Math 20 - Lesson 32

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Math 20 Lesson 32
A Portland Community College mathematics telecourse. A course in arithmetic review. Produced at
Portland Community College. This lesson is a short extension of the last one and itís
just to help us sharpen our feeling of the place value of each of those decimal paces
and to do this we will develop a form called expanded form just as we did with whole numbers
in chapter one. And itís just this simple, take each place digit as a numerator and ask,
whatís the value of its place. That first place is the tenth place then and, which in
this case is plus. The next digit, which is the next numerator, and its place value is
tenth, hundredth so we place that on the bottom and the next numerator and its place value
is tenth, hundredth, thousandth and 4 tenth, hundredth, thousandth, ten-thousandth or,
as you can see, in each additional place to the right I get one more zero on that power
of ten. So as it turns out, the digits are the numerators and the place value is the
denominator and thatís really all this lesson is about. So letís try another one. This
says I have 200 and this says I have no tenths and 8 ones and and, thatís still plus, I
have 6 tenths, then I have no hundredths, and then no thousandths, and then five ten-thousandths.
And these no positions, sometimes we donít both writing them but sometimes we do just
to emphasize that those places are there, thereís just no digit there. Just as here,
there no tenth but sometimes itís written as that, no tenth. Okay, this form we call
the expanded form. It helps us to focus on the nature of this and letís us see that
each digit by sitting in a different place changes its value. So there is a digit value
and a place value which together form the total number value. Now to reverse that, if
we were to write the place value name for this, place value means I want to write this
thing in places, thatís the place part, instead of fractions. So we remember that the first
place is tenth, and we have 8 tenths. The next place is hundredths and we have 2 of
them. And the next place is thousandths and we have 3 of them. No by not specifying those
zero places in this form here, the expanded form, it can be confusing when youíre going
backwards. Since that came first, thereís a tendency to write it first. But this doesnít
say 3 tenths, which is what this position is, it says 3 hundredths which is what the
next position is. There arenít any tenths. So we need to say that and we say it with
a zero and we call this a place holder zero. Now the next place is the thousandths place
but we donít have any thousandths so we need a place holder there. The next spot is ten-thousands
place and there are 5 of them. And the next place is the hundred-thousands place and there
are 6 of those. So we have consistently calling these zeros, place holder zeros, and without
them the number takes on an entirely different meaning. So letís concentrate on the nature
of those zeros for a moment and their places. Now which of these zeros are really necessary
to preserve the value and which ones are perhaps there for looks, they do not really contribute
to the value? Well you can determine that if you begin to think of the place value.
This had no ones and, of course, if I donít have it, thereís no ones here. But if I didnít
have that one, then there would be a tendency to think that was 6 hundredth instead of thousandth.
So that is a necessary zero. But this says I have 6 thousandths but no ten-thousands,
no hundred-thousandths but for that matter you have none of anything else that would
follow either. So as it turns out, these two ñ or this one is the only one that is really
necessary. That if these werenít there and we had this instead, it would be the same
number. Now letís incorporate that into a rule because many beginners get confused on
this point and will leave a zero out that is absolutely essential. This may be rather
obvious but a very important note. Behind the decimal point and to the right of the
last digit, zeros can be appended or dropped without changing the value. Letís say we
have a number like this. No again, those zeros are necessary to preserve the place valueness
of it but behind the decimal point on the end, I can append zeros or take them off at
whim. Thatís strictly for cosmetics. It adds nothing to the value of the number. Except
when we get to round off at a later point in this chapter, these decimals will tell
me something above and beyond the actual value but at this point theyíre not really necessary.
Now here is another zero that is used frequently in business that has no value as far as numberness
is concerned but does help us visually and psychologically. This zero is not necessary.
Yet you find in most business ledgers and billings from computer print outs will have
a zero and sometimes one other here. Now the reason for that is purely psychological. Many
people when they write their decimal will write very fast, somewhat sloppy and sometimes
you donít really see that. Sometimes you will write a decimal this way and youíre
not sure whether thatís a decimal or a zero. It would be nice if we could all be very emphatic
like this but usually weíre in a rush and we donít. So it has been found by many business
people that if we put a zero in front and take that trouble then we are more apt to
notice that legitimate decimal point. So this is purely for visual and psychological help
but logically itís not necessary. Watch this on the billings you get on the next mail out
from the light or water company. Letís play with that idea for a few moments until you
become quite comfortable with it. Here you have a decimal number and our question is,
which of these zeros are really not necessary for the value. Now if I were to throw this
one away, we would know that there is no units with or without the zero. So I really donít
need this but weíve agreed that weíll keep it for cosmetic purposes to really see this.
Now if I were to leave this zero out, we would think that this is 5 tenths because tenths
is the first place value but this tells me there are no tenths and this is 5 hundredths.
So this one is necessary which if you were to throw it away it would change the total
value of our decimal number. And do you see that itís the same thing here? I need that.
This one tells me I have no ñ can you name this place value. Letís see, tenths, hundredths,
thousandths, ten-thousandths, hundred-thousandths, so this says I have no hundred-thousandths
but for that matter, I have no millionths, no ten-millionths, no hundred-millions, and
so on forever. So these end zeros behind the decimal point on the end are really not necessary
except, as we mentioned, when we get to round off it might tell us something about the number
more than the value. But as far as the value is concerned, theyíre really not necessary
they are there for some visual purpose as helping you to see something else and you
will see cases of that in a few lessons. Once more with a different number. Which of these
are necessary and which ones unnecessary? Now here again, this tells me that I have
no ñ letís see, units, tenths, hundredths, thousandths, I have no thousandths, this says
I have no ten-thousandths but for that matter, I have no hundred-thousandths, millionths,
ten-millions, and so on forever towards the front. So as it turns out, these zeros at
the front end of whole numbers are not necessary. By the way, if we have them there they are
not wrong either. They are just value-wise not necessary. But notice itís the same toward
the end. This is tenth, hundredth, thousandth, ten-thousandth, hundred-thousandth, and this
one says that I have no millionths, no ten-millionths, but for that matter, I have no hundred-millionths,
no billionths, and so on in this direction. So all of these decimal zeros on the end behind
the decimal point are not necessary either. So, you know, we have sort of a rule between
decimals and whole numbers together now. The zeros at the end behind the decimal point
are not necessary. Zeros at the front in front of the decimal point are not necessary. But
all other ñ and we could call them in between zeros are very, very necessary. If I didnít
state these two I would read this as tenths, no one hundredths, and 3 thousandths where
this is to be 3 thousandths, and this is 5 hundred-thousandths. So in between zeros are
very, very critical. To not put them in changes the value. The end zeros ñ the end of decimals
and the front of whole numbers are not necessary. Do you recall a similar problem like this
from your whole number division days? Do you remember this problem from not too many lessons
ago? Letís begin to do this problem once again together and see a relationship between
this problem and the point weíre discussing about necessary and unnecessary zeros. First,
realize this zero is necessary since this is a whole number. Because to leave that off
would be to say I have the number 697 as apposed to the number 6970. Just think about money.
Thatís a lit more money than that. That zero is very, very essential. Now notice this is
not an end zero. We said end zeros are not necessary behind the decimal point and that
zero is not behind the decimal point. So it is necessary. So we will see that this divisor
goes into this one approximately twice. Multiplying gives me 68. Subtracting gives me 1. Bring
down my next place number. Do you recall some lessons ago there were some people who say
this does not go into this. So then they would say bring down the next number and this does
go into here, looks like five times. Then they would say 5 times 4 is 20, carry the
2, 5 threes is 15, 17, than say okay, Iím done. This goes into here 25 times. But if
you took 34 times 25 to check it, and until youíre very secure you should always check
it. Watch what happens. 5 fours is 20, carry the 2, 15, 16, 17, two 4s is 8, two 3s is
6, 0, 5, 8, and this is anything but that number. So there must be something wrong with
this. And what happened was when we brought down this 7, it was improper to say that 34
will not go into 17. 34 will go into 17 zero times. Now notice up here some people will
say 34 does not go into 6 so we bring down the next one which is 9, but in fact, 34 will
go into 6. It will go in zero times. But the reason we donít traditionally write this
zero is because on the whole number itís at the front end and hence not necessary.
Knowing that ahead of time, we just donít bother writing it. But in between zeros as
you should become very conscious of in this lesson, is absolutely necessary. Now if we
were to take 205 times 34 and I wonít do it here, to save time Iíve already done it
on the calculator, and the answer is this and it does check. So once again, letís finally
become very, very conscious of this should be obvious fact, and that is simply this,
when youíre writing a number, a whole number or a decimal or combinations of the two, in
between zeros are always very, very necessary. These are what we have been calling place
holder zeros. But behind the decimal point on the end, zeros there are not necessary.
If you have them you can throw them away or if you have some reason, and we will have
reason in the future in many cases, you want them, thatís okay, just stick them in. They
will not change the value. If you stick them in between, they will change the value. So
behind the decimal point on the end zeros are not necessary, however, not harmful to
the value of a decimal number. With a whole number in between zeros are just as important
whether they are here or at the units place. But zeros at the front of whole numbers are
not necessary. And as we pointed it out there are times, however, when we traditionally
will write a zero in front of the decimal point even though technically itís not necessary,
just to emphasize and really force us to see the decimal point. Now it may sound like Iím
beating that point to death but the only reason I do it is for beginners or people that need
heavy review, that is a common mistake to throw away zeros that are very necessary and
to not know when they can put in additional zeros which will not change the value even
though the looks have changed. So it is an important point commonly misunderstood. Well
we have this number before us; letís do a bit of reviewing. Whatís the word name for
this place value name number? First, look at the whole number part and say that by itself.
So thatís the number 201. Remember we donít say and, 201. There that portion is the word
name for that portion. Now the word name for the decimal point is the word and. And to
say the decimal number we pretend like the whole number and the decimal point isnít
there and read this as though it were a whole number. And, of course, if it were a whole
number, that zero is not necessary. So we would say 507. But if we were to quit here,
201 is this, and is this, 507 is this, and this statement is simply telling me that I
have two numbers, perhaps not even being added just two numbers, this number and another
number. So somehow we need the word that will let me know that this is a fraction. In fact,
it is the numerator of a fraction. So now we have to say the denominator. So we have
to give the word name for this end place. So this is tenth, hundredth, thousandth, ten-thousandth,
so 507 ten-thousandths. And please note we need the dash otherwise we think we have ten
thousandths which is different than the single place ten-thousandths. And itís the TH that
tells me weíre in the factional position. Itís very subtle. It was very clever of our
ancestors to invent this rather simple system of place value names that allows us to write
and infinite number, variety of numbers with a very small, finite set of words. Now letís
use the same number to explore another aspect of our growing vocabulary. What do we mean
by the expanded form? Well this is a vehicle by which we are able to tell what happens
to each digit because itís in a separate place. If this 5 were here it would be 5 tenths,
if it were here itís 5 hundredths. If the same 5 were written here it would be 5 thousandths.
So where you write the digit changes its value. So this 2 says that I have 2 hundredths. This
zero is a placeholder that says in addition I have no 10 and this says I have one 1. And
this says I have no tenths, and this one that I have 5 one hundredths, and this one that
I have no one thousandths, and this one that I have 7 ten-thousandths. Sometimes we will
shortcut this by simply saying I have two 100, a 1, 5 hundredths, and 7 ten-thousandths.
Usually when you need this expanded form of a number, particularly in math for elementary
teachers, this form would be negated in favor of this so we can see every place; every single
place from the beginning to the end has been accounted for. Now letís do the reverse.
Letís say we have a number written in that shortened expanded form and we want to go
back to the place value form. Well letís start with a decimal point. Going this way
I have my units and I have 7 units, then I have tenths but I have no tenths so I need
a place holder zero. Then I want to know how many hundredths I have, I have 3 hundredths.
That should be very obvious from two chapters ago. But this say 5 one thousandths, therefore,
to put a 5 here would be an error. This is the number of tenths. Therefore, I have no
tenths up here but to put the 5 here would still be an error because this is my hundredths
place and I have no hundredths here. So again I need my placeholder zero. This is my thousandths
place and I have 5 of those and this is my ten-thousandths place and I have 9 of those.
So this is my place value name for this and the word name would be three hundred seven
and fifty nine ten-thousandths. You see how thatís done again? You read this and its
place value and that gives me the number value. You should be very comfortable with this following
a successful completion of this lesson. Letís us this to recall another idea from the last
chapter which will be called upon in years in the future. This in expanded form tells
me I have 5 tenths. Then I have 2 one hundredths. Then I have 3 one thousandths. And then I
have 4 ten-thousandths. And then I have 8 one hundred-thousandths. Recall from the last
chapter that we could write each of these powers of ten in exponent notation. This is
simply 10 which we say 10 to the first power. This is 10 squared or 10 to the second power.
This is 10 to the third power or 10 cubed. This is 10 to the fourth power. And this is
10 to the fifth power. And recall when these are powers of ten greater than one that these
exponents in effect count the number of zeros following a 1. So 10 to the fifth is 1 followed
by 5 zeros, 10 to the fourth is 1 followed by 4 zeros, and so on. Sometimes when we write
this in expanded notation, you will do it using this kind of notation. Youíll say 5
tenths, plus 2 ten squared, plus 3 ten cubes, plus 4 ten fourths, plus 8 over ten to the
fifth. In more advanced math you will find that this will become the more convenient
notation than this. So once in awhile think back on what weíve covered in previous chapters
because we will very soon begin to tie them together into one subject matter. So once
again, the main purpose of this particular lesson is to go from a decimal notation to
expanded form and back again. Do become very comfortable in that process. Until the next
lesson, this is your host Bob Finnell. Good luck.