Scientific Notation I

Express 0.0000000003457
in scientific notation.

So let's just remind
ourselves what

it means to be in
scientific notation.

Scientific notation will be some
number times some power of 10

where this number right here--
let me write it this way.

It's going to be greater
than or equal to 1,

and it's going to
be less than 10.

So over here, what
we want to put here

is what that leading
number is going to be.

And in general,
you're going to look

for the first non-zero digit.

And this is the
number that you're

going to want to start off with.

This is the only number you're
going to want to put ahead of

or I guess to the left
of the decimal point.

So we could write
3.457, and it's

going to be multiplied
by 10 to something.

Now let's think about
what we're going

to have to multiply it by.

To go from 3.457 to this
very, very small number,

from 3.457, to get
to this, you have

to move the decimal
to the left a bunch.

You have to add a bunch of
zeroes to the left of the 3.

You have to keep moving the
decimal over to the left.

To do that, we're
essentially making

the number much
much, much smaller.

So we're not going
to multiply it

by a positive exponent of 10.

We're going to multiply it
times a negative exponent of 10.

The equivalent is
you're dividing

by a positive exponent of 10.

And so the best way
to think about it,

when you move an
exponent one to the left,

you're dividing by 10, which
is equivalent to multiplying

by 10 to the negative 1 power.

Let me give you example here.

So if I have 1 times 10 is
clearly just equal to 10.

1 times 10 to the
negative 1, that's

equal to 1 times 1/10,
which is equal to 1/10.

1 times-- and let me actually
write a decimal, which is equal

to 0-- let me actually-- I
skipped a step right there.

Let me add 1 times 10 to the 0,
so we have something natural.

So this is one times
10 to the first.

One times 10 to the 0
is equal to 1 times 1,

which is equal to 1.

1 times 10 to the negative
1 is equal to 1/10,

which is equal to 0.1.

If I do 1 times 10
to the negative 2,

10 to the negative 2 is 1
over 10 squared or 1/100.

So this is going to be
1/100, which is 0.01.

What's happening here?

When I raise it to
a negative 1 power,

I've essentially
moved the decimal

from to the right of the
1 to the left of the 1.

I've moved it from
there to there.

When I raise it
to the negative 2,

I moved it two over to the left.

So how many times are we
going to have to move it over

to the left to get this
number right over here?

So let's think about
how many zeroes we have.

So we have to move it one time
just to get in front of the 3.

And then we have to
move it that many more

times to get all of the zeroes
in there so that we have

to move it one
time to get the 3.

So if we started
here, we're going

to move 1, 2, 3, 4, 5,
6, 7, 8, 9, 10 times.

So this is going to be 3.457
times 10 to the negative 10

power.

Let me just rewrite it.

So 3.457 times 10 to
the negative 10 power.

So in general,
what you want to do

is you want to find the
first non-zero number here.

Remember, you want a number
here that's between 1 and 10.

And it can be equal to 1, but
it has to be less than 10.

3.457 definitely fits that bill.

It's between 1 and 10.

And then you just want
to count the leading

zeroes between the
decimal and that number

and include the number
because that tells you

how many times you have
to shift the decimal over

to actually get
this number up here.

And so we have to shift
this decimal 10 times

to the left to get
this thing up here.