Math 20 - Lesson 39

A Portland Community College Mathematics Telecourse.
A course in arithmetic review.
Produced at Portland Community College.
This lesson is going to lead us into a topic
called 'scientific notation.'
An extremely valuable lesson if you're going to go on
into any technology: science, engineering,
or use of any scientific calculator.
But before we go too far,
let's review what we mean by a 'power of ten.'
It's simply any number
which can be written as a product of tens.
See we had a agreed
that if you're multiplying two tens together,
we simply write 10 with a small 2 to the right [ 10² ]
and up high which we call an exponent.
So if I had 10 times itself twice [ 10 · 10 ]
or three tens being multiplied [ 10 · 10 ·10 ]
we would write that as 10 with an exponent of 3 [ 10³ ]
If we had four tens being multiplied [ 10 · 10 ·10 · 10 ]
we would write that as 10 with an exponent of 4 [ 10⁴ ]
and we would verbalize that as 10 to the fourth power.
And by special definition,
10 by itself we would read as 10 to the first power [ 10¹ ]
And the number 1, by special definition,
is simply 10 to the zero power [ 10⁰ ]
Now notice that the powers go up.
On 10 to the zero, there's 1 with no zeros behind it [ 10⁰=1 ]
10 to the first is a 1 followed by one zero [ 10¹ = 10 ]
10 squared is a 1, if you would,
followed by two zeros [ 10² = 100 ] and so on.
Recall this from chapter one.
We'll be using that a lot in this discussion.
Now, for what we're about to develop,
it's going to be very convenient
to make a slight alteration to this power of ten notation.
Of course, we're familiar already
that we can write these powers of tens in decimal form.
In fact, that was the reason
for the invention of the decimal system.
But now let's make this one agreement.
Rather than writing these powers of tens in denominators,
let's write it that way
And use a minus sign in front of the exponent
and the minus sign will mean by agreement
the reciprocal of 10 to the first, [ 1/10¹ = 10⁻¹ ]
the reciprocal of 10 to the second, [ 1/10² = 10⁻² ]
the reciprocal of 10 to the third, [ 1/10³ = 10⁻³ ] and so on.
This is simply an agreement to notation.
Don't look for the logic or the rationale,
it's simply an agreement we're making
that's going to make a very nice system in a few moments.
So in short, the 'negative exponent' will mean by definition
simply the phrase "1 over" or "the reciprocal of."
Recall that as we go on in this lesson.
Now let's write this [ 10⁶ ] in standard notation.
Recall that we had simply said
write a 1 with six zeros behind it
or, if you would, one million. [ 10⁶ = 1,000,000 ]
That means 1 over 1 followed by six zeros [ 1/1,000,000 ]
or one millionth [ 10⁻⁶ = 1/1,000,000 ]
Now if I were to divide this out, you would get zero point one,
two,
three,
four,
five,
and not six zeros but five zeros and a 1 [ 0.000001 ]
these [ 1/1,000,00 and 0.000001 ] are all the same thing.
So we could verbalize this way, if it's a positive power,
write down a 1 and follow it by six zeros [ 1,000,000 ]
If it's a negative exponent, start with a decimal point
and mark off six places to the right,
one,
two,
three,
four,
five,
six,
with the sixth place ending in 1 [ 0.000001 ]
Now at first it seems as though there's a slight difference,
here [ 1,000,000 ] we stick on six 0s, here [ 0.000001 ] five 0s,
but let's look at this from another angle
where we'll see really they're very compatible
in saying about the same kind of a thing.
In both cases let's start with a 1 and, of course,
with all whole numbers the decimal point is just to the right.
Now let's think this way, if the decimal –
if the power of ten is positive,
make this one larger by six places [ 1 _ _ _ _ _ _ ]
And then, of course,
as we create extra places use placeholder zeros [ 1000000 ]
Now let's again start with 1 but let's make it smaller,
let's interpret the negative that way,
to make smaller by six places.
So make smaller by one place,
two,
three,
four,
five,
six [ .000001 ]
So if we view the positive exponent
as make larger by that many places and the negative exponent
as starting from one making smaller by that many places,
we're saying the same thing, make larger by six places,
make smaller by six places.
Okay, recall that, that is going to make
several problems coming up soon very easy.
Now any 1 followed by nothing but zeros
or 1 preceded by nothing but zeros and a decimal point,
we refer to as 'powers of ten.'
If it's larger than one, it'll have a positive power [ ⁺ ]
if it's a decimal less than one,
it will have a negative exponent [ ⁻ ]
Now this one, [ 100,000,000,000 ]
if we were to get up to 1, I'd have to count
for one, two,
three, four,
five, six,
seven, eight,
nine, ten,
eleven places, so we call that 10 to the eleventh power [ 10¹¹ ]
Now note it's not zeros I'm counting it's places.
So here [ 0.000000001] to get back to 1,
and you get back to 1 not here [ 0.00000000_1 ]
but there [ 0.000000001_ ]
just as I did here on that one.
So I'd have to move it one,
two,
three,
four,
five,
six,
seven,
eight,
nine places.
So this [ 0.000000001 ] we will express
as 10 to the negative ninth power [ 10⁻$ ]
So it's getting to that 1 that is a critical thought.
I have to move it nine places to get to 1
and since the number is less than 1, it's a negative exponent.
Here [ 100,000,000_ ] I have to move it
eleven places to get to the 1,
so we call that 10 to the 11th [ 10¹¹ ]
Now having a number multiplied or divided by a power of ten,
occurs very, very frequently in science.
It's almost the rule in science.
But there's a shortcut for doing this
which makes it rather convenient.
Write down the digit of the number you are multiplying
by that power of ten, and the decimal point is there [ 5_35 ]
Now if you're multiplying by a positive power of ten
and think that I wish to make this number [5.35 ] bigger
by that many places, six in this case,
so to make it bigger, if I go this way the number gets bigger,
if I went this way the number is getting smaller.
So I wish to make the number bigger by one,
two,
now to keep moving I need some extra zeros,
three,
four,
five,
six.
Let's check.
Bigger by
one,
two,
three,
four,
five,
six,
and, indeed, if I were to write this [ 10⁶ ]
as 1 followed by six zeros then multiply
by the conventional way times this, [ 5.35 x 10⁶ ]
I would get this. [ 5,350,000. ]
So again, if you're multiplying by a positive power of 6,
you simply make the number bigger by that many places,
inserting zeros if you need to.
Another more formal way of saying this
is to multiply a decimal by a power of ten,
move the decimal point to the right
as many places as the exponent value
and insert placeholder zeros
if extra places are created by that process.
That's saying from a slightly different angle
what we just told you by that example,
that is to multiply by a positive power of ten,
make the decimal number larger by as many places
as the value of the exponent.
What happens when we're multiplying by a negative power of ten?
Well recall a negative power of ten
is 1 over ten to that power [ 10⁻¹ = 1/10⁻¹ ]
times this number [ 4.32 x 10⁻³ ] and, of course,
I can write that now as 4.32 divided by 10 cubed [ 4.32/10³ ]
So first note, multiplying by a negative power of ten
is the same as dividing by its positive equivalent.
Now the shortcut we're going to give you for this procedure
is simply this,
if you're multiplying by a negative power of ten [ 4.32 x 10⁻³ ]
or dividing by a positive power of ten [ 4.32/10³ ]
your number will get smaller by that many places,
three places if you don't look at the negative sign.
So our rule, when multiplying by a negative power of ten
or dividing by a positive power of ten,
make the number smaller by that many places.
So I've made this number [ 4.32 ] smaller
by three decimal places. [ .00432 ]
That rule once again,
to multiply a decimal by a negative power of ten
or to divide by a positive power of ten,
move the decimal point to the left as many places as the exponent
and again insert placeholder zeros if they're needed
because of newly created positions.
Or saying the same thing slightly differently,
but still saying the same thing,
to multiply by a negative power of ten,
which is the same thing as dividing by a positive power of ten,
make the decimal number smaller
by as many places as the exponent value.
An example: I wish to multiply this number [ 7.83 ]
by a negative power of ten. [ 10⁻⁵ ]
The rule we just gave said make it smaller by five places,
smaller by one,
two,
three,
four,
five places,
and I need to insert the placeholder zeros [ .0000783 ]
Here's a number written in the form that we will refer to
from here on out as 'scientific notation.'
Any time you have any number,
will call that number just 'n' for now to stand for 'any number'
where the number itself is greater than or equal to 1
or less than 10, and less than 10,
followed by the product of a power of ten,
positive or negative,
we will call that kind of a number 'scientific notation.'
Scientific notation is a very convenient way
of expressing extremely large or extremely small numbers.
It's so important that all scientific calculators
are programmed, preprogrammed,
to work automatically with scientific notation
when one of its numbers gets too large or too small.
Here are two numbers written in scientific notation,
a decimal number between 1 and 10
followed by a product of the power of ten. [ 6.73 x 10⁵ ]
Again, a decimal number between 1 and 10
followed by a product of the power of ten [ 2.05 x 10⁻⁴ ]
in this case negative. [ 2.05 x 10⁻⁴ ]
Let's write this in place value notation
or scientific notation.
'Place value notation' is the terminology your text uses.
Remember we just stated that
to multiply by a positive power of ten,
you simply make this number [ 6.73 x 10⁵ ]
bigger by five decimal places.
So we started here,
one,
two,
three,
four,
five. [ 673,000 ]
So this scientific notation
is simply another way of saying 673 thousand.
Now if you're multiplying by a negative power of ten [ 10⁻⁴ ]
our shortcut stated to make it smaller by four places,
so smaller by one,
two,
three,
four,
the decimal point goes here. [ .000205 ]
Now let's see,
tenth, [ .0----- ]
hundredth, [ .-0---- ]
thousandth, [ .--0--- ]
ten-thousandth, [ .----2-- ]
hundred-thousandth, [ .----0- ]
millionth. [ .-----5 ]
So this [ 2.05 x 10⁻⁵ ] is another way of saying 205 millionths.
Now at this point, there's not much advantage
of this number [ 673,000 ] over this number [ 6.73 x 10⁵ ]
or vice versa.
This form we call 'scientific notation.'
This form we call either the 'place value name,'
and many scientists will call it the 'standard notation.'
Actually, in science, this [ 6.73 x 10⁵ ] is more of a standard.
This will be used for very large or very small numbers
far more than this form [ 673,000 ] will
Let's try reversing the process,
writing this number [ 827,000,000 ] into scientific notation.
That means we want the decimal point
behind the first non-zero digit [ 8_ ] followed by a power of ten
So our only question is what power of ten
can we place here [ 8.27 x 10-- ]
to justify putting the decimal there [ 8.27 ]
rather than way back here where it legitimately belongs.
Well if it were here, [ 8.27 ]
then to get it back where it should belong
I'd have to make it larger,
that tells me a positive exponent
of one,
two,
three,
four,
five,
six,
seven,
eight.
So if you want to put a number into scientific notation,
put the decimal point behind the first non-zero digit [ 8_27 ]
then ask yourself, if it were there [ 8.27,000,000 ]
how would I have to move it to get it back
to where it really belongs.
We'll have to make it bigger.
That's multiplying by a power of ten
with a positive exponent eight places.
Exponent: eight. [ 10⁸ ]
Let's use that same idea
to write this one [ 0.00000923 ] in scientific notation.
First, scientific notation
says I want the decimal point there. [ 9.23 ]
So I will have to adjust with a product of a power of ten.
Now if it were there [ 9.23 ]
rather than here [ 0.00000923 ] where it really belongs,
to get it back to where it really belongs
I'd have to make it smaller by one,
two,
three,
four,
five,
six places.
So negative exponents says I'm making it smaller by six places.
So this [ 0.00000923 ] we call the 'place value name,'
this [ 9.23 x 10⁻⁶ ] the 'scientific notation.'
In later lessons and this extended lesson,
you will this being used on the calculator
and to express extremely large and extremely small numbers
which would be almost impossible here.
Just one short example,
this [ 10⁻²⁴ cm²] is a number used in measuring
the cross-sectional area of atoms.
It looks very simple here
but let's look at this number in place value notation.
See, almost impossible to write on the paper, as we said.
So the scientific notation is a much better way of saying
this very, very tiny number
and the same thing with extremely large numbers,
which occurs in physics and astronomy and atomic physics,
in particular, very frequently.
Let's clarify a point,
a common error made before perhaps you even make it.
Let's say that for some reason
you wish to write this number [ 48,200,000 ] in scientific notation.
Now by the definition we have just given,
it means we want the decimal point
behind the first non-zero digit.
So I want the decimal point here [ 4.82 ]
rather than back here [ 48,200,000. ] where it properly is.
So we put it behind that first non-zero digit
then we adjust it by multiplying by a power of ten.
So to get this back to where it originally was,
we can see I'd have to make this 4.8 bigger by one,
two,
three,
four,
five,
six,
seven places
and bigger means to make a positive exponent [ 4.82 x 10⁷ ]
That's fairly simple, isn't it?
But there's sometimes there's a tendency
for many students who aren't listening very close
to that phrase which said, to be 'in scientific notation,'
we want the decimal point behind the first non-zero digit.
And there is a tendency then to put the decimal point
simply at the end [ 482. ] of whatever the digits are,
the non-zero digits, and then to adjust by a power of ten
and their reasoning would be,
gee, if it were here [ 482.00,000 ] I'd have to make larger by one,
two,
three,
four,
five,
so they say this. [ 482. x 10⁵ ]
Now it's important to notice that this [ 482. x 10⁵ ]
is a correct equivalent to this. [ 48,200,000 ]
This [ 48,200,000 ] is really equal to this [ 482. x 10⁵ ]
and this [ 482. x 10⁵ ] is equal to this [ 4.82 x 10⁷ ]
But the point you need to realize is that even though
this [ 482. x 10⁵ ] is a correct way of saying it,
we will not allow ourselves to call it 'scientific notation.'
To be called 'scientific notation,'
we insist that the decimal point
be behind the first non-zero digit.
This [482 x 10⁵] we'll just call in general terms a power of ten
and there are many correct ways of writing this
by using powers of ten
but only one that we will allow ourselves
to call 'scientific notation.'
Be very, very clear on that.
'Scientific notation' means the decimal point
is behind the first non-zero digit reading from left to right.
Now as promised on the last lesson,
let me show you something about the scientific calculator
and very large or very tiny numbers.
If you were asked to multiply 32 million times 43 million
on a standard four function calculator,
you will find that you can get this onto the calculator,
after all we have 32, then one,
two, three,
one, two,
three, six zeros, it [ 32,000,000 ] fits.
And then if we say times [x] the next number 43,
one,
two,
three,
four,
five,
six,
that fits too. [ 43,000,000 ]
So the numbers fit on our calculator
but if you were to say equal on most standard calculators,
the answer is so large that it won't fit on the calculator.
So a standard, inexpensive four function calculator
would simply say 'error' or give you some kind of message
that says, hey, I can't do this, it's too large, the answer is.
Watch what happens to a scientific calculator.
As soon as I hit the equal [=] it does something rather strange.
It has 1.376 and there is no doubt that this product
is certainly going to be larger than 1.3,
it's going to be up in the billions and billions.
So what happened?
But notice there's a little blank space there [ 1.376_15 ]
and on a scientific calculator what that blank space says,
without actually writing it out for you,
is that this is times 10.
So interpret that little space as times 10 to that exponent.
So what a scientific calculator will do
so it doesn't lose your problem
because the screen can't hold it,
it'll work the problem anyway
and if it's too large for the screen,
it will automatically click over to scientific notation
and give you the answer in scientific notation
because in standard notation this is much, much, much too long.
Isn't that convenient?
Now watch how if I am multiplying by two very tiny numbers,
the same thing happens.
Here this number [ .0000007 ] again would fit on the screen
and so would this [ .0000008 ] but their product would not;
it would be too many decimal places.
But let's see what the calculator does.
Point one, two,
three, four,
five, six,
seven, see it fits,
times point one, two,
three, four,
five, six,
8,
and that fits but the product will not fit.
So let's hit an equal [=] and see what it does.
Again it gives me the number 5.6,
and notice seven 8s is 56, [ 7 x 8 = 56 ]
but now there is that little blank space with a negative.
Okay, when there is a blank space or a small number
ending your screen output of your calculator,
that means the calculator is trying to tell you
that it's times 10 to some exponent
and the small number or the number following the space
is its exponent.
So now this [ 5.6_-13 ] says if you were to take 5.6
and make it smaller by moving the decimal point 13 places,
that's what this number is in standard notation.
So note if we were forced
to used standard notation for everything,
even calculators could not do a lot of problems.
But if the calculator can work in scientific notation,
at least it can keep the front important part of the decimal
and convert the rest of it to a power of ten
and report our results in scientific notation.
Now let's look at the reverse of that.
Here [ 4.12 x 10¹² ] is a number in scientific notation
and if I was to write this in standard notation
it's much too long to even enter into the calculator.
So does that mean the calculator can't even work with it?
No, again, a scientific calculator
will allow you to enter a number in scientific notation form.
So we would enter this decimal portion here, 4.13 [4][.][1][3]
Now we need some way to tell the calculator
that I'm going to give you a scientific notation number.
So with most calculators, it's the EXP key,
this one [EXP] right here.
On many calculators it might be another key,
so you'll have to find that from your directions.
But on this one it's EXP, so I push that [EXP]
Now note it's got a little blank space [ 4.13_12 ]
that means it needs the power of ten.
So now I simply put in 12.
So 4.3, space, which means times 10, to the twelfth power [4.13_12]
Isn't that nice?
Now how about a negative exponent?
How would we do that?
Let's say this very tiny number is 3.57 x 10⁻⁸
First we enter the decimal portion, which is 3.57
then to tell the calculator
we're entering a scientific notation number,
we push the EXP key; it's ready for the exponent,
so we put in the exponent of 8, but I want it to be negative.
So up here, right here, is a key
which you might not be able to see but it looks like this,
plus/minus [+/-] which means change signs.
So I push that change signs key and now that has become minus 8.
So 3.57 space means times 10 to the minus 8 power [ 3.57_-8 ]
So, in fact,
a scientific calculator can work in scientific notation.
And in science technology
that is almost always the case rather than standard notation.
Of course there's much, much more to this scientific notation
than we possibly have time to explain in this very short lesson.
So in your future science classes and other math classes
do expect to see this being used very, very heavily
and there will be more teaching about this.
This lesson is asking for you
to get into and out of scientific notation
and multiplying and dividing by powers of ten.
If you have that as you go into the next lessons,
the rest of this will become very, very easy.
This is your host.