Math 20 - Lesson 2

A Portland Community College mathematics course.
The course in arithmetic review
produced at the Portland Community College.
This lesson continues to develop a vocabulary
which will be used in later lessons
to explain what would otherwise
seem to be complicated processes.
Here you have the place value name of a number.
Forty five thousand, six hundred, thirty two. [ 45,632 ]
Now let's ask ourselves,
what does a 4 in this place really mean?
I really don't have a 4.
What I have is a 4 times its place value of that spot
which is 10,000. [ 4 x 10,000 ]
And in addition, literally in addition [+ ]
I have a 5 times, [ 4 x 10,000 + 5 x ]
but the 5 is in the one thousand's position.
And in addition to that, I have a 6 times,
It's in the one hundred's place position, [ + 6 x 100 ]
and in addition to that, I have a 3 tens,
3 times 10, [ + 3 x 10 ]
and in addition to that I have 2 ones. [ 2 x 1 ]
This way of looking at it forces us
to look at the individual pieces.
Pretty much as though
if you were repairing a carburetor or souping it up,
you'd have to tear it apart and look at the pieces,
which is what we're doing here.
We could have shortcutted this a little bit
by writing this 4 times 10,000 as 40,000, [ 4 x 10,000 = 40,000 ]
which is what the 4 means [ 40,000 ] by being in this position.
And this 5 times 1,000 is 5,000, [ 5 x 1,000 = 5,000 ]
and see, by placing the 5 in the thousand's position,
it ceases to be 5 and it becomes 5,000.
This 6, by virtue of its position, has become 600.
[6 x 100 = 600 ]
This 3 by being in the 10's position has become 30.
[ 3 x 10 = 30 ]
And the 2 has become still 2. [ 2 x 1 = 2]
This way of expressing a number,
we call
the 'expanded form' of that place value name.
Sometimes rather than writing it in numeral form
we'll write in it word form and get
this:
This is just another way of saying with words
what we are saying here with place value notation.
So in future lessons,
as we need to look at the parts of a number,
this is one of the ways we will choose to do it,
and this will help us to stress the value that a digit gets
by occupying a certain position.
This lesson also wishes to introduce you to new symbols
which are used to denote the relationship between two numbers.
You're already used to this in the case of the equal symbol [=]
which tells us that this expression
and this expression stand for the same number.
But often you have two expressions,
which don't stand for the same number.
So for those cases we need a new symbol.
135 is greater than the number 80, [ 135 > 80 ]
and that's what this symbol will mean: [>] greater than.
And 2 is a lesser number than 5
so this symbol [<] will mean: less than.
[>] Greater than.
And a convenient way of remembering that
is that this spread
is greater than this spread.
So this is a greater number than that.
And this symbol [<] means less than,
and this spread is less than this spread.
Some instructors like to tell their students
that the point points to the smaller number
and that allows you to remember it rather nicely.
A device which will be used frequently
to explain how numbers relate and work together
is to array the numbers on a line
going from the smaller progressively towards the larger.
When we show numbers in this manner,
we'll refer to this as a number line.
Now as a definition on the number line
any number to the left of another number
by definition now will be the smaller number
even if we don't know what it is.
So in this case if 'a' stands for 'any number,'
and 'b' stands for 'any other number,'
and all that we know about them
is that this is to the left of that on the number line,
then by agreement, this will be less than that.
If the number a is less than number b,
then of course the number b is greater than the number a,
so we can say it this way:
This number line will be a very convenient device
when you have completed this course
and begin to move into algebra.
You're going to see us
begin to use letters to stand for numbers quite frequently,
which seems to frighten a beginning algebra student
until they realize they're not really letters;
they're just marks on a piece of paper,
which mean the phrase 'some number' and nothing else.
For instance, we could use the letter 'r' to stand for rate
and of course that's the number once you measure it.
We could use 'v' to stand for the number measure of velocity.
We could use capital 'A' or little 'a' for 'area.'
We could use 'a' for 'amount.'
What will be important is to realize they're simply marks
that stand for the phrase 'some number.'
So if I use a capital 'A,' that will be as different in my use
as 'a' from a small 'a' as a 'b' or 'm' is.
So don't confuse these.
In algebra and in arithmetic,
these [A and a] are two entirely different marks
when referring to some number.
And the last point this lesson wishes to address
is to come to grips with how to round off.
This symbol [≈] we'll use frequently throughout these tapes,
and it simply means 'approximately equal to.'
Some books, rather than using the double squiggles,
will use an equal sign with a dot above it [≐]
In either case, they mean 'approximately equal to.'
Now if I were to say that my city has 387,256 people,
by the time you are watching this tape, this number has changed.
As a matter of fact
our population is changing every single minute.
It will change in groups of ten every single day,
in hundreds by the week.
So perhaps the only number that will remain stable
over several years is up here in the ten thousands position.
So we perhaps would like to have this number
just rounded to that stable position
at the nearest ten thousand
or let's say, the state population to the nearest ten thousand.
So let's come to grips now
with the round off rules we will use throughout this course.
So let's add to our formal notes
the rules of round off as we will use them.
The first step is to determine
the place at which you need to or wish to round off.
In the text exercise this will simply be given to you.
In our case we were asked to round
to the nearest ten thousand position.
To the nearest ten thousands position.
See?
Ones,
tens,
hundreds,
thousands,
ten thousands.
Next look to just that one digit to its right.
Look to that one digit to its right.
All the others have nothing to do with it.
So don't consider them at all at this stage.
Just that one digit.
Up to this point, the amount that we're going to keep
starting from the front working back to the place
we wish, at which we wish to round.
Now looking at the digit to its immediate right,
if it is 5 or more, round up.
That is, add one.
In this case the digit immediately to the right
of the round off position is 5 or more.
So I will round this up,
which means that this 8 is changed to 9.
Now in this position down here,
the digit immediately to the right of the round off position
is not 5 or more.
So in that case, if 4 or less
and by rounding down, we'll mean leave it as it is.
This digit is 4 or less, so I leave that as it is.
Then replace all digits to the right by zeros.
So these digits here are replaced by zeros.
These digits here are replaced by zeros.
So that we can now say the number:
12,354,958 Twelve million, three hundred fifty four,
12,354,958 nine hundred fifty eight.
is approximately equal to 12,350,000. [12,354,958 ≈ 12,350,000]
twelve million, three hundred fifty thousands [ 12,350,000 ]
See, we need those zeros to keep the magnitude of the number
because if they weren't there, this number for sure
is nowhere close to being approximately equal to that.
Same thing here.
The number 387,256 is approximately equal to 390,000.
Now we use approximations almost every day.
When you give population of your city,
you're approximating it.
When you give the distance from the earth to the sun,
92,800,000 ninety two million, eight hundred thousand,
that's merely an approximation.
in this case, rounded to the nearest one hundred thousand.
These are the round off rules used most frequently
in this, and most courses you'll take.
There are other kinds of round off rules
in some more advanced courses of study.
These are the ones, however, used most frequently,
especially with calculators.
So it's important that you learn them as quickly as possible.
Quickly let's review them.
Determine the place at which you wish to round off.
Look to adjust that one digit to its right.
If it is 5 or more, round up.
If it is 4 or less, round down,
and replace all digits
to the right of that round off position by zeros.
It's very important that you learn and use these rules
as soon as possible.
Let's relate that to calculators.
Now and then later in the course we will be using calculators,
and it's important to realize
that some calculators round off naturally,
that is, it's built right into the calculators,
and other calculators do not,
so if you use a calculator,
let's check it now to see if it rounds off or not.
Let's take a peek ahead to something we hope you know,
and that is conversion of fractions to decimals,
and that simply means 2 divided by 3.
And if you were to do this by hand without the calculator,
you would get .666 and these 6s go on forever.
Now in later chapters
we're going to look at this in more detail,
so we're actually peeking ahead
to something that we will study later.
But let's assume we can do that for a moment,
and since this is a review course,
we presume that you do in fact know this,
but just perhaps need to review.
We will assume it just for the minute here.
No matter where you quit, the digit before it is 5 or more,
so if we wanted to round here, since this is 5 or more,
we would add 1 to this and make it 7.
So, where we quit,
if we rounded it would round up to 7.
So let's check that with your calculator.
This on the calculator punch a 2, then punch a division [2][÷]
then punch a 3 [3] and see what we get.
See if we get the string of 6s
or see if on the end of our calculator we get the 7.
If we get 6s all the way to the end,
that means the calculator is not going to round.
If we get a 7, that tells us that that particular calculator
is going to round off by these rules.
I'm going to use that idea to check these two calculators
to see if either of them rounds off.
So on this small, inexpensive, four-function calculator,
I'll punch two, [2]
divide by [÷]
three, [3]
equals, [=]
and you see, it has .666 all the way to the end.
If this were rounding at this position,
it should have rounded this to a 7.
So this calculator does not round off
at the last indicated place.
This calculator, let's see [2], divide by [÷], [3] equals [=],
and see it has .666667.
So this is a calculator that I know
that the last digit is possibly a rounded digit.
So now as I use this calculator
I'll remember this one doesn't round. This one does.
You might wish to check your calculator.
During this lesson
we will concentrate on rounding whole numbers, not decimals.
Let's say you're buying a second-hand car
and the price tag is 4,325 Four thousand three hundred twenty five
See here's the place value name
and what you heard me say:
four thousand three hundred twenty five is the word name,
hence our last lesson.
If we wanted to round it to the nearest hundred
our first chore is to find the hundreds place.
So here is units tens, hundreds.
Look only at the digit just before it and ask: Is this 5 or more?
The answer is no.
Then I leave this hundreds digit alone,
everything in front of it alone,
and everything behind it I turn to zero.
So if somebody were to say
how much are you paying for that car?
You would say [≈] about $4300,
and that's, in fact, the way we communicate. Isn't it?
Usually we don't tell them the price
right down to the nearest dollar.
We give it to the nearest hundred
because that's sort of where the money
becomes rather important to us.
Another car might be priced at $3289,
and if we wish to round that to the nearest hundred,
again we find the nearest hundreds position
units,
tens
hundred
Look at the one just before it.
It is 5 or more, so we'll add 1 to this digit, making it 3,
leaving the ones in front of it alone,
and the ones behind it we again, replace by zero.
So we'd say, this car cost approximately $3300,
[$3,300] or 3 thousand 300 dollars,
and you can see that this number is really closer to 3300 dollars
than it is 3200 dollars.
After all, $89 is almost the next hundred, not the one down,
so you see this rounding off begins to make sense.
Where you have to be careful with round off
is those situations where you might be tempted
not to follow this simple rule
that is determine where you want to round.
If this is 4 or more, leave these alone,
and replace these by zero.
If on the other hand
the place before where you want to round is 5 or more
you add 1 to this, leave these alone,
and below that again, you replace by zero.
Consider this rather strange situation.
We have been given the number 1,999,875
[1,999,875] one million, nine hundred and ninety nine thousand,
[1,999,875] eight hundred seventy five
and we desire to round it to the nearest thousand.
So there is my unit, ten
hundred
thousand.
So we wish to round here.
We look at the digit just to its right, which is 5 or more.
So we wish to round add 1 to this digit
and these [,875] will be replaced by zeros.
See if I add 1 to 9, that forces me to get 10,
which is zero, carry the 1,
but that forces me to another 10,
which is zero, carry the 1.
That forces me to another 10, which is zero carry the 1.
So to the nearest one thousand
this number is approximately equal to two million. [2,000,000]
So these zeros were place holder zeros
acquired by rounding these up.
These were zeros, which simply occurred
because of the computation.
One of the difficulties of rounding off is,
if you just saw this number,
you can't tell by looking at the number
whether it was rounded here, here, here, or here.
All you know is the number is close to 2,000,000.
Somebody would have to verbally tell you
that that's the number to the nearest thousand.
So if the digit to the right
of where you wish to round is 5 or more,
you add 1 to this always
and simply let the chips fall as they will, so to speak.
So be very careful when you see numbers like this.
And also never forget
to replace the numbers at which you are rounding,
below which you're rounding, by place holder zeros;
otherwise, you're saying that this number is close to 2,000
and that's patently a false statement.
So what you finish with must be close to,
that's what this symbol [≈] says, to the number you started with.
If that isn't so,
then you know you have made a mistake in digit replacement
by place holder zeros.
So be cautious of that idea.
Before we leave each other,
let's review several other ideas
that were important to this lesson.
Let's say we have two sheets, a green and orange one,
each covering a number.
We won't let you see the number,
but we do tell you this is true about the two numbers.
That even though we don't know what they are,
what can we say about this one relative to this one?
Or in short, what does this mark say?
And this mark says that this number is less than this number.
So this number could have perhaps been 25,
then whatever this number is,
if this is less than it
and this number must be greater than that.
Say 31.
So you will have many problems in your lesson
giving you statements like this
asking you is this a true statement, or not?
So this is saying 25 new symbol less than 31, yes.
That's true.
Is this statement true?
Well, let's read it.
459, and now it's just a matter
this is the greater spread than this,
so we're asking: Is 459 greater than 1008? [ 459 > 1008 ]
And of course the answer is no.
So this part of our lesson
is simply to get you used to this symbol,
which reads 'greater than,' [ > ]
and this symbol [ < ] which reads 'less than.'
Backing off one more step in this particular lesson,
let's not forget what we mean by the phrase 'expanded form,'
and that's to write the value of each digit by itself.
So this digit, because it's in the thousand's place,
is now worth not 5,
but 5000.
And. 'And' of mathematics translates to 'plus' [+]
This 3 because it's in the hundred's place, is now worth 300,
'and' [+] 8 is in the ten's place, so it's now worth 80
'and' [+] 6.
Some people want to look upon it in this form.
Others will actually want to emphasize the thousands,
hundreds, tens, etc., so they might say 5 one thousands,
3 one hundreds, [ 3 x 100 ]
8 tens, [ 8 x 10 ]
and 6 ones. [ 6 x 1 ]
We will refer to each of them as 'expanded form.'
So if we ask for expanded form, give us either this or this.
Both will be acceptable as a meaning to this phrase.
Let's turn that question around
and ask ourselves what the place value name is
for this expanded form.
So if we were to concentrate on places,
we have first the one's place,
tens,
hundreds,
one thousand,
ten thousand,
hundred thousand,
one million etc.
So this says I have 8 ones,
7 hundreds, but here's my hundreds place,
and 5 ten thousands.
So here's my one thousand 10 thousand.
And I have 5 of those.
And of course these in between spots we have to fill
with a place holder zero
so we won't lose track of our sizeness, which we call 'magnitude'
So now reading this we have 50 thousand, 7 hundred, 1.
Expanded form,
place value form,
or if you will, place value name,
and word name.
Same number, just expressed three different ways.
Whichever one will be used,
well, that will depend upon what we're going to be doing.
And in the future you will know
from the context of the situation which form we need.
Sound like a bunch of words? You bet.
That's exactly what this and the last lesson is,
working with vocabulary
so using that very precise vocabulary
we can explain to you in the near future
some rather complicated processes.
But with this highly technical vocabulary,
we can make the explanations somewhat simple.
So play with these words and these kinds of problems
and we will use them very, very shortly.
So this is your host, Bob Finnell, until the next lesson.
Good luck.