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Transcript:

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.

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.