### Video Transcript

In this video, we will learn how to
express numbers in scientific notation, sometimes referred to as standard form. We will look at how we can write
very large and very small numbers in this notation. When numbers have a very large or
very small absolute value, writing them means writing a lot of digits, for example,
241300000. At the other extreme, we have
0.000000764. Scientific notation is a compact
way of writing numbers of this form.

Let’s begin by looking at a
definition of scientific notation. A number written in scientific
notation is written in the form 𝑎 multiplied by 10 to the power of 𝑏, where the
absolute value of 𝑎 is less than 10 and greater than or equal to one. This means that 𝑎 could take any
of the following values: four, seven, 3.2, negative 6.3, etcetera. The exponent, or power, 𝑏 is a
positive integer for large numbers and a negative integer for small numbers.

The following are examples of
numbers written in scientific notation. Four multiplied by 10 to the power
of seven, 3.2 multiplied by 10 to the power of eight, and 2.14 multiplied by 10 to
the power of negative seven. 0.6 multiplied by 10 to the power
of eight and 23 multiplied by 10 to the power of negative five are not in scientific
notation. This is because their value of 𝑎
is either less than one or greater than or equal to 10. 23 is greater than 10, and 0.6 is
less than one. We will now look at some questions
that involve expressing numbers in scientific notation.

There are approximately 200 million
cows in India. Express this number in scientific
notation.

Let’s firstly consider how we would
write the number 200 million. The number one million is written
as a one followed by six zeros. This means that 200 million can be
written as 200 followed by another six zeros. Altogether, we have two followed by
eight zeros. A number is written in scientific
notation if it is written in the form 𝑎 multiplied by 10 to the power of 𝑏. The absolute value, or modulus, of
𝑎 must be less than 10 and greater than or equal to one. The value of 𝑏 can be a positive
or negative integer.

In order to write a number in
scientific notation, we firstly put a decimal point after the first nonzero
digit. In this question, this is the first
digit, two. As the digit after this is a zero,
our value for 𝑎 will be 2.0, or just two. In order to find the value of the
exponent 𝑏, we need to work out what we multiply two by to give us 200000000. We can do this using place value or
by counting how many times the decimal point moves till it gets to the end of our
number.

In this question, there are eight
bounces, or eight place values. This means that our exponent is
eight. As already mentioned, 2.0 is the
same as two. So, the number 200 million in
scientific notation is two multiplied by 10 to the power of eight. This is approximately the number of
cows in India.

We will now look at another
question using scientific notation to estimate large quantities.

The ship Titanic that sank in the
Atlantic had a mass of about 47450000 kilogrammes. Which one of the following could be
used as an estimate for this value? Is it A) five multiplied by 10 to
the power of six? B) Five multiplied by 10 to the
power of seven? C) Five multiplied by 10 to the
power of eight? D) Four multiplied by 10 to the
power of seven? Or E) four multiplied by 10 to the
power of eight?

Our five answers are all written in
scientific notation in the form 𝑎 multiplied by 10 to the power of 𝑏. We know that they’re in scientific
notation as the absolute value of 𝑎 is less than 10 and greater than or equal to
one. The value of 𝑏 is an integer. We need to convert 47450000
kilogrammes into scientific notation and then see which of our answers is the best
estimate.

In order to write a number in
scientific notation, we begin by finding the first nonzero digit. In this question, it is a four. We put a decimal point after this
number and then write all the other nonzero digits. In this case, we have 4.745. This will be multiplied by 10 to
some power. The exponent will be positive when
dealing with large numbers and negative when dealing with small numbers. In this case, it will be a positive
integer.

We need to work out how many times
we would multiply 4.745 by 10 to get the answer 47450000. We can do this using place value or
by counting the number of times the decimal point would move to the right. In this case, the answer is
seven. 47450000 written in standard form
is 4.745 multiplied by 10 to the power of seven. Looking at our five possible
answers, we can immediately see that options A, C, and E are incorrect as the
exponent is not seven.

In order to round our value of 𝑎,
4.745, to the nearest integer, we look at the number in the tenths column. As this is a seven, we will round
up. So, our estimate becomes five
multiplied by 10 to the power of seven. The correct answer is B, an
estimate of 47450000 kilogrammes is five multiplied by 10 to the power of seven
kilogrammes.

We’ll now look at a question
converting a small number into scientific notation.

What is 0.00007 written in
scientific notation?

A number is written in scientific
notation if it is written in the form 𝑎 multiplied by 10 to the power of 𝑏, where
the absolute value of 𝑎 must be less than 10 and greater than or equal to one and
𝑏 is a positive or negative integer. In this question, we want to write
0.00007 in scientific notation. Our first step is to find the first
nonzero digit in our number. In this question, there is only
one, the seven. We place a decimal point after this
digit. Have there been other digits after
this seven, we would’ve written these after the decimal point, for example,
7.124. In this case, there are no nonzero
digits after the seven, so we have 7.0, or just seven.

We will multiply this by 10 raised
to some power. Our value for 𝑏 will be negative
for small numbers. This is because raising 10 to a
negative power actually means we will be dividing. 10 to the power of negative four is
the same as one over 10 to the power of four. So, we’re actually dividing by 10
to the power of four, which will make the number smaller.

To work out the value of 𝑏, we
need to find the number of place values the numbers have moved or the number of
times the decimal point has moved to the left. To get from 7.0 to 0.00007, we move
our decimal point five places to the left. Therefore, our exponent is negative
five. As 7.0 is the same as seven,
0.00007 written in scientific notation is seven multiplied by 10 to the power of
negative five.

We will now look at a second
example of writing a small number in scientific notation.

Write 0.000853 in scientific notation.

A number is written in scientific
notation when it is in the form 𝑎 multiplied by 10 to the power of 𝑏. We know that the absolute value of
𝑎 must be less than 10 and greater than or equal to one. We know that 𝑏 must be an
integer. It is a positive integer for large
numbers and a negative integer for small numbers. In this question, we want to write
the small number 0.000853 in scientific notation. We begin by finding the first
nonzero digit, in this case, eight. We put a decimal point after this
number and then write any further nonzero digits. In this question, our value of 𝑎
is 8.53. This is less than 10 and greater
than or equal to one.

We need to multiply this by 10
raised to some power. In this case, the power or exponent
will be negative as to get from 8.53 to 0.000853, we are actually dividing. Multiplying by 10 to the power of
negative six is the same as dividing by 10 to the power of six. To work out the value of this
exponent, we need to work out how many places our digits have moved, or how many
times the decimal point has moved. In this case, it is four. Therefore, our exponent is negative
four. The number 0.000853 written in
scientific notation is 8.53 multiplied by 10 to the power of negative four.

Our next question involves solving
a real-world problem.

The length of an object is seven
millimetres. Express this length in metres,
giving your answer in scientific form.

Let’s firstly recall some of our
conversions of metric units of length. There are 10 millimetres in one
centimetre. There are 100 centimetres in one
metre. Combining these facts tells us that
there are 1000 millimetres in one metre. This means that in order to convert
a value from millimetres to metres, we need to divide by 1000. Seven divided by 1000 is equal to
0.007. This means that seven millimetres
is the same as 0.007 metres.

This number is not in scientific
form as it would need to be written 𝑎 multiplied by 10 to the power of 𝑏, where
the absolute value of 𝑎 is less than 10 and greater than or equal to one and 𝑏 is
an integer. 0.007 written in scientific
notation is seven multiplied by 10 to the power of negative three. This is because the digits have
moved three places to get from 7 to 0.007.

We could’ve worked this out
directly from the calculation seven divided by 1000. 1000 is the same as 10 to the power
of three, or 10 cubed. Dividing by 10 to the power of
three is the same as multiplying by 10 to the power of negative three. Once again, we get the answer seven
multiplied by 10 to the power of negative three. This is the length of the object in
metres.

Our final question involves working
out a calculation and then writing it in scientific notation.

Express three multiplied by 300 in
scientific notation.

There are several ways we could
approach this question. We will look at two different
methods. Our first method involves
multiplying three by 300 first. This gives us 900. We know that a number is written in
scientific notation if it is in the form 𝑎 multiplied by 10 to the power of 𝑏,
where the absolute value of 𝑎 is less than 10 and greater than or equal to one. As there is only one nonzero digit
in this question, this will be the value of 𝑎.

900 written in scientific notation
will be equal to nine multiplied by 10 raised to some power. 10 squared is equal to 100. And nine multiplied by 100 is equal
to 900. Three multiplied by 300 expressed
in scientific notation is nine multiplied by 10 squared.

An alternative method would be to
rewrite three multiplied by 300 as three multiplied by three multiplied by 100. Three multiplied by three is equal
to nine. 100 is equal to 10 to the power of
two, or 10 squared. Once again, we get the answer nine
multiplied by 10 squared.

We will now look at the key points
in this video on scientific notation. A number written in scientific
notation is written in the form 𝑎 multiplied by 10 to the power of 𝑏. The absolute value of 𝑎 must be
less than 10 and greater than or equal to one. And 𝑏 will be a positive or
negative integer. It will be positive for large
numbers and negative for small numbers.

To write a number in scientific
notation, we firstly find 𝑎 and then find 𝑏. 𝑎 is the number with exactly the
same digits as the original number but such that the absolute value of 𝑎 is less
than 10 and greater than or equal to one. 𝑏 is the number of places the
decimal point needs to move from its position in 𝑎 to its position in the original
number. 𝑏 will be positive for large
numbers and negative for small numbers. For example, the number 237000
written in scientific notation is 2.37 multiplied by 10 to the power of five. In the same way, 0.00067 is 6.7
times 10 to the power of negative four.