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Transcript:

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.

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.