### Video Transcript

In this video, we’re gonna be
working with numbers in scientific notation. And some of you may know that as
standard form. Basically, we’re talking about
numbers in the format where we got something times 10 to the power of something
else. And it’s basically a way of writing
very large or very small numbers in a more compact format.

So, scientific notation is when we
put numbers in this particular format, a number times 10 to the power of some other
number. Now, that first number, represented
by 𝑎 in this general formula, has to be between one and 10. Well, specifically, it can be equal
to one, so it’s greater than or equal to one, but it can’t be quite as big as
10. It could be 9.999 recurring, but it
can’t be quite as big as 10. So, it’s a number between one and
9.9 recurring.

And the other number, represented
by this 𝑏, is a positive or negative integer which represents the exponent, or
power, of 10. Well, the best way to understand
this is to have a look at a couple of examples. So, let’s see an example.

1.5 times 10 to the power of three,
or 10 to the third power.

Now, multiplying by 10 to the power
of three, or 10 cubed, is like multiplying by 10, then multiplying by 10, then
multiplying by 10 again. So, that’s 1.5 times 10 times 10
times 10. Now, there is a shorthand way of
thinking about this which makes it very easy to work with scientific notation. Now, this isn’t strictly
mathematical, but this shorthand method does make it a lot easier to work with.

Multiplying by 10 is the same as
taking that decimal point and moving it one place to the right. So, 1.5 becomes 15 point, well,
15.0. Now, if we multiplied by 10 again,
we’re taking that decimal point and moving it one place to the right. Well, we didn’t have a number
there, but remember 15 is the same as 15.0. So, in this position here, there
would’ve been a zero, and we can put a decimal point over here. So, now, we’ve multiplied by 10
twice, and we’ve got one five zero point, so 150.

We’ve just got to multiply by 10
once more. And doing that moves the decimal
point yet another place to the right. So, that moves to here, and we’ve
got our zero that fills in here. So, 1.5 times 10 times 10 times 10
is 1500. Now, it’s important not to leave
your number in this format because that looks a real mess. This is nice and clear. 1.5 times 10 to the power of three
is 1500. Remember, 1.5 times 10 was 15, 15
times 10 was 150, and 150 times 10 is 1500. Okay let’s look at another
example.

9.09 times 10 to the power of
five.

Well, multiplying by 10 to the
power of five is like multiplying by 10 then by 10 then by 10 then by 10 then by 10
again. So, we’re multiplying by 10 five
times. So, let’s look at our shorthand way
of actually working that out. When we multiplied 1.5 times 10 to
the power of three, we ended up with three of these little arrows here. So, multiplying by 10 to the power
of five, we’re gonna end up with five little arrows over here.

That means that this decimal point
is going to move one, two, three, four, five places to the right. And we need to fill in some
zeros. We need a zero here, a zero here,
and a zero here. We could write one after the
decimal point as well if you want. So, our answer is 909000.

Now, when you’re doing your working
out, you wouldn’t normally bother writing this out or any of this. You would just use this kind of
like short notation here, which is rough working out, and then write your answer
correctly in a nice easy uncluttered format. Okay, hopefully, that’s fairly
clear. Let’s look at a couple of examples
now with negative powers of 10.

For example, 7.2 times 10 to the
power of negative four.

Now, in that exponent negative
four, the four tells us we’re multiplying by 10 four times and the negative tells us
to flip that fraction. So, instead of 10, it’s one over
10. So, 7.2 times 10 to the power of
negative four means 7.2 times a tenth times a tenth times a tenth times a tenth. Now, multiplying by a tenth is the
same as dividing by 10. So, that means we’re starting off
with 7.2 and then we’re dividing that by 10, dividing by 10 again, dividing by 10
again, and dividing by 10 again.

Let’s just jump straight to our
shorthand notation for doing this then. 7.2 divided by 10 is gonna be 0.72,
so effectively that decimal point is moving left one place here. Now, we’ve got to do that four
times, so one, two, three, four. So, our decimal point ends up
here. We need to put in some zeros here
as placeholders and a zero in front of the decimal point so we know it’s
0.00072.

Now, these zeros here were very
important because they were placeholders that told us that the seven should be in
the one ten thousandth column and the two should be in the one one hundred
thousandths column. And again, it’s important to write
out our answer clearly without all that horrible working out scribbled all over it,
so 0.00072. Now, remember, all this moving the
decimal point around is not necessarily mathematically correct in the way that we’re
looking at it, but it’s a nice shorthand way of working out the answer to these
questions. Okay let’s have a look at one more
example.

3.05 times 10 to the negative
six.

Well, times 10 to the negative six,
remember, means that we’re dividing the negative power, means we’re dividing by 10
six times. So, we start off with
~~3.005~~ [3.05] and we divide by 10, we divide by 10, we divide by 10,
we divide by 10, we divide by 10, and we divide by 10 again. So, I’m dividing by 10 lots of
times. I need to leave a bit of space to
the left-hand side of my number here.

Okay, divide by 10 once, the
decimal point moves to here, twice, three times, four times, five times, six times,
which means our decimal point’s moved here. Now, I can fill in the zeros. There’s one here. There’s one here. There’s one here, one here, one
here. And I’m gonna to put a zero in
front of the decimal point as well. So, that’s 0.00000305. And that’s our answer written out
nice and clear.

Okay, then, so, what’s all this
scientific notation about? In the examples we’ve shown you,
there’s not a massive amount of difference between writing that and writing
that. That’s because the examples I’ve
given you were relatively easy just to get the idea across. What if I was to ask what’s the
distance from Earth to Proxima Centauri, the nearest star other than the sun?

Well, it’s over 40 quadrillion
metres away from the Earth. Now, that’s quite a big number to
write out. And once we start talking on an
astronomical scale, there are even bigger numbers than that to deal with. Now, it starts to become more
efficient to write something like 4.0208 times 10 to the power of something
metres. But what is that something?

Well, if we’ve got a decimal point
here, in order to move it to the end of this number over here, it’s got to jump over
one, two, three, four, five, six, seven, eight, nine, 10, 11, 12, 13, 14, 15, 16
places. So, that’s 4.0208 times 10 to the
power of 16 metres. And in working that out, remember,
this number here has to be bigger than or equal to one, but it must be less than
10. So, we’ve made it four point
something. And in this case, that left us with
10 to the 16th power.

And at the other end of the
spectrum, we’ve got very very small things. So, for example, the diameter of a
proton is about a millionth of a nanometre, or 0.000000000000001 metres. Now, there must be a quicker and
easier way to write that. Well, the digit on the end here is
just a one, so we can write it as one times 10 to the power of negative something
metres. So, what is that something?

Well, if we wrote a one with a
decimal point after it, like this, we would have to move that decimal point one,
two, three, four, five, six, seven, eight, nine, 10, 11, 12, 13, 14, 15, places to
the left. So, we can write the number as one
times 10 to the negative 15 metres. Okay, let’s have a look at some
typical questions you might see in a test.

Write 3.141579265 times 10 to the
power of four as an ordinary number.

Now, if you live somewhere that
calls this standard form, then a number not written in standard form would be called
an ordinary number. But if you live somewhere where
they call this format scientific notation, then they’d call the ordinary number
version standard form. How confusing’s that? Anyway, let’s not worry about that
for now. You know where you live. You know what you call these things
hopefully.

So, 3.141579265 times 10 to the
power of four, in this case, we’re multiplying by 10 positive four times. We’re multiplying by 10 four times,
so that decimal point is gonna have to move one, two, three, four places to the
right. And after we’ve multiplied that
number by 10 four times, we get 31415.79265.

How about this one then?

Write 6.2 times 10 to the power of
seven as an ordinary number.

Well, we’re gonna write down the
6.2, and then we’re gonna multiply it by 10 seven times. So, that’s one, two, three, four,
five, six, seven. So, that’s gonna be 62000000.0. Well, obviously, in this case we
can leave off the point zero, and that gives us an answer of 62 million.

Now, write 9.603 times 10 to the
negative three as an ordinary number.

So, we’re gonna take 9.603, and
we’re gonna multiply it by 1 over 10 three times, which is dividing by 10 three
times. So, this decimal point is gonna
move to the left. It’s gonna make that number
smaller. Divide by 10 once, twice, three
times, so my decimal point’s gonna go here. I can then put in my holding zeros,
and we can see that the answer is 0.009603.

Now, let’s have a look at one or
two slightly more challenging questions.

Work out the value of three times
10 to the four times 1.2 times 10 to the power of seven, giving your answer in
scientific notation.

Well, with multiplication, it
doesn’t actually matter what order we multiply these things together in, so I can
get rid of those parentheses straight away. So, that’s the same as three times
10 to the four times 1.2 times 10 to the power of seven. Now, we can multiply the 10 to the
four and the 10 to the seven together, and we can multiply the three by the 1.2. Now, three times 1.2 is 3.6. And if I multiply by 10 four times
and then another seven times, I’ve multiplied by 10 11 times. That’s 10 to the power of 11. So, that’s equal to 3.6 times 10 to
the power of 11.

Now, remember, to be in proper
scientific notation, the number must be in this format, 𝑎 times 10 to the power of
𝑏, where 𝑎 is a number between one and 10. It can be equal to one, but it
can’t be equal to 10. And 𝑏 is a positive or negative
power exponent of 10. Now, in our case, the 𝑎 value,
this 3.6, that is in the right range. It’s bigger than or equal to one,
but it’s less than 10. And we have, in this case, a
positive power, or exponent, of 10. So, our answer is 3.6 times 10 to
the power of 11.

Right then, just one final question
before we go.

Work out the value of five times 10
to the 5 times 8.2 times 10 to the power of 11, giving your answer in scientific
notation.

Well, again, multiplication is
commutative, so it doesn’t matter whether we’ve got the parentheses and what order
we write those in. So, I’m going to do five times 8.2
times 10 to the power of five times 10 to the power of 11, in that order. Now, five times 8.2 is 41. And if I multiplied by 10 to the
power of five, that’s multiplying by 10 five times, and then by 10 to the power of
11, that’s multiplying by 10 another 11 times, that’s a total of 16 times. So, it’s times 10 to the power of
16. So, that’s equal to 41 times 10 to
the power of 16.

But wait a minute! That’s not our answer! This number here has to be between
one and 10. 41 isn’t between one and 10. But look, I could write it as 4.1
times 10 to the power of something. Well, in fact, in this case it
would be times 10 to the power of one because I need to multiply 4.1 by 10 once to
turn it into 41. So, 41 is equal to 4.1 times 10 to
the power of one.

And that’s the number that has to
be multiplied by 10 to the power of 16. Well, we can see that we’ve got 4.1
times 10 once times 10 another 16 times, which means that’s times 10 17 times in
total. And there we have it. We’ve now got our number in proper
scientific notation. That first term is 4.1. That is between one and 10. And we’ve got times 10 to the power
of 17.