourse.
A course in arithmetic review. Produced at Portland Community College. As with fractions,
many people have hesitancy about ordering decimals, deciding which one is larger, which
is smaller. You would be surprised at how many people think this is the larger because
it begins with a 9 and this only with a 1 but as it turns out, this is the larger one.
Now after all, decimals are fractions. Letís write these. This is 125 thousandths but this
one is 936 and see the average person says, see thatís larger than that. But this is
936 ten-thousandths. And recall from fractions days we could not order fractions unless they
had the same denominator. And I can make this one thousand become ten-thousand by multiplying
top and bottom by 10 but if I do that, this top has become a hundred, one thousand, 250
ten-thousandths and this one, of course, is still 936 ten-thousandths. So in this form
of having common denominators is quite obvious that this is the larger number. So the secret
was in the form of having common denominators we only look at the top. Well in decimals
to say common denominators is the same thing as saying letís have the same number of places.
So this one has four places, this one only three. But do you remember that a tape or
so ago we said that with a decimal on the end you can append or take off zeros to your
convenience. In this case, it is one of our conveniences to append a zero. Now if the
two have the same number of decimal places, itís the same thing fraction-wise as having
common denominators. Now we can look at the two and we can see that 1250 is larger than
936 so that is the larger. So the secret for ordering decimals is to be sure that they
all have the same number of places, appending zeros if necessary to accomplish this. Then
pretend as though the decimal isnít there and order them by the way you see it and 1250
is larger than 936. Now it becomes even more obvious when we have a series of decimals
if we aligned them vertically and keep the decimal points in a line. So 1.2, this going
to be .056, this one would be 1.35, and the last one .0092. So keep the decimal points
in a line and keep aligned places, tenths in a line, hundredths in a line, thousandths
in a line, and ten-thousandths. Now if it will help you to visualize things better,
on the end append zeros either on paper or mentally until they all have the same number
of places. And now pretend like the decimal points arenít there and ask yourself which
is the smallest here. Well you can see quickly itís this one down here. This is the smallest,
then the next smallest would be the 560 here, then the next smallest 1350, then, of course,
that makes this the largest. With very little practice you can do this mentally without
going through this trickery but this is useful until youíre comfortable in beginning to
think in terms of size. Letís do another one while reviewing our symbols. Recall what
this symbol means? You see it more and more in your later math. This means greater than
that. So the question is this decimal greater than that decimal. Well this decimal has 1,2,3,4
places, so this one has 1,2,3, so we append with another zero. Now if we look at them
without the decimal points, this side seems to have 1250, whereas this side has only 375.
So indeed this one is bigger than this one. So the answer to this question is yes. That
is true. Now letís recall this symbol means less than. So what weíre asking is .48 less
than .051 and again, an inexperienced person would think this is 48 and this is 51 and
say of course itís true. But theyíre not taking into account the decimal system. Into
compared decimal numbers, they must have the same number of places. So thereís two, thereís
three, so I append a third place zero. Now ask the question, this is 480, this one is
only 51 so this is much bigger than that. So the answer to our question is no, thatís
not a true statement. Three service stations advertise unleaded gasoline. The prices are
as listed here. Which is the most economical? That is, which is least? And our technique
simply suggests to us to write each price with the decimal points lined up, keeping
the place values in a line, tenths, hundredths, thousandths, ten-thousandths. Then, if it
will assist you, append zeros on the end behind the decimal point, which you canít do with
whole numbers. Then at that point simply pretend like the decimal point isnít there and ask
yourself which of these numbers is the least. And one can see since whole numbers are easier
to judge, this one would be the least, and that is A. So A has the best price. Now, of
course, you can just see because of your sensitivity to numbers thatís well and good. This is
a device which helps you bridge the lack of that sensitivity until experience gives it
to you. Letís do another problem like that but at the same time developing some ideas
and language skills that will help us go from this lesson into addition and subtraction
of decimals which will follow this lesson. First, in order to find the order if we donít
have sensitivity to size just by looking at them, an inexperienced person would say this
is the smallest, this is next smallest, third smallest, and this is the largest. If you
didnít have that sensitivity at first we said this which Iím going to correct in just
a moment, we said to write each one lining up the decimals places, the decimal point.
So 1.005 and this is lining up the decimal point 2,3 and, of course, the zero in front
isnít necessary and the next one is point, lining up the decimal points, 507, and the
next one is .053. Now what we need to correct is the idea that itís really not the decimal
point that weíre lining up itís the place values. In fact, we want all the digits in
the tenths place lined up. We want all the digits in the hundredths place lined up. We
want all the digits in the thousandths place, and so on. Same thing with the whole numbers,
we want the digits in the ones place, then tens, then hundreds if we had them. So itís
really not the decimal points weíre lining up itís the places. The decimal points are
merely there to help me find where the places get started from. The reason I say that is
there is a tendency to make a careless mistake by some people, instead of writing .23 like
this, to write it perhaps like that and just leave a vacant spot. But I canít have the
hundredths place in the thousandths place here. All the hundredths places must be together
in the same position. Itís very important that you follow what Iím saying. Then append
zeros, if it will help your thinking, then also, if it will help your thinking, pretend
like the decimal points arenít there. So if there decimal points werenít there and
this zero is not necessary, nor is this, nor is this, or this. So therefore, we could see,
just knocking off temporarily ñ temporarily unnecessary decimal points, that this is most.
Well actually I couldnít knock this off so itís really the number.053. Itís the smaller
so it will be less than the next one and we can see now that 230 is certainly smaller
than 507 so this is my next smaller one. And, of course, I donít need that zero anymore
so there is no point in writing it. That will be smaller than my next one. And we can see
that 507 is certainly smaller than 1005. So this would be less than .507 and we have one
more left over which is 1.005. So we have these rearranged from the smaller to the larger
correctly and we said, to get our thinking onto paper, line up the decimal points but
we want to now realize that, in fact, weíre lining up the tenths, the hundredths, and
the thousandths, and so on. So in actuality itís decimal places of like values that must
be lined up. Keep that strongly at the back of your mind. In fact, letís carry that notion
right back to our whole number addition which we did at the beginning of this course. Remember
if we were adding a long string of numbers many of us learned the rule that we must write
these vertically and, as it was stated then, we must line up the right ends. Now I hope
youíre beginning to realize itís not just the right ends we want lined up, we want all
the units in a line and all the tens in a line, and all the hundreds, and all the thousands.
So, in fact, it was not the ends we wanted lined up that was just to help us get started
writing, itís the place values we want lined up. Therefore, if one were careless, as some
beginners tend to be with a situation like this, you might tend to write something like
this. You spread your digits around and you suddenly realize thatís illegal. I cannot
add tenths to hundredths and I cannot add hundreds to thousands. I can only add digits
from like places. What weíre saying now is itís the same with decimals. Thatís why
in chapter one, we made such a big deal, and we are now again, on place values, because
all the instructions that we give you are in terms, in fact, of place values. Letís
take a moment out and take a peek into a future lesson and youíll see why this is so important.
When weíre adding decimals rather than saying as we did with whole numbers to line up the
ends, weíll start our conversation by saying line up the decimal points vertically. So
this would be 1.03, this would be 0. ñ but now we want to realize is that itís not just
the decimal points we want lined up, we want the tenths lined up, we want the hundredths
lined up, and we want the thousandths in the next column. Of course there wasnít any thousandths
here just hundredths but if I wanted to help my visualization I could insert this extra
zero that weíve talked about and now we have 30 thousandths which is the same thing as
3 ten-hundredths. Same thing here, beginning our thinking by lining up the decimal points,
weíre carrying it through to line up the tenths, hundredths, thousandths, and now ten-thousandths
and if you need or desire this appended zero thatís fine, just give it to yourself. Now
we add and youíre going to see here the extreme advantage over decimal fractions to pure fractions.
Because once we have line it up, all the like place values, then the rule youíre going
to have in another lesson or so is simply pretend like the decimal point isnít there
ñ of course, weíre really keeping them all in a line including the decimal point in the
answer, we keep it there and add as though this was not a fraction problem but really
a whole number problem. So weíre just adding as though we had a whole number problem but,
in fact, we have done a fraction problem. There is the great beauty of the invention
of the decimal system. It allows us to work fractions, which are quite messy, as though
they were whole numbers, which is quite easy. So here again note, we start our thinking
by saying line up the decimal points, which by the way we will do for addition and subtraction
but not multiplication and division, they have a different rule. But then following
through by actually lining up the like place values then temporarily forgetting that I
have a decimal point and then adding as though I have a whole number and Iím done working
the fractions. Isnít that neat? Now note the similarities between what we just did
by peeking ahead and this current lesson. In order decimal numbers weíre going to visually
line up starting with the decimal points lining up the like place values. So this is a 2 and
5 tenths but here again be careful, this whole number does have a decimal point. Itís just
that itís understood to be behind it and we donít write it there unless we need to.
Well now we need to because we want to line up the like places, so the units with units,
the tens in a separate column. Now here again, lining up the point, 092. Now thatís a units
digit, tenths digit, hundredths digit. Now with some experience you wonít bother writing
these zeros down, you will simply go down here, add, add the next column, if youíre
to add but if you are to order as we are here, as a beginner you will probably still find
it more convenient to append with zeros. Now for a brief moment we pretend like the decimal
points arenít there and, of course, in this case if the decimal points weíre there we
wouldnít have to worry about that zero either. Now itís really easy to find the smallest,
in this case it would be this one, which in its original form which we have to go back
to in the final analysis, is the smallest, which means it would be less than the next
smallest which we can see now is going to be this one which in its original form was
2.5 which would be smaller than the next which we now see 7 is certainly smaller than 15,
so itís original form is 7.05 which would be less than now the last one which is simply
15. And again make a mental note that all whole numbers indeed are also decimals and
the decimal is always here, never anywhere else always here on the right end, thatís
again a common mistake for many beginners. Itís always on the right end. And we avoided
our fractions. So note thereís a pattern here similar to what we just did for adding
and as you will find later, similar to subtraction. So be very, very cautious of the subtle similarities
as well as the differences so that you wonít get confused in the future. To show you the
importance of this, here is a problem like situations which occur very, very frequently
in all of science and technology, particularly chemistry. Letís say that you have a vial
which is supposed to contain this many grams of a particular substance, in this case charcoal.
Letís say a person, Jerry here, wishes to double check this so he weighs it on an extremely
sensitive scale and reading as well as he can, he determines that he has 4.21 grams,
which is not the same as this. Now if this is correct, is Jerry weighing too light or
too heavy. And so if we can to compare this weíre really asking is this smaller than
this or is this greater than this. And what Iím doing on paper ultimately we would just
do mentally, lining up the decimal points, that is the decimal places, theyíre the same
here, theyíre the same here. And this is the first one thatís bigger than this one
so this is the larger number so Jerry is weighing too heavily. We will look at this situation
very, very frequently and very closely in the next lesson because it might be that Jerryís
scale is incapable of reading past the nearest hundredth of a gram because thatís minutely
sensitive. So heíd need better scales to get down to here. So if his scales could only
read to the nearest hundredth and this is what it really was, we will ask this in the
next lesson, is this a reasonable measure of this when I have less accuracy. So our
next lesson, in fact, is going to be approximation and round off accuracy. Because, in fact,
when youíre measuring things in reality you never measure them exactly, thatís physically
impossible. So you can only measure things to certain amounts of accuracy depending upon
the kind of instrument youíre using at that particular moment. So bear that to the back
of your mind. That will become an important point in not only a future lesson in this
course but in all of, all of science. Hereís another, Margo and Josh each measured the
diameter of a coin and got 0.81 and 0.807 centimeter, respectively. Which measurement
is the smaller? Now again, a beginner might see that 807 is larger than 81 and say, hey
this is smaller, this is larger but here again letís line up the decimal places. So 0.81
is Margoís measure and Joshís is .80, lining the hundreds with the hundreds, 7. And now
if we wish to insert that extra zero thatís fine and we can see that, in fact, Joshís
is smaller not larger. Again, for those of you who are taking this program that might
eventually take a science class, this is a very frequent classroom experiment at the
beginning of the course. Itís where the instructor gives everyone a very accurate measurement
scale, centimeters, calipers, whatever, and send the same object all the way around the
room and ask everyone to measure it as carefully as they can and write it now. Now the same
object, the same kind of measurement skills and yet you will find that it is the rule
rather than the exception that most people will not measure precisely the same and your
science class will discuss why that is so. But again, you will find that they will be
close and itís important for us to determine later how close and in this lesson of the
different measures which is the smallest and which is the largest. And also be aware that
sometimes youíre trying to order things not from the smallest to the largest but from
the largest down to the smallest. So again, you line up beginning with the decimal places
but actually lining up the places not the point. So 1 ten, 1 unit rather, and 2 tenths,
no units, no tenths, 3 hundredths, 4 thousandths, and no units, 2 tenths, no hundredths, no
thousandths, 5 ten-thousandths and if it will help you to insert enough zeros so that you
can visually compare these easier. This is what we meant in the last tape by these zeros
being for looks only, cosmetics if you will. It just helps you to orient your thinking
in value they serve no purpose whatsoever. But now pretending the decimal points arenít
there, we can see, in fact, this is the largest 1.2. And with that out of the way our next
largest is this one which is .98. With it out of the way I can see this is my next largest
.2005. And finally that leaves us with the smallest which is .034. And notice in all
of these cases I did not write the zero in front of the decimal point but again I can
or I donít have to, the value remains the same. So you will find, I trust, that this
is an extremely simple lesson but you will also find that itís a remarkably important
one for the future. Till your next lesson, this is your host Bob Finnell. Good luck.