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Math 20 Lesson 34 A Portland Community College mathematics telecourse.

A course in arithmetic review. Produced at Portland Community College. As with fractions,

many people have hesitancy about ordering decimals, deciding which one is larger, which

is smaller. You would be surprised at how many people think this is the larger because

it begins with a 9 and this only with a 1 but as it turns out, this is the larger one.

Now after all, decimals are fractions. Letís write these. This is 125 thousandths but this

one is 936 and see the average person says, see thatís larger than that. But this is

936 ten-thousandths. And recall from fractions days we could not order fractions unless they

had the same denominator. And I can make this one thousand become ten-thousand by multiplying

top and bottom by 10 but if I do that, this top has become a hundred, one thousand, 250

ten-thousandths and this one, of course, is still 936 ten-thousandths. So in this form

of having common denominators is quite obvious that this is the larger number. So the secret

was in the form of having common denominators we only look at the top. Well in decimals

to say common denominators is the same thing as saying letís have the same number of places.

So this one has four places, this one only three. But do you remember that a tape or

so ago we said that with a decimal on the end you can append or take off zeros to your

convenience. In this case, it is one of our conveniences to append a zero. Now if the

two have the same number of decimal places, itís the same thing fraction-wise as having

common denominators. Now we can look at the two and we can see that 1250 is larger than

936 so that is the larger. So the secret for ordering decimals is to be sure that they

all have the same number of places, appending zeros if necessary to accomplish this. Then

pretend as though the decimal isnít there and order them by the way you see it and 1250

is larger than 936. Now it becomes even more obvious when we have a series of decimals

if we aligned them vertically and keep the decimal points in a line. So 1.2, this going

to be .056, this one would be 1.35, and the last one .0092. So keep the decimal points

in a line and keep aligned places, tenths in a line, hundredths in a line, thousandths

in a line, and ten-thousandths. Now if it will help you to visualize things better,

on the end append zeros either on paper or mentally until they all have the same number

of places. And now pretend like the decimal points arenít there and ask yourself which

is the smallest here. Well you can see quickly itís this one down here. This is the smallest,

then the next smallest would be the 560 here, then the next smallest 1350, then, of course,

that makes this the largest. With very little practice you can do this mentally without

going through this trickery but this is useful until youíre comfortable in beginning to

think in terms of size. Letís do another one while reviewing our symbols. Recall what

this symbol means? You see it more and more in your later math. This means greater than

that. So the question is this decimal greater than that decimal. Well this decimal has 1,2,3,4

places, so this one has 1,2,3, so we append with another zero. Now if we look at them

without the decimal points, this side seems to have 1250, whereas this side has only 375.

So indeed this one is bigger than this one. So the answer to this question is yes. That

is true. Now letís recall this symbol means less than. So what weíre asking is .48 less

than .051 and again, an inexperienced person would think this is 48 and this is 51 and

say of course itís true. But theyíre not taking into account the decimal system. Into

compared decimal numbers, they must have the same number of places. So thereís two, thereís

three, so I append a third place zero. Now ask the question, this is 480, this one is

only 51 so this is much bigger than that. So the answer to our question is no, thatís

not a true statement. Three service stations advertise unleaded gasoline. The prices are

as listed here. Which is the most economical? That is, which is least? And our technique

simply suggests to us to write each price with the decimal points lined up, keeping

the place values in a line, tenths, hundredths, thousandths, ten-thousandths. Then, if it

will assist you, append zeros on the end behind the decimal point, which you canít do with

whole numbers. Then at that point simply pretend like the decimal point isnít there and ask

yourself which of these numbers is the least. And one can see since whole numbers are easier

to judge, this one would be the least, and that is A. So A has the best price. Now, of

course, you can just see because of your sensitivity to numbers thatís well and good. This is

a device which helps you bridge the lack of that sensitivity until experience gives it

to you. Letís do another problem like that but at the same time developing some ideas

and language skills that will help us go from this lesson into addition and subtraction

of decimals which will follow this lesson. First, in order to find the order if we donít

have sensitivity to size just by looking at them, an inexperienced person would say this

is the smallest, this is next smallest, third smallest, and this is the largest. If you

didnít have that sensitivity at first we said this which Iím going to correct in just

a moment, we said to write each one lining up the decimals places, the decimal point.

So 1.005 and this is lining up the decimal point 2,3 and, of course, the zero in front

isnít necessary and the next one is point, lining up the decimal points, 507, and the

next one is .053. Now what we need to correct is the idea that itís really not the decimal

point that weíre lining up itís the place values. In fact, we want all the digits in

the tenths place lined up. We want all the digits in the hundredths place lined up. We

want all the digits in the thousandths place, and so on. Same thing with the whole numbers,

we want the digits in the ones place, then tens, then hundreds if we had them. So itís

really not the decimal points weíre lining up itís the places. The decimal points are

merely there to help me find where the places get started from. The reason I say that is

there is a tendency to make a careless mistake by some people, instead of writing .23 like

this, to write it perhaps like that and just leave a vacant spot. But I canít have the

hundredths place in the thousandths place here. All the hundredths places must be together

in the same position. Itís very important that you follow what Iím saying. Then append

zeros, if it will help your thinking, then also, if it will help your thinking, pretend

like the decimal points arenít there. So if there decimal points werenít there and

this zero is not necessary, nor is this, nor is this, or this. So therefore, we could see,

just knocking off temporarily ñ temporarily unnecessary decimal points, that this is most.

Well actually I couldnít knock this off so itís really the number.053. Itís the smaller

so it will be less than the next one and we can see now that 230 is certainly smaller

than 507 so this is my next smaller one. And, of course, I donít need that zero anymore

so there is no point in writing it. That will be smaller than my next one. And we can see

that 507 is certainly smaller than 1005. So this would be less than .507 and we have one

more left over which is 1.005. So we have these rearranged from the smaller to the larger

correctly and we said, to get our thinking onto paper, line up the decimal points but

we want to now realize that, in fact, weíre lining up the tenths, the hundredths, and

the thousandths, and so on. So in actuality itís decimal places of like values that must

be lined up. Keep that strongly at the back of your mind. In fact, letís carry that notion

right back to our whole number addition which we did at the beginning of this course. Remember

if we were adding a long string of numbers many of us learned the rule that we must write

these vertically and, as it was stated then, we must line up the right ends. Now I hope

youíre beginning to realize itís not just the right ends we want lined up, we want all

the units in a line and all the tens in a line, and all the hundreds, and all the thousands.

So, in fact, it was not the ends we wanted lined up that was just to help us get started

writing, itís the place values we want lined up. Therefore, if one were careless, as some

beginners tend to be with a situation like this, you might tend to write something like

this. You spread your digits around and you suddenly realize thatís illegal. I cannot

add tenths to hundredths and I cannot add hundreds to thousands. I can only add digits

from like places. What weíre saying now is itís the same with decimals. Thatís why

in chapter one, we made such a big deal, and we are now again, on place values, because

all the instructions that we give you are in terms, in fact, of place values. Letís

take a moment out and take a peek into a future lesson and youíll see why this is so important.

When weíre adding decimals rather than saying as we did with whole numbers to line up the

ends, weíll start our conversation by saying line up the decimal points vertically. So

this would be 1.03, this would be 0. ñ but now we want to realize is that itís not just

the decimal points we want lined up, we want the tenths lined up, we want the hundredths

lined up, and we want the thousandths in the next column. Of course there wasnít any thousandths

here just hundredths but if I wanted to help my visualization I could insert this extra

zero that weíve talked about and now we have 30 thousandths which is the same thing as

3 ten-hundredths. Same thing here, beginning our thinking by lining up the decimal points,

weíre carrying it through to line up the tenths, hundredths, thousandths, and now ten-thousandths

and if you need or desire this appended zero thatís fine, just give it to yourself. Now

we add and youíre going to see here the extreme advantage over decimal fractions to pure fractions.

Because once we have line it up, all the like place values, then the rule youíre going

to have in another lesson or so is simply pretend like the decimal point isnít there

ñ of course, weíre really keeping them all in a line including the decimal point in the

answer, we keep it there and add as though this was not a fraction problem but really

a whole number problem. So weíre just adding as though we had a whole number problem but,

in fact, we have done a fraction problem. There is the great beauty of the invention

of the decimal system. It allows us to work fractions, which are quite messy, as though

they were whole numbers, which is quite easy. So here again note, we start our thinking

by saying line up the decimal points, which by the way we will do for addition and subtraction

but not multiplication and division, they have a different rule. But then following

through by actually lining up the like place values then temporarily forgetting that I

have a decimal point and then adding as though I have a whole number and Iím done working

the fractions. Isnít that neat? Now note the similarities between what we just did

by peeking ahead and this current lesson. In order decimal numbers weíre going to visually

line up starting with the decimal points lining up the like place values. So this is a 2 and

5 tenths but here again be careful, this whole number does have a decimal point. Itís just

that itís understood to be behind it and we donít write it there unless we need to.

Well now we need to because we want to line up the like places, so the units with units,

the tens in a separate column. Now here again, lining up the point, 092. Now thatís a units

digit, tenths digit, hundredths digit. Now with some experience you wonít bother writing

these zeros down, you will simply go down here, add, add the next column, if youíre

to add but if you are to order as we are here, as a beginner you will probably still find

it more convenient to append with zeros. Now for a brief moment we pretend like the decimal

points arenít there and, of course, in this case if the decimal points weíre there we

wouldnít have to worry about that zero either. Now itís really easy to find the smallest,

in this case it would be this one, which in its original form which we have to go back

to in the final analysis, is the smallest, which means it would be less than the next

smallest which we can see now is going to be this one which in its original form was

2.5 which would be smaller than the next which we now see 7 is certainly smaller than 15,

so itís original form is 7.05 which would be less than now the last one which is simply

15. And again make a mental note that all whole numbers indeed are also decimals and

the decimal is always here, never anywhere else always here on the right end, thatís

again a common mistake for many beginners. Itís always on the right end. And we avoided

our fractions. So note thereís a pattern here similar to what we just did for adding

and as you will find later, similar to subtraction. So be very, very cautious of the subtle similarities

as well as the differences so that you wonít get confused in the future. To show you the

importance of this, here is a problem like situations which occur very, very frequently

in all of science and technology, particularly chemistry. Letís say that you have a vial

which is supposed to contain this many grams of a particular substance, in this case charcoal.

Letís say a person, Jerry here, wishes to double check this so he weighs it on an extremely

sensitive scale and reading as well as he can, he determines that he has 4.21 grams,

which is not the same as this. Now if this is correct, is Jerry weighing too light or

too heavy. And so if we can to compare this weíre really asking is this smaller than

this or is this greater than this. And what Iím doing on paper ultimately we would just

do mentally, lining up the decimal points, that is the decimal places, theyíre the same

here, theyíre the same here. And this is the first one thatís bigger than this one

so this is the larger number so Jerry is weighing too heavily. We will look at this situation

very, very frequently and very closely in the next lesson because it might be that Jerryís

scale is incapable of reading past the nearest hundredth of a gram because thatís minutely

sensitive. So heíd need better scales to get down to here. So if his scales could only

read to the nearest hundredth and this is what it really was, we will ask this in the

next lesson, is this a reasonable measure of this when I have less accuracy. So our

next lesson, in fact, is going to be approximation and round off accuracy. Because, in fact,

when youíre measuring things in reality you never measure them exactly, thatís physically

impossible. So you can only measure things to certain amounts of accuracy depending upon

the kind of instrument youíre using at that particular moment. So bear that to the back

of your mind. That will become an important point in not only a future lesson in this

course but in all of, all of science. Hereís another, Margo and Josh each measured the

diameter of a coin and got 0.81 and 0.807 centimeter, respectively. Which measurement

is the smaller? Now again, a beginner might see that 807 is larger than 81 and say, hey

this is smaller, this is larger but here again letís line up the decimal places. So 0.81

is Margoís measure and Joshís is .80, lining the hundreds with the hundreds, 7. And now

if we wish to insert that extra zero thatís fine and we can see that, in fact, Joshís

is smaller not larger. Again, for those of you who are taking this program that might

eventually take a science class, this is a very frequent classroom experiment at the

beginning of the course. Itís where the instructor gives everyone a very accurate measurement

scale, centimeters, calipers, whatever, and send the same object all the way around the

room and ask everyone to measure it as carefully as they can and write it now. Now the same

object, the same kind of measurement skills and yet you will find that it is the rule

rather than the exception that most people will not measure precisely the same and your

science class will discuss why that is so. But again, you will find that they will be

close and itís important for us to determine later how close and in this lesson of the

different measures which is the smallest and which is the largest. And also be aware that

sometimes youíre trying to order things not from the smallest to the largest but from

the largest down to the smallest. So again, you line up beginning with the decimal places

but actually lining up the places not the point. So 1 ten, 1 unit rather, and 2 tenths,

no units, no tenths, 3 hundredths, 4 thousandths, and no units, 2 tenths, no hundredths, no

thousandths, 5 ten-thousandths and if it will help you to insert enough zeros so that you

can visually compare these easier. This is what we meant in the last tape by these zeros

being for looks only, cosmetics if you will. It just helps you to orient your thinking

in value they serve no purpose whatsoever. But now pretending the decimal points arenít

there, we can see, in fact, this is the largest 1.2. And with that out of the way our next

largest is this one which is .98. With it out of the way I can see this is my next largest

.2005. And finally that leaves us with the smallest which is .034. And notice in all

of these cases I did not write the zero in front of the decimal point but again I can

or I donít have to, the value remains the same. So you will find, I trust, that this

is an extremely simple lesson but you will also find that itís a remarkably important

one for the future. Till your next lesson, this is your host Bob Finnell. Good luck.

A course in arithmetic review. Produced at Portland Community College. As with fractions,

many people have hesitancy about ordering decimals, deciding which one is larger, which

is smaller. You would be surprised at how many people think this is the larger because

it begins with a 9 and this only with a 1 but as it turns out, this is the larger one.

Now after all, decimals are fractions. Letís write these. This is 125 thousandths but this

one is 936 and see the average person says, see thatís larger than that. But this is

936 ten-thousandths. And recall from fractions days we could not order fractions unless they

had the same denominator. And I can make this one thousand become ten-thousand by multiplying

top and bottom by 10 but if I do that, this top has become a hundred, one thousand, 250

ten-thousandths and this one, of course, is still 936 ten-thousandths. So in this form

of having common denominators is quite obvious that this is the larger number. So the secret

was in the form of having common denominators we only look at the top. Well in decimals

to say common denominators is the same thing as saying letís have the same number of places.

So this one has four places, this one only three. But do you remember that a tape or

so ago we said that with a decimal on the end you can append or take off zeros to your

convenience. In this case, it is one of our conveniences to append a zero. Now if the

two have the same number of decimal places, itís the same thing fraction-wise as having

common denominators. Now we can look at the two and we can see that 1250 is larger than

936 so that is the larger. So the secret for ordering decimals is to be sure that they

all have the same number of places, appending zeros if necessary to accomplish this. Then

pretend as though the decimal isnít there and order them by the way you see it and 1250

is larger than 936. Now it becomes even more obvious when we have a series of decimals

if we aligned them vertically and keep the decimal points in a line. So 1.2, this going

to be .056, this one would be 1.35, and the last one .0092. So keep the decimal points

in a line and keep aligned places, tenths in a line, hundredths in a line, thousandths

in a line, and ten-thousandths. Now if it will help you to visualize things better,

on the end append zeros either on paper or mentally until they all have the same number

of places. And now pretend like the decimal points arenít there and ask yourself which

is the smallest here. Well you can see quickly itís this one down here. This is the smallest,

then the next smallest would be the 560 here, then the next smallest 1350, then, of course,

that makes this the largest. With very little practice you can do this mentally without

going through this trickery but this is useful until youíre comfortable in beginning to

think in terms of size. Letís do another one while reviewing our symbols. Recall what

this symbol means? You see it more and more in your later math. This means greater than

that. So the question is this decimal greater than that decimal. Well this decimal has 1,2,3,4

places, so this one has 1,2,3, so we append with another zero. Now if we look at them

without the decimal points, this side seems to have 1250, whereas this side has only 375.

So indeed this one is bigger than this one. So the answer to this question is yes. That

is true. Now letís recall this symbol means less than. So what weíre asking is .48 less

than .051 and again, an inexperienced person would think this is 48 and this is 51 and

say of course itís true. But theyíre not taking into account the decimal system. Into

compared decimal numbers, they must have the same number of places. So thereís two, thereís

three, so I append a third place zero. Now ask the question, this is 480, this one is

only 51 so this is much bigger than that. So the answer to our question is no, thatís

not a true statement. Three service stations advertise unleaded gasoline. The prices are

as listed here. Which is the most economical? That is, which is least? And our technique

simply suggests to us to write each price with the decimal points lined up, keeping

the place values in a line, tenths, hundredths, thousandths, ten-thousandths. Then, if it

will assist you, append zeros on the end behind the decimal point, which you canít do with

whole numbers. Then at that point simply pretend like the decimal point isnít there and ask

yourself which of these numbers is the least. And one can see since whole numbers are easier

to judge, this one would be the least, and that is A. So A has the best price. Now, of

course, you can just see because of your sensitivity to numbers thatís well and good. This is

a device which helps you bridge the lack of that sensitivity until experience gives it

to you. Letís do another problem like that but at the same time developing some ideas

and language skills that will help us go from this lesson into addition and subtraction

of decimals which will follow this lesson. First, in order to find the order if we donít

have sensitivity to size just by looking at them, an inexperienced person would say this

is the smallest, this is next smallest, third smallest, and this is the largest. If you

didnít have that sensitivity at first we said this which Iím going to correct in just

a moment, we said to write each one lining up the decimals places, the decimal point.

So 1.005 and this is lining up the decimal point 2,3 and, of course, the zero in front

isnít necessary and the next one is point, lining up the decimal points, 507, and the

next one is .053. Now what we need to correct is the idea that itís really not the decimal

point that weíre lining up itís the place values. In fact, we want all the digits in

the tenths place lined up. We want all the digits in the hundredths place lined up. We

want all the digits in the thousandths place, and so on. Same thing with the whole numbers,

we want the digits in the ones place, then tens, then hundreds if we had them. So itís

really not the decimal points weíre lining up itís the places. The decimal points are

merely there to help me find where the places get started from. The reason I say that is

there is a tendency to make a careless mistake by some people, instead of writing .23 like

this, to write it perhaps like that and just leave a vacant spot. But I canít have the

hundredths place in the thousandths place here. All the hundredths places must be together

in the same position. Itís very important that you follow what Iím saying. Then append

zeros, if it will help your thinking, then also, if it will help your thinking, pretend

like the decimal points arenít there. So if there decimal points werenít there and

this zero is not necessary, nor is this, nor is this, or this. So therefore, we could see,

just knocking off temporarily ñ temporarily unnecessary decimal points, that this is most.

Well actually I couldnít knock this off so itís really the number.053. Itís the smaller

so it will be less than the next one and we can see now that 230 is certainly smaller

than 507 so this is my next smaller one. And, of course, I donít need that zero anymore

so there is no point in writing it. That will be smaller than my next one. And we can see

that 507 is certainly smaller than 1005. So this would be less than .507 and we have one

more left over which is 1.005. So we have these rearranged from the smaller to the larger

correctly and we said, to get our thinking onto paper, line up the decimal points but

we want to now realize that, in fact, weíre lining up the tenths, the hundredths, and

the thousandths, and so on. So in actuality itís decimal places of like values that must

be lined up. Keep that strongly at the back of your mind. In fact, letís carry that notion

right back to our whole number addition which we did at the beginning of this course. Remember

if we were adding a long string of numbers many of us learned the rule that we must write

these vertically and, as it was stated then, we must line up the right ends. Now I hope

youíre beginning to realize itís not just the right ends we want lined up, we want all

the units in a line and all the tens in a line, and all the hundreds, and all the thousands.

So, in fact, it was not the ends we wanted lined up that was just to help us get started

writing, itís the place values we want lined up. Therefore, if one were careless, as some

beginners tend to be with a situation like this, you might tend to write something like

this. You spread your digits around and you suddenly realize thatís illegal. I cannot

add tenths to hundredths and I cannot add hundreds to thousands. I can only add digits

from like places. What weíre saying now is itís the same with decimals. Thatís why

in chapter one, we made such a big deal, and we are now again, on place values, because

all the instructions that we give you are in terms, in fact, of place values. Letís

take a moment out and take a peek into a future lesson and youíll see why this is so important.

When weíre adding decimals rather than saying as we did with whole numbers to line up the

ends, weíll start our conversation by saying line up the decimal points vertically. So

this would be 1.03, this would be 0. ñ but now we want to realize is that itís not just

the decimal points we want lined up, we want the tenths lined up, we want the hundredths

lined up, and we want the thousandths in the next column. Of course there wasnít any thousandths

here just hundredths but if I wanted to help my visualization I could insert this extra

zero that weíve talked about and now we have 30 thousandths which is the same thing as

3 ten-hundredths. Same thing here, beginning our thinking by lining up the decimal points,

weíre carrying it through to line up the tenths, hundredths, thousandths, and now ten-thousandths

and if you need or desire this appended zero thatís fine, just give it to yourself. Now

we add and youíre going to see here the extreme advantage over decimal fractions to pure fractions.

Because once we have line it up, all the like place values, then the rule youíre going

to have in another lesson or so is simply pretend like the decimal point isnít there

ñ of course, weíre really keeping them all in a line including the decimal point in the

answer, we keep it there and add as though this was not a fraction problem but really

a whole number problem. So weíre just adding as though we had a whole number problem but,

in fact, we have done a fraction problem. There is the great beauty of the invention

of the decimal system. It allows us to work fractions, which are quite messy, as though

they were whole numbers, which is quite easy. So here again note, we start our thinking

by saying line up the decimal points, which by the way we will do for addition and subtraction

but not multiplication and division, they have a different rule. But then following

through by actually lining up the like place values then temporarily forgetting that I

have a decimal point and then adding as though I have a whole number and Iím done working

the fractions. Isnít that neat? Now note the similarities between what we just did

by peeking ahead and this current lesson. In order decimal numbers weíre going to visually

line up starting with the decimal points lining up the like place values. So this is a 2 and

5 tenths but here again be careful, this whole number does have a decimal point. Itís just

that itís understood to be behind it and we donít write it there unless we need to.

Well now we need to because we want to line up the like places, so the units with units,

the tens in a separate column. Now here again, lining up the point, 092. Now thatís a units

digit, tenths digit, hundredths digit. Now with some experience you wonít bother writing

these zeros down, you will simply go down here, add, add the next column, if youíre

to add but if you are to order as we are here, as a beginner you will probably still find

it more convenient to append with zeros. Now for a brief moment we pretend like the decimal

points arenít there and, of course, in this case if the decimal points weíre there we

wouldnít have to worry about that zero either. Now itís really easy to find the smallest,

in this case it would be this one, which in its original form which we have to go back

to in the final analysis, is the smallest, which means it would be less than the next

smallest which we can see now is going to be this one which in its original form was

2.5 which would be smaller than the next which we now see 7 is certainly smaller than 15,

so itís original form is 7.05 which would be less than now the last one which is simply

15. And again make a mental note that all whole numbers indeed are also decimals and

the decimal is always here, never anywhere else always here on the right end, thatís

again a common mistake for many beginners. Itís always on the right end. And we avoided

our fractions. So note thereís a pattern here similar to what we just did for adding

and as you will find later, similar to subtraction. So be very, very cautious of the subtle similarities

as well as the differences so that you wonít get confused in the future. To show you the

importance of this, here is a problem like situations which occur very, very frequently

in all of science and technology, particularly chemistry. Letís say that you have a vial

which is supposed to contain this many grams of a particular substance, in this case charcoal.

Letís say a person, Jerry here, wishes to double check this so he weighs it on an extremely

sensitive scale and reading as well as he can, he determines that he has 4.21 grams,

which is not the same as this. Now if this is correct, is Jerry weighing too light or

too heavy. And so if we can to compare this weíre really asking is this smaller than

this or is this greater than this. And what Iím doing on paper ultimately we would just

do mentally, lining up the decimal points, that is the decimal places, theyíre the same

here, theyíre the same here. And this is the first one thatís bigger than this one

so this is the larger number so Jerry is weighing too heavily. We will look at this situation

very, very frequently and very closely in the next lesson because it might be that Jerryís

scale is incapable of reading past the nearest hundredth of a gram because thatís minutely

sensitive. So heíd need better scales to get down to here. So if his scales could only

read to the nearest hundredth and this is what it really was, we will ask this in the

next lesson, is this a reasonable measure of this when I have less accuracy. So our

next lesson, in fact, is going to be approximation and round off accuracy. Because, in fact,

when youíre measuring things in reality you never measure them exactly, thatís physically

impossible. So you can only measure things to certain amounts of accuracy depending upon

the kind of instrument youíre using at that particular moment. So bear that to the back

of your mind. That will become an important point in not only a future lesson in this

course but in all of, all of science. Hereís another, Margo and Josh each measured the

diameter of a coin and got 0.81 and 0.807 centimeter, respectively. Which measurement

is the smaller? Now again, a beginner might see that 807 is larger than 81 and say, hey

this is smaller, this is larger but here again letís line up the decimal places. So 0.81

is Margoís measure and Joshís is .80, lining the hundreds with the hundreds, 7. And now

if we wish to insert that extra zero thatís fine and we can see that, in fact, Joshís

is smaller not larger. Again, for those of you who are taking this program that might

eventually take a science class, this is a very frequent classroom experiment at the

beginning of the course. Itís where the instructor gives everyone a very accurate measurement

scale, centimeters, calipers, whatever, and send the same object all the way around the

room and ask everyone to measure it as carefully as they can and write it now. Now the same

object, the same kind of measurement skills and yet you will find that it is the rule

rather than the exception that most people will not measure precisely the same and your

science class will discuss why that is so. But again, you will find that they will be

close and itís important for us to determine later how close and in this lesson of the

different measures which is the smallest and which is the largest. And also be aware that

sometimes youíre trying to order things not from the smallest to the largest but from

the largest down to the smallest. So again, you line up beginning with the decimal places

but actually lining up the places not the point. So 1 ten, 1 unit rather, and 2 tenths,

no units, no tenths, 3 hundredths, 4 thousandths, and no units, 2 tenths, no hundredths, no

thousandths, 5 ten-thousandths and if it will help you to insert enough zeros so that you

can visually compare these easier. This is what we meant in the last tape by these zeros

being for looks only, cosmetics if you will. It just helps you to orient your thinking

in value they serve no purpose whatsoever. But now pretending the decimal points arenít

there, we can see, in fact, this is the largest 1.2. And with that out of the way our next

largest is this one which is .98. With it out of the way I can see this is my next largest

.2005. And finally that leaves us with the smallest which is .034. And notice in all

of these cases I did not write the zero in front of the decimal point but again I can

or I donít have to, the value remains the same. So you will find, I trust, that this

is an extremely simple lesson but you will also find that itís a remarkably important

one for the future. Till your next lesson, this is your host Bob Finnell. Good luck.