Math 20 - Lesson 31

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Math 20 Lesson 31
A Portland Community College Mathematics Telecourse. A course in arithmetic review. Produced at
Portland Community College. Before we review the workings of decimals, let’s pause for
a couple of short lessons to examine what are decimals and why do we write them that
way. First, it, a decimal, is a special notation for fractions whose denominators are powers
of ten. Now do you recall what we mean by powers of ten? Recall, if you will, that a
power of ten is any number that can be written as a product of tens. So we have four tens
being multiplied together, that’s 10 to the 4th power and that multiplication has
a particular short cut which makes it of some value. You simply write down a 1 and follow
it by four zeroes and so forth. So refresh yourself as to what we do mean by the power
of ten. We’ll be using it quite a bit during this chapter. Now if we were to put the power
of tens into the denominator, 1 over 10 to the first or 1/10, we’re going to agree
to write that as .1. Now this is just an agreement by people a long time ago but it proves to
be very convenient. So 1 over ten squared or 1/100, we agree to write as .01. And 1/1000
we agree to write as .001 and so on. Now let’s find a pattern here which might make it easier
to remember. If you look at the power of ten in the denominator or the devisor position,
count the number of zeroes, 1, 2, 3, 4 and start at this point which we will call a decimal
point and count over 1, 2, 3, 4, 1, 2, 3, that 4th position is where our 1 goes. This
is simply an invention by our ancestors but it proves to be very, very helpful. Without
this our calculators and computers would not have been possible. So you see how it works?
This notation is really a way of writing a fraction without a fraction, in a sense. So
we will read this as 100. So if I want three of those hundreds, I will put 3 in that position.
If I want seven of them, I put a 7 in that position. So the important idea is that the
second position here is our hundredths position for the denominator, which is always 100.
Therefore, if we wanted to write this fraction using that notation, we recall that 1/10 is
that first position after or to the right of the decimal point. Now if I want 5/10 rather
than 1/10, I put the 5 in that first position which we’ll call the tenths position. Now
generalizing that, if 1/1000 is .001, that is the third spot in the thousands fraction.
Then what would we – how would we express 8/1000? Well it is that third spot that we
want because it is the thousands spot. Instead of having 1000s, we want 8000s. So we will
put an 8 there. Are you beginning to see the idea? Each of these places names a fractional
denominator. The first place is the tenth place. Second place is the hundredth place.
Third place is the thousandths place and so on. So as it turns out, the decimal system
is an extended place value system were now this first place to the right, we’ll call
it tenth place, hundredth place, thousandth place, ten-thousandth place, hundred-thousandth
place, etc. This is something you should memorize if you have not already. Notice that word-wise
it’s the TH that tells us that we are in the fractional place. So to the right of the
decimal point will be these specialized powers of ten fractions, to the left will be our
traditional whole numbers. So a number like this if we wanted to say it, this is really
5 in the tenths place, 6 in the hundredths place, 7 in the thousandths place, so 5 tenths,
6 one hundredths, and 7 one thousandths. Or if we want to say it all as one number, we
simply do this. Pretend like the decimal point isn’t there and say it without it. So if
we were to say it without it that would be 567. Then say the place name of this right-most
place. So tenth, hundredth, thousandths, that’s the thousandths place, so now we say 567 thousandth.
567 thousandth and the thousandth, TH, I am in the fraction place opposed to a whole number.
Again, to get the word name from its place value name or form, say or write the number
without the decimal point. Then say the place value of the right-most place. Let’s try
that with one more number. This form of number we call the place value name. So for this
place value name we wish to know the word name. That is if you had to say it over the
telephone, how would you say it, other than, of course, .00107, which is the way we generally
do it. Okay, we would say it without the decimal point. So without the decimal point, these
front zeroes are not necessary. So we would say 107. Then name the right-most digit’s
place name, so tenth, hundredth, thousandth, ten-thousandth, hundred-thousandth, so 107
hundred thousandth. Note that the TH tells me I’m in the fractional place and the hyphen
here tells me this is all one place, the hundredth-thousandths place. Again, tenth, hundredth, thousandth,
ten-thousandth, hundred-thousandth. And to go from our word name back to the place value
name, we forget this for a moment and write this, two hundred fifteen, 215. Now sort of
reversing our order, we want the ten-thousandth place. So let’s just think this way, tenth
– now this isn’t really a tenth, this is just a gimmick that works – tenth, hundredth,
thousandth, ten-thousandth. So I will need at least one place here and then my decimal
point. Now let’s say it properly, tenth, hundredth, thousandth, ten-thousandth, so
215 ten-thousandths. Now some teachers would prefer to discuss it this way, forget this
and establish the ten-thousandth place. So there’s tenth, hundredth, thousandth, ten-thousandth.
I want 215 of them but they must end at the end so that forces me to put the 215 here
and a zero up here. Now if you were to reverse that and write it this way, 215 and stick
a zero here, you wouldn’t get it because if we said this without the decimal point
you have 2150 ten-thousandths and this was not to be 2150, it was 215 ten-thousandths.
So this place holder zero must go to the front. Now the genius of this system is that the
decimal form of the powers of ten fractions, which is what we call decimals, will allow
us to add, subtract, multiply, and divide with them by rules which are similar to whole
number rules, which are a lot easier than the rules of fractions, which I’m sure you
are convinced of after our last chapter. And as it will turn out, the decimal number combined
with whole numbers allows us to treat the mixed number as though it’s one number.
It’s a very convenient notation. It is truly an act of genius of our ancestors to have
developed it. So as with the whole numbers, it’s very important that you learn the place
names of all of these places now to the right of the decimal point and it always starts
with tenth. It is very important point. Make a strong note of this. We begin the first
spot with tenth, then hundredth, then thousandth, then ten-thousandth, then hundred-thousandth,
then millionth, then ten-millionth, and so on, exactly like with whole numbers but in
reverse direction and we must start with tenth, where as whole numbers start with one, very
important to note that. So looking at this with our whole numbers, now see watching this
in conjunction with the whole number system we start with ones, tens, hundreds, and move
this way. With decimals we don’t start with one, we start with tenth then hundredth. So
here’s one then ten and tenth, hundreds, hundredth, thousands, thousandths, and so
on. As it will turn out, the decimal point we will verbalize by the word and. Okay, now
this system is a very important system to have and memorize, particularly those of you
who will go on to computer science or math for elementary teachers or, of course, mathematics
in general. Now coming back and putting the decimal together with the whole number, we
read as 56 and 28 hundredths. A number like this in place value form we would write as
206 and now the decimal portion you could pretend like it’s not there and that would
be the number 206 again and now we name this right-most digit place, tenth, hundredth, thousandth, ten-thousandth,
so 206 ten-thousandth. Again, 206 stop and 206 ten-thousandths. The TH tells me I’m
in the fractional spot the hyphen tells me this is all one place and the number that
comes before it tells me how many of those places I have, 206 of those ten-thousandth
place. Now let’s reverse it, going from a place value, a word name to the place value
name. So five thousand five and – the and is a whole number between the divider and
the decimal, so five thousand five, 5005. The and is a decimal point. Now let’s find
the place we want to stop at, ten-thousands, tenth, hundredths, thousandths, ten-thousandths
and we want only five of those ten-thousandths. So I want five of those ten-thousandths. So
these placeholders must be filled with zeroes so that we don’t mistake this for another
place value. Okay, so that’s it, the beginnings of a very important number system. One of
the geniuses of our civilization. Just as we hope you are beginning to realize that
every mark in mathematics says something or else we wouldn’t use it, so we hope you
begin to realize that that’s the way it is with language too. Notice the difference
between these two just because of the insertion of this hyphen. First, we notice that each
of them has THS on the end which tells me I’m in the fractions or decimals situation
but here says I want to go out to the thousandths place so here’s ten, hundredth, thousandths,
which means I don’t even need this, and I want one hundred of them. So if I want a
hundred to end here, I would have to write one hundred thousandths, which, of course,
as a fraction would be this, which, of course, we can also see reduces to 1/10. However,
down here the hyphen says that I don’t want the one thousandth place I want the hundred-thousandths
place. So here is my tenth, hundredths, thousandths, ten-thousandths, hundred-thousandths and at
the hundred-thousandths place I want just one of them and, of course, in this case I
would have to back fill in placeholder zeroes to remind myself this is down here and not
somewhere else. So here I have a 1/100000 which won’t reduce anymore and we see that
these are entirely different numbers. This number, as a matter of fact, is ten thousand
times larger than this one. Hence the importance of realizing what this hyphen is really saying,
that all of this is one place where as here we want this as the place and this is the
number of those places. You see each mark tells us something distinctly different and
that’s part of the difficulties for beginners in mathematics as well as in English composition
is that we’ve grown up with our spoken language which tends to be a little bit sloppy and
a little bit creative because of that, by the way. However, in the precision of technical
writing and mathematics, every single mark must mean something distinctly different or
we ask, why would we use it. So be careful on the writing of these numbers. Just to illustrate
the subtlety and importance in minor changes in form, let’s look at these three problems
two different ways. If we were to write each of these in fractional forms, we would first
pretend like the decimal point isn’t there. So, of course, that zero in front is just
for emphasis. So we have the number 457 but the decimal point says that this is a special
kind of a fraction. Specifically, it’s over a 1 followed by a 1, 2, 3 places. Or if you
wanted to say it, you’d say this is the tenth place, hundredths place, thousandths
place, so 457 thousandths. Whereas on the next one without the decimal point we still
have a number, 457, but over a 1 followed by 1, 2, 3, 4 places. Now please note, I am
not counting zeroes, I am not counting numbers, I am counting decimal places starting at the
decimal point. So there are four decimal places, hence four zeroes following the one in the
denominator. Or saying it verbally, this is a tenth place, hundredth, thousandth, ten-thousandths,
so we have 457/10000. And the last one we still have without the decimal point 457 but
now shortcut-wise over one followed by counting decimal places now starting at the decimal
point. So there’s 1, 2, 3, 4, 5, 6, 1, 2, 3, 4, 5, 6, or verbalizing it, tenth place,
hundredths, thousandths, ten-thousandths, hundred-thousandths, millionths. So here we
have 457/1000000. So to a beginner, you look at these and see that they all have 457 and
sort of feel that they’re all the same and yet they are distinctly different numbers
because of the different placement of that 457 in each case and that decimal point is
what helps me locate that placement. Then the zeroes between the decimal point and that
first non-zero digit we call place holders. It makes sure we are seeing this as a ten-thousandths
place rather than a thousandths place, which we wouldn’t see if we didn’t have that
zero. So these we call place holders as well as a zero in front which is simply there for
emphasis. Were we to leave the front zeros off, technically it’s the same number but
we have found psychologically and visually that when we leave it off people tend not
to see the decimal point. Where as when there is a units placeholder in front, this seems
to emphasize the presence of the decimal point. So this is here for effect, these are here
out of necessity. Do you see that? It’s very important. And if we were to write the
word name for each of these, the differences between each representation is just as subtle.
Watch this. If we want the word name for this, first we omit the decimal point and say the
number as it appears to us 457, 457, and then we name the place of the right-most digit.
So the right-most digit, 7, is in the tenths, hundredths, thousandths place. So this 457
is right in sitting in the thousandths place. And notice that the TH is what tells me I’m
in the fractions place rather than the whole numbers place. Now let’s do the second one
and compare with the results of the first one. We still have without the decimal point
457, which I wrote, but now we need the place value of the right-most digit. So this is
tenths, hundredths, thousandths, ten-thousandths, so we have 457 ten-thousandths. Notice again
that the THS tells me I’m in the fractional part of my number but the hyphen here is very
important. That tells me I want to go clear to the ten-thousandths as opposed to just
the thousandths place which we had above it. So again the importance of a TH and the importance
of a hyphen in our word names and in our place value names, the importance of the placeholder
zeroes. Now let’s follow that through with our third example. Again, omitting the decimal
point, we have a number, 457 but now the decimal point tells me I must name this by the end
digit place. So here again we have the tenths, hundredths, thousandths, ten-thousandths,
hundred-thousandths, and millionths. So we have 457 millionths and again the importance
of the THS. Are you beginning to get a feel for this new kind of way of writing called
decimals? Of course, for most of you it really isn’t new, is it. We’re just reviewing
what you’ve grown up with since elementary school and now trying to really nail down
subtle points that you might have missed in the past because subtle points are going to
be all important to us as we move into more advanced uses of these decimals than perhaps
you remember encountering in elementary school or high school. And now let’s tie that back
together with our whole numbers now at the conclusion of this lesson. If we want the
place value name for this verbal statement, well the first part simply says two hundred
six to the and. The and is going to serve verbally to tell me where the decimal point
is going to be, so I simply have a number, 206. The and tells me where to locate the
decimal point. Then I have 206 ten-thousands. So the TH tells me I am in the fractional
place. The hyphen tells me that all of the denotes the exact place. So finding the exact
place, ten-thousands, there’s the tenth place, hundredths, thousandths, ten-thousandths.
So I want the number, 206, to end at this spot which tells me 6 goes there, 0 goes there,
2 goes there, but I still have some front-most places that I need to hold somehow otherwise
we don’t know it’s there. So that’s the purpose of placeholder zero in decimals.
Just as in my whole numbers, this zero is a placeholder or without it I would have thought
that was 26. So without this I would think that was 206 thousandths and I need a placeholder
here. I have a placeholder here too and that tells me I have no tens in this 206, just
the two hundreds and six. So I needed placeholders in these locations to keep the numbers in
the appropriate places. Are you beginning to get an appreciation for the very subtle
but simple system invented by our ancestors? You will actually appreciate this so much
when we begin to review addition, subtraction, multiplication, and divisions of decimals
because at that point it will allow us to treat fractions and mixed numbers as though
they were whole numbers, thus simplifying a lot of work. Could you finish this lesson
by giving the word name for this? Let’s do it just talking-wise. First, up to the
decimal point, that’s one thousand seven and this says and going o the end, again we
have one thousand seven but we’re in a fractions place, so there’s tenths, hundredths, thousandths,
ten-thousandths, hundred-thousandths, millionths, ten-millionths. So we have a one thousand
seven ten millionths. So putting it all together, one thousand seven and one thousand seven
ten-millionths. There, is it coming together? This is your host, Bob Finnell. Until our
next lesson together, good luck.