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Math 20 - Lesson 41 A Portland community college mathematics telecourse.

A course in arithmetic review. Produced at Portland community college. Can you see that

just by the title alone that this lesson is going to be simply a subset of the previous

lesson? After all, fractions are implied divisions and we already had that. So the last lesson

which was a bit long will make this lesson to be that much shorter. So we simply read

this as 3 divided by 8 and write it accordingly and then we write the decimal points on the

end of the whole number. Since I'm dividing by a whole number the decimal point goes straight

up. I can append as many zeros as I wish and then simply divide. To save you time I will

do it separately. We find after three divisions it comes out evenly hence we have a terminating

decimal representative of the decimal. Again, all fractions are in disguise a division.

Another one, four divided by eleven so setting it up in its division form we get this and

doing the division we begin to get an unending quotient. However we can see the repetition,

the repetition is at 36 so we could write this as .36 with the bar over it or, whoever

is using this might make a statement like this: "Round to the nearest thousandth." So

in a case like that we go out to the nearest thousandth look only at the digit next to

it and that's 5 or more so we add one to the thousandth position which is four. Here we

have a rounded approximation of 4/11 accurate to the nearest thousandth. Some people will

show it with a symbol like this and others with a symbol like that. Both of these are

symbols which mean approximately equal to. Again, a fraction is really a division in

disuse. Another one- two and five sixteenths 2 5/16, that’s one way of saying it. Another

way of saying it is two and five divided by 16. (2. 5÷ 16) Decimal point is on the end,

shoot it straight up, add as many zeros as you think are in necessary and begin to divide.

In this case we get a terminating zero. So we can say that 2 5/16 is 2.3125 and in the

case, exactly. Here is an interesting fact regarding repeating or terminating decimals.

The fact is this- a fraction whose denominator has only 2's and or 5's as prime factors (that

is dividers) will result always in a terminating decimals. All others will be unending, repeating

decimals. Interesting fact. Let's look at some examples. If you divide 7 by 32 (7 ÷ 32)

it will come out to be .21875. It goes out quite a ways but at this point it fits; it

terminates. Notice that its dominator prime factors to 25 which means that two is the

only prime number that divides 32. The same thing with eleven five hundredths 11/500.

If you were to divide that out it would come out to .022 and it quits. If you were to prime

factor it you would get two to the second power times five to the third power (22 · 53

) hence the only prime numbers that will divide 500 is 2 and 5 and that's what our special

note said. If 2 and 5 are the only divisors (prime number wise) of the dominator it will

always terminate. Sometimes it will terminate rather soon, sometimes it will go quite a

ways before it terminates but it will, always, terminate. Here's a rather cute shortcut for

some fractions. Let's say we wished to change this one into a decimal. But first note if

I were to multiply top and bottom by 5- the top of course would be 35 and the bottom becomes

100; a power of 10. Then in that case we recall back a couple of lessons, we can simply make

the top smaller by two decimal places or .35 and sure enough if you divide 7 by 20 you

get .35, so sometimes you can sort of watch the denominators and ask if there is any number

you can multiply it by to make it a power of 10 because if I can, all I need to do is

move the decimal point instead of dividing and that is quite easy. That kind of fraction

occurs quite frequently in business fractions. Can you see that here? If I were to multiply

500 by 2 it becomes 1000 and 2 times the top is easy to do mentally. 10 carry the one,

three, six. But if I am multiplying by 1000 that is the same thing as making the top smaller

by one, two, three places which is .63 because in fact I don't need that end zero now. So

in your work, in other classes, particularly bookkeeping and some of those, watch for that

kind of a denominator and frequently you will be in a position to do this with a calculator.

Here they want it rounded to the nearest ten thousandth. So that's six and here's six and

thirty-seven divided by 89. So on your calculator 37 ÷ 89 = and you get of course a very long

decimal answer. But according to our instructions we don't need all that, we need it rounded

to the nearest ten thousandth so let's see: tenth, hundredth, thousandth, ten thousandths,

look only at the one to its right. It's four or less so I can drop it so there I have converted

that mixed number to an approximate decimal, accurate to the nearest ten thousandth. Let's

play with a couple of problems where this kind of skill is really very necessary. Consider

this one: A shop micrometer reads to the nearest thousandth of an inch. A bolt is supposed

to be 27/32 of an inch in diameter. What would that be on the micrometer? Well I don't have

a micrometer here but perhaps if you have seen one you can get the idea. Here's a bolt

and a micrometer is basically a tool like this C-clamp which you would tighten up on

the bolt and as you were doing that you would have a scale here that reads these turn around

in terms of the distance that it is going down. Because of that you can get this to

read very very accurate to the nearest thousandth of an inch rather routinely and in some specialized

cases to the nearest ten thousandth of an inch. The point is many standardized bolts

come marked with a diameter of 32/64 but most standard micrometers will be calibrated to

read in the decimal system. So you are trying to check the bolt to see if it really is this

diameter. Then what would you expect to see on the micrometer? So our problem is basically

this stated arithmetically and that is to simply convert 27/32 to a decimal accurate

to the nearest thousandth. Now to do that all we have to do is remember that this symbol

means "divided by." Let's also note this, 32 is divisible only by the prime number 2.

That means if I were to divide 27 by 32 it will terminate at some point. Well this read

27 divided by 32. Decimal point is on the end, go straight up, I want it to the nearest

thousandth so here is tenth, hundredth, thousandth, however in round off situations you always

go one digit further to see if I round up or back. In doing this division you would

get these four digits. Actually there is one more there but since we want it only to the

nearest thousandth all we need is one more digit and since that's five or more we would

add one to this. So that means if you were trying to read this and if this were actually

the diameter of that bolt then on the micrometer the best we could see is .844. It would be

closer to 4 than 3 because 7 is closer to 4 than it is down to 3. So you see by this-

Whenever, in a text book, you see a problem that looks like this, it seems to be rather

sterile and out of this world, they really do come from real world situations. Another

one. There are decimal rulers which read to the nearest hundredth of an inch then in such

a ruler what would 7/16 read as? Because most standard school rulers will read to the sixteenth

or 32nd but you might look around a shop the next time you are near one and see if you

can find one of these decimal rulers; they are fairly common. We simply again, remember

that this means divided by so that reads seven divided by sixteen 7 ÷ 16 and what the problem

is stating is that the ruler cannot read anything beyond 100 so whatever is beyond here really

is irrelevant to the ruler. But we do need one more digit here to find out if the ruler

will read to the next highest or the one down below so performing this division either by

hand or by calculator you would get .437 and so on but again we only need one more digit

to tell this is more than 5 therefore we round this up. So we can now claim that 7/16 is

approximately equal to the decimal number .44. That is closer to .44 than

it is to .43 so on our decimal ruler this is actually what we would see. We would see

that edge of our measuring item almost against the next four in the hundredths place; just

a squidge under it. Here's an interesting problem. A science book, most of them, gives

the value of this symbol which we call pi as 3.1416. We use this number a whole lot

when working with spheres and circles. Whereas another book says to use 22/7. Can one really

do this? As we are really asking is this - is 22/7 really equal to 3.1416?

Assuming this is correct, that is accurate to the nearest ten thousandths. Well let's

check this. You might recall many many elementary schools use this as a value of pi before you

got into decimals in the later grades. Let's divide 22 by 7 (22/7) and see if it comes

out to be this. So, 22 divided by 7, decimal point is here and goes straight up and this

actually divided and if I do so I get this quotient. Let's compare them and you will

actually get a feel of what to round off really means. We see they agree at the whole number.

They agree at the tenth. They agree at the hundredth but at the thousandth they disagree

however if I were to round this decimal value of Pi to the nearest thousandth this would

round to 2 of course this one would round to three doesn't it? So what we can really

say is that 22/7 is approximately equal to Pi and we can say more than that

we can tell you exactly how approximate it is. So the correct statement is that 22/7

is not equal to Pi however it is correct to say that 22/7 is approximately equal to Pi

to the nearest hundredth. So if this amount of accuracy is good enough you could use this

fraction as an approximator. If you need more accuracy than that then this would be not

good enough. By the way while we are here let me show you a cute device to remember

Pi to as many places as you need within normal usage. If you remember the phrase "May I have

a large container of coffee?” (a phrase many people use), and count the letters - three, point, one,

four, one, five, nine, two, six. (3.145926) out to that many places in fact that is what

Pi is equal to. Whereas 22/7 if you were to work that with more accuracy you would find

that it is approximately equal to that decimal (3.1428571) Again you can see that they agree

to the nearest hundredth. Beyond that, Pi and 22/7 are entirely different numbers. So

if you can remember this phrase, (an easy one to remember), it will give you Pi to quite

a few places. Let's see if we can verbalize this. Tenth, hundredth, thousandth, ten thousandths,

hundred thousandths, millionths, ten millionths so this phrase gives you Pi to the nearest

ten millionths. More accuracy then you would even use in most sciences. By the way a question

frequently asked of me in class at this point is "Where does that Pi, that number, come

from? How does that fit into mathematics?" Well if you don't know it's very simple and

very interesting. Well if you were to take any circle and start on the rim and measure its distance

around which we call its circumference; let's call that capital C. Then measure the distance

across the circle that goes through the center which we call the diameter and call that D

and divide these two (C/D). Depending on how accurate the circle is drawn you would come

out close to that value of Pi 3.1416 well actually 159 etc. . . . Of course that rounds

to six at the nearest ten thousandths. The more accurate that circle was the closer it

would be to Pi. The size of the circle had nothing to do with it. If you were to take

a smaller circle and measure its circumference (c) and its diameter which would also be smaller

(d) and divide these two, the ratio would again be that number Pi. So the

definition Pi is nothing more than the ratio or the number you get when you divide the

length of the circumference by the diameter and using very advanced math we can prove

that this number which goes on and on forever, is approximately equal to that number. And

interestingly enough that number, 3.1415926 etc. . . . is not a repeating decimal and

is not a terminating decimal. It's a very interesting number and advanced mathematics

spends a lot of time studying and using it. But never the less, here we have a decimal

number that goes on forever and very frequently we need that number and we of course cannot

go on forever so we must specify where we are going to quit and that is nothing more

than this and the last lesson is about and that is rounding decimal numbers. Therefore

it is necessary that by the completion of this lesson that you realize to ask a simple

question like changing a fraction (two of them here) into a decimal is it not enough?

Because if you were to do that by performing the division, in this case 7/19 (seven divided

by nineteen) or in this case 3/40 (three divided by forty) we now need to realize that as I

do either one of these that I realize that I will get either a decimal that goes on forever,

(with a repeating cycle of course somewhere in it), or a decimal that might go on a very

long way. For instance in this particular case since 40 (the denominator) has only 2's

and 5's as prime divisors, this will be a terminating decimal somewhere. It might go

out a long way and this one will never terminate. It will keep on giving me digits forever and

ever. So whenever I am converting a fraction to a decimal and I intend to use that answer

I have to quit somewhere. So at this point you realize that it is not enough to say simply

to change to a decimal. You must specify where I am to quit, that is where am I to round?

How much accuracy is needed? And that has to be specified to you external to the problem,

there are no math rules telling you where to round off; it depends what you want to

use the decimal equivalent for. So in this particular case let's assume that somehow

someone has determined it needs to be accurate to the nearest ten thousandths. Therefore

as we perform the division we will go out to the nearest tenth, hundredth, thousandth,

ten thousandth and one extra digit (we don't need this now) to find out if I round up or

down. In this case, if I were to follow through with that I would get this and the next digit

is next than four or less so I round down so 7/19 to the nearest ten thousandths is

the decimal point .3684 and it's very important to realize that this is not exactly equal

to that. It's closer to this than .3683 or .3685 that it is the closest approximation

to that place. Then doing the same thing to this fraction that we wish to have as a decimal

accurate to the nearest ten thousandth we would get a situation where it actually terminated

before I got to the nearest ten thousandths however as I wrote this more likely I would

not terminate it here but actually to insert a zero because when I say that it terminates,

what I am really meaning is that I get zeros as my repeating digit from that point on.

So in a sense a terminating decimal is one in which zero is the reporting portion. If

you were to give an answer to somebody that means you are going to give them this without

this so they don't know that it came from a fraction that just happened to terminate

at the nearest thousandths. If you needed ten thousandths you need to tell the person

that you are going to give this to that hey, that ten thousandths digit is a legitimate

and accurate zero. So sometimes even though a zero does not contribute to a value of a

number it is necessary to keep it in order to convey how much accuracy you have to the

person to whom you are going to have this number. And do remember that if whole numbers,

mixed numbers are involved that the whole number part doesn't enter into the computation.

It's simply the whole number of the decimal. Then to find the decimal portion we work only

with the proper fraction portion so 3/11 three divided by eleven. In this case let's see

if a pattern comes out. We haven't followed through too many to find that pattern but

if we wanted the nearest hundredth we would go only to the hundredth and then one extra

to see where the round off it. So let's see eleven into thirty is twice, twenty-two, eight,

eleven into eighty is seven, seventy- seven, three, eleven into 30 is twice, then at this

point you can see I've got 30 here, then 80, 30 here, then 80, so in fact it's the 27 that

is going to repeat but in fact we want it only to the nearest hundredth. The digit next

to it is 4 or less so I drop them so this is approximately equal to 2.27 (? 2.27). If,

however, I were to say this, 2.27 with a bar over it, for now we know that the bar means

to go on with this pattern forever then we can say more than being approximately. We

can even say it's exactly equal. So that's a convenient notation isn't it? So this lesson

is a very simple one, one of the simpler you have had. I give you lots of practice of division

and of course rounding off. But it's one that's used a whole lot in shops, in industry and

to be surely in science. In the next lesson we'll show you yet another way of showing

a fraction as a decimal in an exact way; a method you perhaps haven't seen before but

is used frequently in stock markets. So, until then this is your host Bob Finel- Good Luck.

A course in arithmetic review. Produced at Portland community college. Can you see that

just by the title alone that this lesson is going to be simply a subset of the previous

lesson? After all, fractions are implied divisions and we already had that. So the last lesson

which was a bit long will make this lesson to be that much shorter. So we simply read

this as 3 divided by 8 and write it accordingly and then we write the decimal points on the

end of the whole number. Since I'm dividing by a whole number the decimal point goes straight

up. I can append as many zeros as I wish and then simply divide. To save you time I will

do it separately. We find after three divisions it comes out evenly hence we have a terminating

decimal representative of the decimal. Again, all fractions are in disguise a division.

Another one, four divided by eleven so setting it up in its division form we get this and

doing the division we begin to get an unending quotient. However we can see the repetition,

the repetition is at 36 so we could write this as .36 with the bar over it or, whoever

is using this might make a statement like this: "Round to the nearest thousandth." So

in a case like that we go out to the nearest thousandth look only at the digit next to

it and that's 5 or more so we add one to the thousandth position which is four. Here we

have a rounded approximation of 4/11 accurate to the nearest thousandth. Some people will

show it with a symbol like this and others with a symbol like that. Both of these are

symbols which mean approximately equal to. Again, a fraction is really a division in

disuse. Another one- two and five sixteenths 2 5/16, that’s one way of saying it. Another

way of saying it is two and five divided by 16. (2. 5÷ 16) Decimal point is on the end,

shoot it straight up, add as many zeros as you think are in necessary and begin to divide.

In this case we get a terminating zero. So we can say that 2 5/16 is 2.3125 and in the

case, exactly. Here is an interesting fact regarding repeating or terminating decimals.

The fact is this- a fraction whose denominator has only 2's and or 5's as prime factors (that

is dividers) will result always in a terminating decimals. All others will be unending, repeating

decimals. Interesting fact. Let's look at some examples. If you divide 7 by 32 (7 ÷ 32)

it will come out to be .21875. It goes out quite a ways but at this point it fits; it

terminates. Notice that its dominator prime factors to 25 which means that two is the

only prime number that divides 32. The same thing with eleven five hundredths 11/500.

If you were to divide that out it would come out to .022 and it quits. If you were to prime

factor it you would get two to the second power times five to the third power (22 · 53

) hence the only prime numbers that will divide 500 is 2 and 5 and that's what our special

note said. If 2 and 5 are the only divisors (prime number wise) of the dominator it will

always terminate. Sometimes it will terminate rather soon, sometimes it will go quite a

ways before it terminates but it will, always, terminate. Here's a rather cute shortcut for

some fractions. Let's say we wished to change this one into a decimal. But first note if

I were to multiply top and bottom by 5- the top of course would be 35 and the bottom becomes

100; a power of 10. Then in that case we recall back a couple of lessons, we can simply make

the top smaller by two decimal places or .35 and sure enough if you divide 7 by 20 you

get .35, so sometimes you can sort of watch the denominators and ask if there is any number

you can multiply it by to make it a power of 10 because if I can, all I need to do is

move the decimal point instead of dividing and that is quite easy. That kind of fraction

occurs quite frequently in business fractions. Can you see that here? If I were to multiply

500 by 2 it becomes 1000 and 2 times the top is easy to do mentally. 10 carry the one,

three, six. But if I am multiplying by 1000 that is the same thing as making the top smaller

by one, two, three places which is .63 because in fact I don't need that end zero now. So

in your work, in other classes, particularly bookkeeping and some of those, watch for that

kind of a denominator and frequently you will be in a position to do this with a calculator.

Here they want it rounded to the nearest ten thousandth. So that's six and here's six and

thirty-seven divided by 89. So on your calculator 37 ÷ 89 = and you get of course a very long

decimal answer. But according to our instructions we don't need all that, we need it rounded

to the nearest ten thousandth so let's see: tenth, hundredth, thousandth, ten thousandths,

look only at the one to its right. It's four or less so I can drop it so there I have converted

that mixed number to an approximate decimal, accurate to the nearest ten thousandth. Let's

play with a couple of problems where this kind of skill is really very necessary. Consider

this one: A shop micrometer reads to the nearest thousandth of an inch. A bolt is supposed

to be 27/32 of an inch in diameter. What would that be on the micrometer? Well I don't have

a micrometer here but perhaps if you have seen one you can get the idea. Here's a bolt

and a micrometer is basically a tool like this C-clamp which you would tighten up on

the bolt and as you were doing that you would have a scale here that reads these turn around

in terms of the distance that it is going down. Because of that you can get this to

read very very accurate to the nearest thousandth of an inch rather routinely and in some specialized

cases to the nearest ten thousandth of an inch. The point is many standardized bolts

come marked with a diameter of 32/64 but most standard micrometers will be calibrated to

read in the decimal system. So you are trying to check the bolt to see if it really is this

diameter. Then what would you expect to see on the micrometer? So our problem is basically

this stated arithmetically and that is to simply convert 27/32 to a decimal accurate

to the nearest thousandth. Now to do that all we have to do is remember that this symbol

means "divided by." Let's also note this, 32 is divisible only by the prime number 2.

That means if I were to divide 27 by 32 it will terminate at some point. Well this read

27 divided by 32. Decimal point is on the end, go straight up, I want it to the nearest

thousandth so here is tenth, hundredth, thousandth, however in round off situations you always

go one digit further to see if I round up or back. In doing this division you would

get these four digits. Actually there is one more there but since we want it only to the

nearest thousandth all we need is one more digit and since that's five or more we would

add one to this. So that means if you were trying to read this and if this were actually

the diameter of that bolt then on the micrometer the best we could see is .844. It would be

closer to 4 than 3 because 7 is closer to 4 than it is down to 3. So you see by this-

Whenever, in a text book, you see a problem that looks like this, it seems to be rather

sterile and out of this world, they really do come from real world situations. Another

one. There are decimal rulers which read to the nearest hundredth of an inch then in such

a ruler what would 7/16 read as? Because most standard school rulers will read to the sixteenth

or 32nd but you might look around a shop the next time you are near one and see if you

can find one of these decimal rulers; they are fairly common. We simply again, remember

that this means divided by so that reads seven divided by sixteen 7 ÷ 16 and what the problem

is stating is that the ruler cannot read anything beyond 100 so whatever is beyond here really

is irrelevant to the ruler. But we do need one more digit here to find out if the ruler

will read to the next highest or the one down below so performing this division either by

hand or by calculator you would get .437 and so on but again we only need one more digit

to tell this is more than 5 therefore we round this up. So we can now claim that 7/16 is

approximately equal to the decimal number .44. That is closer to .44 than

it is to .43 so on our decimal ruler this is actually what we would see. We would see

that edge of our measuring item almost against the next four in the hundredths place; just

a squidge under it. Here's an interesting problem. A science book, most of them, gives

the value of this symbol which we call pi as 3.1416. We use this number a whole lot

when working with spheres and circles. Whereas another book says to use 22/7. Can one really

do this? As we are really asking is this - is 22/7 really equal to 3.1416?

Assuming this is correct, that is accurate to the nearest ten thousandths. Well let's

check this. You might recall many many elementary schools use this as a value of pi before you

got into decimals in the later grades. Let's divide 22 by 7 (22/7) and see if it comes

out to be this. So, 22 divided by 7, decimal point is here and goes straight up and this

actually divided and if I do so I get this quotient. Let's compare them and you will

actually get a feel of what to round off really means. We see they agree at the whole number.

They agree at the tenth. They agree at the hundredth but at the thousandth they disagree

however if I were to round this decimal value of Pi to the nearest thousandth this would

round to 2 of course this one would round to three doesn't it? So what we can really

say is that 22/7 is approximately equal to Pi and we can say more than that

we can tell you exactly how approximate it is. So the correct statement is that 22/7

is not equal to Pi however it is correct to say that 22/7 is approximately equal to Pi

to the nearest hundredth. So if this amount of accuracy is good enough you could use this

fraction as an approximator. If you need more accuracy than that then this would be not

good enough. By the way while we are here let me show you a cute device to remember

Pi to as many places as you need within normal usage. If you remember the phrase "May I have

a large container of coffee?” (a phrase many people use), and count the letters - three, point, one,

four, one, five, nine, two, six. (3.145926) out to that many places in fact that is what

Pi is equal to. Whereas 22/7 if you were to work that with more accuracy you would find

that it is approximately equal to that decimal (3.1428571) Again you can see that they agree

to the nearest hundredth. Beyond that, Pi and 22/7 are entirely different numbers. So

if you can remember this phrase, (an easy one to remember), it will give you Pi to quite

a few places. Let's see if we can verbalize this. Tenth, hundredth, thousandth, ten thousandths,

hundred thousandths, millionths, ten millionths so this phrase gives you Pi to the nearest

ten millionths. More accuracy then you would even use in most sciences. By the way a question

frequently asked of me in class at this point is "Where does that Pi, that number, come

from? How does that fit into mathematics?" Well if you don't know it's very simple and

very interesting. Well if you were to take any circle and start on the rim and measure its distance

around which we call its circumference; let's call that capital C. Then measure the distance

across the circle that goes through the center which we call the diameter and call that D

and divide these two (C/D). Depending on how accurate the circle is drawn you would come

out close to that value of Pi 3.1416 well actually 159 etc. . . . Of course that rounds

to six at the nearest ten thousandths. The more accurate that circle was the closer it

would be to Pi. The size of the circle had nothing to do with it. If you were to take

a smaller circle and measure its circumference (c) and its diameter which would also be smaller

(d) and divide these two, the ratio would again be that number Pi. So the

definition Pi is nothing more than the ratio or the number you get when you divide the

length of the circumference by the diameter and using very advanced math we can prove

that this number which goes on and on forever, is approximately equal to that number. And

interestingly enough that number, 3.1415926 etc. . . . is not a repeating decimal and

is not a terminating decimal. It's a very interesting number and advanced mathematics

spends a lot of time studying and using it. But never the less, here we have a decimal

number that goes on forever and very frequently we need that number and we of course cannot

go on forever so we must specify where we are going to quit and that is nothing more

than this and the last lesson is about and that is rounding decimal numbers. Therefore

it is necessary that by the completion of this lesson that you realize to ask a simple

question like changing a fraction (two of them here) into a decimal is it not enough?

Because if you were to do that by performing the division, in this case 7/19 (seven divided

by nineteen) or in this case 3/40 (three divided by forty) we now need to realize that as I

do either one of these that I realize that I will get either a decimal that goes on forever,

(with a repeating cycle of course somewhere in it), or a decimal that might go on a very

long way. For instance in this particular case since 40 (the denominator) has only 2's

and 5's as prime divisors, this will be a terminating decimal somewhere. It might go

out a long way and this one will never terminate. It will keep on giving me digits forever and

ever. So whenever I am converting a fraction to a decimal and I intend to use that answer

I have to quit somewhere. So at this point you realize that it is not enough to say simply

to change to a decimal. You must specify where I am to quit, that is where am I to round?

How much accuracy is needed? And that has to be specified to you external to the problem,

there are no math rules telling you where to round off; it depends what you want to

use the decimal equivalent for. So in this particular case let's assume that somehow

someone has determined it needs to be accurate to the nearest ten thousandths. Therefore

as we perform the division we will go out to the nearest tenth, hundredth, thousandth,

ten thousandth and one extra digit (we don't need this now) to find out if I round up or

down. In this case, if I were to follow through with that I would get this and the next digit

is next than four or less so I round down so 7/19 to the nearest ten thousandths is

the decimal point .3684 and it's very important to realize that this is not exactly equal

to that. It's closer to this than .3683 or .3685 that it is the closest approximation

to that place. Then doing the same thing to this fraction that we wish to have as a decimal

accurate to the nearest ten thousandth we would get a situation where it actually terminated

before I got to the nearest ten thousandths however as I wrote this more likely I would

not terminate it here but actually to insert a zero because when I say that it terminates,

what I am really meaning is that I get zeros as my repeating digit from that point on.

So in a sense a terminating decimal is one in which zero is the reporting portion. If

you were to give an answer to somebody that means you are going to give them this without

this so they don't know that it came from a fraction that just happened to terminate

at the nearest thousandths. If you needed ten thousandths you need to tell the person

that you are going to give this to that hey, that ten thousandths digit is a legitimate

and accurate zero. So sometimes even though a zero does not contribute to a value of a

number it is necessary to keep it in order to convey how much accuracy you have to the

person to whom you are going to have this number. And do remember that if whole numbers,

mixed numbers are involved that the whole number part doesn't enter into the computation.

It's simply the whole number of the decimal. Then to find the decimal portion we work only

with the proper fraction portion so 3/11 three divided by eleven. In this case let's see

if a pattern comes out. We haven't followed through too many to find that pattern but

if we wanted the nearest hundredth we would go only to the hundredth and then one extra

to see where the round off it. So let's see eleven into thirty is twice, twenty-two, eight,

eleven into eighty is seven, seventy- seven, three, eleven into 30 is twice, then at this

point you can see I've got 30 here, then 80, 30 here, then 80, so in fact it's the 27 that

is going to repeat but in fact we want it only to the nearest hundredth. The digit next

to it is 4 or less so I drop them so this is approximately equal to 2.27 (? 2.27). If,

however, I were to say this, 2.27 with a bar over it, for now we know that the bar means

to go on with this pattern forever then we can say more than being approximately. We

can even say it's exactly equal. So that's a convenient notation isn't it? So this lesson

is a very simple one, one of the simpler you have had. I give you lots of practice of division

and of course rounding off. But it's one that's used a whole lot in shops, in industry and

to be surely in science. In the next lesson we'll show you yet another way of showing

a fraction as a decimal in an exact way; a method you perhaps haven't seen before but

is used frequently in stock markets. So, until then this is your host Bob Finel- Good Luck.