Explanation of scientific notation

Explanation of Scientific Notation
Scientific notation uses powers of 10 to state the position of a number in relation to a
decimal place. Powers of 10 are quite easy, its just a case of writing zeros, or, more to the
point, avoiding having to write them. Most of us have encountered powers through our
everyday lives. An engine has cubic centimetres of capacity, we buy litres of milk, a piece
of paper is a number of cm squared.
Squaring
Squaring a number is usually the first thing we learn about the powers of numbers. A
power is just another way of saying how many times a number is multiplied by itself. Let’s
think of a square tile in your kitchen or bathroom.
It has the same height and width. That’s what makes it square, otherwise it would be just
another rectangle. For a tile of 5cm height and 5cm width, we could calculate how much
flat space it takes up, it’s surface area, by multiplying it’s height by it’s width. Surface area
is a mathematical term, and can be thought of the space a shape takes up in 2 dimensions.
The number 2 is key to calculating the area of a square. In this case 5cm x 5cm = 25 square
cm. See, even the answer has the word square in it.
An explanation of Scientific Notation – Stuart Last - 2017
1
5cm
5cm

The unit of “square centimetres” is very descriptive. It tells us the count of squares of 1
unit in a square of a larger number of units. In this case we are talking about centimetres
so in a square of 5cm by 5cm there are 25 squares of size 1cm by 1cm. I can show that by
slicing up our 5cm x 5cm square.
After a deft bit of slicing, I have a stack of rectangles that are 1cm high and 5cm wide.
There are 5 rectangles showing that if we add 5 rows of 5 we get a block of 25 1cm blocks.
Gluing it all back together, you can count the 25 square centimetres arranged in a square
of 5 cm by 5cm. Our multiplication of 5cm x 5cm, or 5cm
2
, definitely = 25 square cm.
Another way to show this calculation would be to say 5cm multiplied by itself or 5cm
2
. The
little 2 in “superscript” tells us how many time to multiply a number by itself. In this case
it’s 2 times, so we can write it as 5cm x 5cm, which is exactly the same as calculating the
surface of a square.
2
1 square cm
5 square cm
5 square cm
5 square cm

Cubing
In squaring we worked out how much space an object in 2 dimensional space takes up. A
cube is a square shown in 3 dimensional space. The space that a 3 dimensional object
takes up is called volume. For example a game die may have a single face surface area of
2cm * 2cm, or 2cm
2
. In 3 dimensional terms though, we can add depth to height and
width.
So for a cube object like a game die, we can add it’s depth. As a cube’s dimensions are all
the same we can write it’s volume as 2cm * 2cm * 2cm which equals 8 cubic cm. But look,
there are 3 lots of 2cm, so we can also write it as 2cm
3
as a way of saying that we need to
multiply 2 by itself 3 times. This is known as cubing, but we can also say we have raised
2cm to the power of 3.
After 3 dimensions, we run out of dimensions that we are able to naturally perceive,
although physicists are postulating many more dimensions that we are just not really
aware of. However, we can use the power of powers very much to our advantage, in
particular the powers of 10.
3
2cm
2cm
2cm

Powers of 10.
Powers of 10 has a sort of magical property; one that can make a scientists life a lot easier.
Lets look at our square and cube again with dimensions of 10cm.
A square piece of paper with height of 10cm and width of 10cm would have an area of
10cm x 10cm or, because 10 is multiplied by itself twice, 10cm
2
. This gives us 100 square
cm. Have a quick look at the numbers here.
10cm x 10 cm = 10cm
2
= 100 square cm.
We had 10 and multiplied by 10. There’s 2 zeros. The result was 100, which has 2 zeros,
and, if we look at the powers of 10 in the calculation, we can see 10
2
also has a 2 in it.
Coincidence? Maybe. Let’s try another.
A cube of dimensions 10cm wide, by 10cm high, by 10cm deep. that’s 10cm x 10cm x10cm
or, using our powers notation 10
3
which says multiply 10 by itself 3 times. The answer is
1000 cubic centimetres.
4
10cm
10cm
10cm
10cm
10cm

10cm x 10cm x 10cm = 10cm
3
= 1000 cubic cm
Lets look at those numbers again. In 10cm x 10cm x 10cm there are 3 zeros. In our answer
of 1000 there are 3 zeros, and in our representation of 10
3
we see the number 3 next to a
zero.
So lets ramp it up even more. Let’s look at 10
6
. This means 10 multiplied by itself 10 times.
10
6
= 10 x 10 x 10 x 10 x 10 x 10 = 1, 000, 000
We could write this out as 10 x 10 x 10 x 10 x 10 x 10 and try to figure it out, but we know
from the examples above that we could just write down a 1 followed by 6 zeros. Which
gives us 1, 000, 000. By all means check this. It’s good science to seek empirical evidence.
Because the powers of 10 also explain how many zeros follow the 1 this almost magical
property of the powers of 10 can be used is surprising ways.
The powers of 10 are powerful indeed.
Now, if you can cast your mind back to your 10 times table, you may be able to recall that
1 x 10 = 10
2 x 10 = 20
3 x 10 = 30
4 x 10 = 40
5 x 10 = 50 …. and so on.
If we write the same table with 10 as powers of 10 we see 10
1
. That is 1 occurrence of 10 in
the chain of multiplications which could be written as 10 x but there is nothing to multiply
it by so it evaluates to 10.
1 x 10
1
= 10
2 x 10
1
= 20
5

3 x 10
1
= 30
4 x 10
1
= 40
5 x 10
1
= 50 …. and so on
Thinking back to our patterns from squares and cubes what we see is that, for example, 3 x
10
1
gives us 3 followed by 1 zero.
Now let’s think about what would happen if we had 10
2
. 10
2
evaluates to 100 or (10 x 10).
We know this because, from our experiments in squares and cubes, the power of 2, when
combined with 10, tells us how many zeros there will be. Lets apply this logic to a times
table.
1 x 10
2
= 100 = 1 x 100
2 x 10
2
= 200 = 2 x 100
3 x 10
2
= 300 = 3 x 100
4 x 10
2
= 400 = 4 x 100
5 x 10
2
= 500 = 5 x 100
Hopefully, you will see a pattern developing. The rule that the power of 10 decrees the
number of zeros has applied to the number we are multiplying 10
2
by. For example, 5 x 10
2
is the same as saying 5 with two zeros after it.
Let’s try it with 5 x 10
3
. 10
3
is another way of saying 1 with three zeros after it, or the
number 1000. 5 x 10
3
adds 3 zeros to the end of 5. Think of it another way, 5 x 10
3
can also
be thought of as saying 5 x 1 with 3 zeros after it.
A brief history of zero.
Yes this – will be a very brief history, but it’s important to us to understand what a zero
represents. The history of maths, as a conscious activity of calculation and recording, can
be traced back to the first civilizations in the Middle East. These early civilizations are a
6

hotbed for many of the foundational scientific ideas that we hold so true today. In circa
2000 BC, the Babylonians found a method to account for the productivity of farm land (yup
the accountants came first kids). They devised a method for recording volume and area by
way of marking up values, using the tip of a reed. This was known as cuneiform (see
Wikipedia entry on Babylonian Numerals
(1)
). This decimal system didn’t require a zero,
because they never counted nothings, they only counted things that existed.
It wasn’t until the 7
th
century AD, in India, that the use of 0 in a positional numbering
system was developed. Most of us who have had the benefit of a maths education will be
familiar with the use of positional numbering, although we may no know it as such. The
decimal system uses the numbers 0 to 9 to identify the number of units, 10, 100s, 1000s,
10,000s, 100,000s etc as follows:
The number 273098 can be shown by listing the count of each decimal place.
Hundred
thousands
Ten
thousands
thousands hundreds tens units
2 7 3 0 9 8
You could also think of the same number as being written as
Hundred
thousands
Ten thousands thousands hundreds tens units
2 x 100,000 7 x 10,000 3 x 1,000 4 x 0 9 x 10 8 x 1
Equals 200,000 Equals 70,000 Equals 3,000 Equals 0 Equals 90 Equals 8
And we can stack these up as a long addition to get the sum of all the numbers
200,000
70,000
3,000
7

0
90
8
273,098
Unsurprisingly, we have arrived at the number we started with, but maybe you noticed
something about the way the values of each place were worked out. The hundred
thousands column, for example was calculated by writing 2 x 100,000 = 200000. Using
what we have already learned about the powers of 10, we could also have written this as 2
x 10
5
. The powers of 10 has been used to describe how many zeros follow the number
being multiplied. Each column shows an order of power more than the previous column,
like this:
Hundred
thousands
Ten thousands thousands hundreds tens units
100,000 10,000 1,000 100 10 1
10
5
10
4
10
3
10
2
10
1
10
0
Also notice that we introduced the concept of 10 to the power of zero, or 10
0
. From what
we have learned about the powers of 10, we know, and have proved that the value of the
power to which 10 is raised defines the number of zeros in the number. 10 without any
zeros is one. This is actually true of any number raised to the power of zero. They all equal
1.
Try this on your calculator, or type the following into Google:
Calculate 267 ^ 0
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On the UK English Keyboard, the “^” is achieved by holding down the shift key whilst
typing 6. The result will be 1. You can use any number raised to the power of 0 and will
still get the same result. This could be different for different keyboard language layouts,
but it’s a fairly common symbol. Typing “Calculate” followed by a formula into Google, will
bring up the Google calculator, so it’s a pretty useful resource. It also makes it easy to
write in or copy very complex formula and paste it in to the search field, saving a lot of
calculator tapping.
It’s all about the decimal
The introduction of decimal places, and the use of zero to show nothing for a certain place
was a very powerful step indeed. It allowed for a standardised method for recording
complex values with a small number of characters, and to show the concept of zero.
Until now, I have only been writing about whole numbers: numbers that can be written
without a fraction or decimal place. Now, we are going to throw in a curve ball and look at
decimal places. The place system itself allows the use of separate columns for units, 10s,
100s, thousands and an infinity of further places extending to the left of units, each column
being an order of power greater than the previous one.
The place system also works the other way, to the right of units. Let’s put up our number
again, and add some columns to the right.
Hundred
thousands
Ten
thousands
Thousands Hundreds Tens Units Tenths Hundredths Thousandths
100,000 10,000 1,000 100 10 1 0.1 0.01 0.001
10
5
10
4
10
3
10
2
10
1
10
0
10
-1
10
-2
10
-3
Decimals are common in daily use. Just have a look at the price tags when you next go
shopping. If you see an item priced at £8.95 you already know it’s is going to be £8 plus
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some fraction of a pound, represented by the .95. We would say it as “eight pounds and
ninety-five pence” but I want to look a little more closely at what this price represents.
In the same way that the power increases by an order of 1 as we move left, the opposite
happens as we move right. The opposite of a plus is a minus, so as we move right past the
units, the power of 10 start to show negative numbers. The negative number simply
indicates the direction we are moving, and the number is the number of places we are
moving in that direction. For example 10
-1
means we have moved 1 place to the right of
the units position. We know this as a tenth, and we can write a tenth as a decimal.
0.1
Let’s think about what this number represents. Firstly, from what we have done before, we
already know that it shows 10
-1
. We also now know that this shows the number of tenths,
or how many tenth parts of 1 we are using. We can show this as a fraction by writing
1
10 , or even show a pizza divided in to 10 equal slices and pulling out one of them, but
that just makes me hungry.
The main meaning I want to focus on, though, is that 0.1 shows 10
-1
, and that it represents
how many zeros are shown to the left of the number 1, including the zero number of units.
To extend this concept just a little further, when we have a positive power of 10, it indicates
the number of 0s that will appear to the right of 1, and a negative power of 10 shows how
many 0 will appear to the left of the number 1.
So 10
6
will show a 1 followed by 6 zeros and 10
-6
will show a 1 preceded by six zeros,
including the zero units.
10
6
= 1,000,000 - we know this as 1 million and it has a 1 followed by 6 zeros
10
-6
= 0.000001 – we know this as 1 millionth and it has a 1 preceded by 6 zeros,
including the zero units.
10

The introduction of the decimal place has given us an amazing amount of power in what
we can do with powers of 10. We can now move to the incredibly large, to the incredibly
small, without having to write loads of zeros. Just see how much easier it is to write and
read 10
30
or 10
-45
rather than all those zeros.
Thinking back to that pesky 95 pence again. I can write that down as 95 pence, but shops
will usually show it as £0.95. I’m going to write that down using a modified positions table
positions table used earlier.
Ten
thousands
10
4
Thousands
10
3
Hundreds
10
2
Tens
10
1
Units
10
0
Decimal
.
Tenths
10
-1
Hundredths
10
-2
Thousandths
10
-3
0 0 0 0 0 . 9 5 0
I can see from the table that £0.95 represents no units, 9 tenths and 5 hundredths, or, if
you prefer, 95 hundredths. That makes good sense because there are 100 pennies in £1.00,
and we are only showing 95 of them.
New notation and terminology
This is a great time to introduce a new rule about dots in maths. When a dot appears at
the bottom of the line, like a full stop, it is a decimal place. For example “1.5” can be read as
“1 point 5” or “1 decimal 5”. When we use a dot and it’s mid line, it’s another way of writing
multiplication or “x”. The reason we use the dot notation for multiplication is to avoid
confusing the algebraic use of x with a multiplication. Imagine trying to write an equation
2 multiplied by x. 2xx is very ambiguous so instead we would write 2⋅x or simply 2x.
Both of these mean 2 lots of x or 2 multiplied by x so I’m going to use this dot notation
from now on.
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2x 2=2⋅2=4
I also want to add a new term here. The exponent. This is just another, and more official
way of saying “powers of”. Most modern texts use the term “exponent” instead of
“powers of”, and it’s pretty much universal in the maths and science community, so I
should start using it too.
Exponents = Powers of
Exponents of 10 – the new maths super-heroes
How else can we use the exponents of 10 to make our lives easier. Firstly, let me look at
words that are in common use, especially since the explosion in use of computers, mobile
phones and the internet. I’m pretty sure you have heard the words Gigabyte, Kilometre,
megabyte, centimetre, decade, nanosecond. Even though these words have their basis in
science, their usage has become common place to explain the magnitude of values that
are being used. By magnitude, I mean the size of the values. These magnitudes are
directly tied to the exponents of 10, and make it easier to talk about very large or very
small numbers by describing the size of the units we are talking about. As an example, and
as a shout out to those who appreciate ice-cream, would you prefer a centilitre or decalitre
of ice-cream? Lets find out.
Prefix Symbol Factor Exponent
exa E 1000000000000000000 1018
peta P 1000000000000000 1015
tera T 1000000000000 1012
giga G 1000000000 109
mega M 1000000 106
kilo k 1000 103
hecto h 100 102
deca da 10 101
(none) (none) 1 100
deci d 0.1 10−1
centi c 0.01 10−2
milli m 0.001 10−3
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micro μ 0.000001 10−6
nano n 0.000000001 10−9
pico p 0.000000000001 10−12
femto f 0.000000000000001 10−15
atto a 0.000000000000000001 10−18
Before my computer nerd brethren shout out, a quick caveat to the above. Often times,
when we are talking about prefixes in computer terms, we are not discussing the value in
decimal terms, as shown in the table above, but in binary terms. The concept of binary,
bits and bytes is outside the scope of this exploration of exponents, but, if you are curious
to see the full details of the difference, it’s worth having a quick look at the Wikipedia entry
on scientific prefixes (https://en.wikipedia.org/wiki/Unit_prefix). It’s just worth noting that
when your broadband provider or mobile networks talks about your bandwidth, even
though it’s related to computer technology, they are commonly using the decimal
interpretation of the prefixes as it offers a smaller value. Sneaky huh!!!
That aside, a quick look at the table shows the relationship between the prefix, it’s symbol,
the magnitude and it’s exponent. All of these can be used to describe a value, but some
are clearer, and easier to handle than others. This, then, is the whole purpose of scientific
notation, so let’s get on with it.
Rounding around
So I’ve spent quite a while talking about exponents and how the powers of 10 can be used
to save us writing out far too many zeros, and also making our calculations easier to read.
But it’s not practical to only work with powers of 10. There are lots of other numbers, both
incredibly large and incredibly small, that aren’t multiples of 10. 547892, is quite different
from 100,000.
Looking at 547892, we may want to think about how many of those individual numbers are
significant. Will subtracting 2 from the number really have a noticeable impact when we
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are talking about hundreds of thousands. It’s quite common for us humans, even the more
sciency types, accountancy types and statistical types to ignore the smaller parts of a
number and focus only on those values which have a real impact.
We can round a number up or down to focus on the parts of a number which have real
impact. For example, I could write 547892, as 548,000, and it would be a fair approximate
of the value, especially when dealing with values of a similar magnitude. In this case I
have, correctly, reduced the number to 3 significant figures, but maintained the
magnitude of the numbers removed. I have rounded up. The hundreds, tens and units of
the number weren’t, in this example, considered significant and so were reduced to zero,
but the digit in the thousands position has been raised by one to account for 892 being
almost 1000.
Rather than arbitrarily rounding up or down, the most common method of rounding is to
look at the position of the smallest significant digit and the largest digit of amount to be
rounded. , and act upon that. Lets take another look at 547,892 being shown to 3
significant figures. Here’s my approach:
For 547,892 we are looking to show only the 3 left most digits and to round the
remaining digits.
547,892 can also be written as 547,000 + 892.
The left most digit of the remainder, 892, is 8, is in the hundreds position and is
greater than or equal to 5, so is closer to 1,000.
We can replace 892 with 1,000 in our sum which can now be written as 547,000 +
1,000 which equals 548,000.
In this example, I have been able to maintain the impact of the digits being discarded, even
after removing them from the final number. If we had 547392 instead, I would have
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chosen to round down, as 3 is less than 5, so indicates the number of hundreds we have is
closer to 0 than 1000. Again, the impact of the digits being discarded is maintained in the
final number, whilst the number focuses on the significant figures.
The crux of this rule, then, is that if a digit is equal to or greater than 5, round it up, if it is
less than 5, round it down. After a little practice, this approach becomes second nature.
Moreover, most spreadsheets and programming languages will let you define the number
of significant figures, and how roundings are to be carried out. Instead of arbitrarily
rounding everything up, or rounding everything down, we have rounded to the nearest
1000.
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97% of statistics are made up on the spot
`
As can be seen from the above, somewhat exaggerated and admittedly contrived example,
rounding and significant figures can be misused to present a picture far away from the
fact. The two lessons here are don’t believe everything you read, especially when
“statistics show”, and, if you want to be a good scientist, you need to use an appropriate
16
We are going to look at a town with a population of 10,000. As with
any town, there is some unemployment and, in this case 960 people
are unemployed. The town council has been asked to publish the
statistics in the local paper so their statistician chats with the council
leaders, and they ask him to make it look good.
Under these instructions, our statistician decides to show the figure
rounded down to the nearest 1000. This means that any number
less than 1000, will be round down, or reduced down to the nearest
thousand. This suits our council leaders just fine. Why? Well
because if we round 960 down to the nearest 1000, we get 0. 960 is
not quite 1000, so has been demoted to zero. As a result the
Council reports that unemployment, out of a population of 10,000 is
at 0%, even though 960 people are, in fact unemployed.
Under the same report, the Council has also been asked to show it’s
savings for the year. Out of a budget of £10,000, they have
managed to save only £120. Our statistician is asked to produce a
report on the saving made rounded up to the nearest thousand.
Rounding £120 up to the nearest £1000 shows a saving of £1000, or
10% of the budget. That’s quite a difference from £120 or less than
2% of the budget.

number of significant figures, and make sure your roundings are fair. If not, you may be
asked to prove your figures.
Scientists expect to have their work studied and rigorously tested by their peers. It’s how
theories gain validity and its this basis of scientific method that makes the outcomes so
robust. With reputation being such an important aspect of a scientists career, a great deal
of care is taken in how the results of experimentation and data analysis are presented.
17

Exponents and their superhero powers
Well we’ve looked at the exponents of 10, significant figures and roundings. Now it’s time
to bring them all together and wield the superpower of exponents.
Exponents, that share the same base make multiplication easy. Each exponent has a base
and a power to which the base is raised. For example the exponent 10
3
has base of 10 and
a power of 3. As we know:
10
3
=10⋅10⋅10=1000
(remember that a mid line dot is a multiplication symbol)
So what if we need to multiply numbers together. If they share the same base, we can
shortcut all the trying to figure out what the numbers actually are, then multiplying them
together with just some simple addition.
Consider this formulae:
a=10
3
⋅10
4
What is the value of a? Lets write it out in full using all the times we have to multiply by 10.
I’ll use brackets to group the exponents together.
(10⋅10⋅10)⋅(10⋅10⋅10⋅10)
Now take a look at this. We can actually ignore the brackets because it’s all multiplication.
So we have
10⋅10⋅10⋅10⋅10⋅10⋅10
Which is 10 multiplied by itself 7 times which is 10
7
. A quick look back a the formula
a=10
3
⋅10
4
and I can see that the powers of 3 and 4, when added together, give a power
of 7.
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It’s such a great shortcut. If we are multiplying exponents together, and they share the
same base, we can just add the powers together, to get the new value, not complicated
huge multiplications required. No wonder scientists love scientific notation.
But there’s more. In maths, if you can do a thing, generally speaking, you can also do the
opposite. What is the opposite of multiplication? Division. To divide exponents with the
same base, simply subtract the powers. And, because you can do addition and subtraction
at the same time, working from left to right, you can carry out complex multiplications and
divisions of exponents with the same base just by adding and subtracting the powers.
So, 10
8
÷10
3
⋅10
15
−10
10
is basically the same as saying 10
(8-3+15-10)
. By carrying out the
simple sum in the brackets we get 10
10
. That’s really saved a lot of messing about with
really big numbers.
The science of scientific notation
It doesn’t end there, though. The magic of exponents is about to get really wild.
So far, I have been deliberately dealing with pure base 10 exponents. I have touched, very
briefly on significant figures, and roundings and it’s now time to bring that in to play.
Remember, or look back if you like, that we looked at the number 547892. I wrote this
number down as 3 significant figures, and rounded the 892 element to the nearest 1000,
giving us 548000. Notice that there are 3 zeros at the end of that number. Also remember
that 1000 has 3 zeros, but can also be written as 10
3
.
We can express any number as a multiplication problem. For example 10 can be written as
2⋅5 or 1⋅10 or 5⋅2 . Similarly, we can express 500 as 2⋅250 or, more
importantly for scientific notation, as 5⋅100 .
We can take 5⋅100 a step further and expand it even more by expressing 100 and as an
exponent. As we know 100 can be written as 10
2
. By substituting the number 100 with
10
2
our number 500 can also be written as:
19

5⋅10
2
and this is 500 written as scientific notation. Let’s try it with a bigger number.
6,000,000,000 (spoken as 6 billion) is a large number, and contains a lot of zeros. By using
scientific notation we can make this number a lot easier to work with.
6000000000 can be written as 6⋅1000000000 .
by writing 1000000000 as an exponent (with 9 zeros) we get 10
9
.
so to express 6000000000 as an exponent we write 6⋅10
9
Isn’t that much easier to read?
Our superpowers aren’t over yet. Because multiplication, and division can be carried out at
the same time in the order of operations, we can multiply both our exponents together by
reodering the problem, and carrying out some simple arithmetic. If you are not sure about
the orders of operation, Google BODMAS, BEDMAS of PEMDAS – they’re all the same
thing and will help make your maths life a lot easier.
Lets multiply our 2 exponents together.
6⋅10
9
⋅5⋅10
2
because we can carry out all multiplications at the same time we we can group the
exponents together.
6⋅5⋅10
9
⋅10
2
lets calculate the exponents first by adding the powers of 10.
6⋅5⋅10
11
now let’s multiply the 6 and 5 together
30⋅10
11
20

but look at the number 30. We have another power of 10, or 10
1
. We can add that 1 power
of 10 to our exponent and remove it from our 30.
3⋅10
12
By using exponents we have taken a rather daunting looking problem and made it really
simple for us to calculate. Imagine how horrible the problem would look written out in full:
6000000000⋅500=3000000000000 which alsoequals 3⋅10
12
I think that using scientific notation is already making our lives quite a lot easier.
The point of the decimal point
Lets rewrite the positional table.
10
4
10
3
10
2
10
1
10
0
. 10
-1
10
-2
10
-3
10
-4
10000s 1000s 100s 10s 1s . 10ths 100ths 1000ths 10000ths
As you can see the further we move away from the decimal point in the middle, the more
zeros exist in our number to the right and to the left.
To explore this I want to look at what happens to the decimal place in relation to the 1 as
we move left and right.
At 1 we can also write 1.0, which is written to 1 decimal place. The decimal place is bang in
the middle, between the 1 and zero. There are no zeros between 1 and the decimal point.
1 is also written as 10
0
.
Lets write 10 to 1 decimal place. Written as 10.0 the decimal place has moved in relation to
the one. It has jumped one position to the right away from the one. There is 1 zero
between the 1 and the decimal point. 10 is also written as 10
1
.
Okay, lets continue and look at 100. Written to 1 decimal place, we have 100.0, and the
decimal place has again jumped 1 place to the right in relation to the 1. There are 2 zeros
between the 1 and the decimal point. 10 is also written as 10
2
.
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Hopefully you can see a direct relationship between the digit 1, the decimal place, and the
number of zeros between them.
How about smaller numbers. Lets start with 1 again. 1 can be written to 1 decimal place as
1.0. The decimal place is smack bang between the 1 and zero, and there are no zeros
between the 1 and the zero. 1 can also be written as 10
0
.
1 tenth can be written to 1 decimal place as 0.1 . The 1 and zero have switched places as
the decimal place has jumped 1 space to the left., and the one is next to the decimal place.
There are no zeros between the 1 and decimal point. 1 tenth can also be written as 10
-1
.
Hmmm, okay! A little different from the positive powers of 10. Lets continue to make the
numbers smaller.
1 hundredth can be written to 2 decimal places as 0.01. Now the decimal place has jumped
another place to the left, in relation the digit 1. There is now 1 zero between the 1 and the
decimal point. 1 hundredth can also be written as 10
-2
.
Finally let’s quickly checkout 1 thousandth. We can write 1 thousandth as 0.001 to 3
decimal places. The decimal place has jumped another place the left, and there are now 2
zeros between the digit 1 and the decimal point. 1 thousandth can be written as
10
-3
.
So even moving to the exponents of 10 less than 1, there is a pattern. The way I like to
think of it is that we have to have a zero to the left of the decimal place to indicate no
units. That convention really helps us as it makes sure that we have the same number of
zeros to the left of the 1 digit as indicated in the exponents power. We just have to
remember to put a decimal place between the left most zero and the next one for the
number to make sense.
Lets try a couple of problems to see the decimal place moving and the number of zeros
between the digit 1 and the decimal place.
The number 2 million can be written as 2⋅10
6
. What is that number written as a
decimal? First, lets write down the number 2 down as a decimal.
22

2.0
As you can see the decimal point is right up against the 2, and there are no zeros between
the digit 2 and the decimal point. Now lets multiply 2 by 10
6
by moving the decimal place
6 places to the right, and filling the gaps with zeros.
2000000.0
Now the decimal point has jumped six places to the right, and there are 6 zeros between
on and the decimal point.
Here’s another problem to work on in a similar manner.
The number 3 thousandths can be written as 3⋅10
− 3 .
What is that number written as a
decimal? First I am going to write down 3 as a decimal.
3.0
Now I am going to move the decimal 3 places to the left and add zero’s to fill the gaps.
.0030
Finally I can tidy up this number by adding a zero in front of the decimal point to show no
units, and remove the trailing zero as we are working to 3 decimal places.
0.003
That’s fantastic. Not only can exponents help us by making numbers easier to read and
handle, they can also make multiplication and division of large numbers a snap, and now,
they actually tell you how many zeros are supposed to be in your answer giving you a very
handy sanity check.
Full-on notate like a scientist.
Our final step in our journey to notate like a scientist is to work with our decimal place.
Scientific notation’s convention is that must have one and only one number to the left of
the decimal place. I want to show you how to take a very large or very small number, and
write it in scientific notation. The first number I am going to deal with is one of the
23

numbers I have been using in a study into the change in average distance of the Moon
from the Earth. The current distance of the Moon from the Earth is reported by NASA to
be 384,400km. Now that is slightly more than a quick stroll on a Sunday morning, so let’s
see how we need to treat this number.
My study into the distance needs to account for very small changes over time, so I want to
use 4 significant figures, which is the same number of significant figures that NASA has
supplied the distance in. How very helpful and precognitive of them.
I now need to write this number in scientific notation, but I can only have one number to
the left of the decimal place, as per the scientific notation convention. What I will do first is
rewrite the number but with only 1 significant figure; the three. I’m not actually going to
do any rounding here, just put the other numbers to the side for now.
300,000km
Now I can write that number as an expression with an exponent of 10 with a power of 5.
3⋅10
5
km
But I want to keep all my significant figures. So what I can do is substitute the numbers I
put to one side back into the expression as decimals of 3⋅10
5
.
3.844⋅10
5
km
And that’s our scientific notation of 384,400km. I can also look at it another way. In order
to have only 1 number to the left of the decimal place, I have to moved the decimal place
of 384, 400.0km to the left 5 times, like this.
3.844000
I can now ditch any trailing zeros, as the will not figure in my exponential expression.
3.844
24

Now, because I moved the zero to the left 5 times, effectively dividing by 10
5
, I will need to
multiply this complete number by 10
5
in order to restore it’s value. So I tag that on to the
end of the expression.
3.844⋅10
5
km
and that is my scientific notation for 384,400km.
Now to the very small.
The annual change in the distance of the Moon from the Earth, is 4cm a year. But I need to
do my work in like terms, and km is a very common measurement in physics. On top of
that, whilst 4cm in the space of a year is not a very big number, over billions of years, that
small distance will actually add up to be quite significant, so it’s prudent for me to convert
cm to km to ensure I am working with consistent measurements.
So 4cm in terms of km is 0.00004km. I want to write this in scientific notation. If I look at
the number I can see there are 4 zeros between the 4 and the decimal point, but I want to
shift the 4 another additional place to the left. Alternatively stated, I want to move the
decimal place, 5 places to the right so that the 4 is on the left of it.
4.00000
Now, as with the big number, I can ditch the zeros, as these will be explained by the
exponent.
4
Finally, I can add the exponent. Remember that this is the multiplying factor that will
return our number back to it’s original state. So to move the 4 back to it’s position 5
decimal places to the right, we are going to have to multiply it by 10
-5
. Let’s add that back
in to our expression.
4⋅10
−5
Let me work through another really small, slightly more complicated number to reinforce
the use of decimal places in scientific numbers.
25

Let’s use 0.000007645 for no other reason that it happened to be what I randomly tapped
in on my keypad.
Firstly I would like to show this number to 3 significant figures. So I am going to lose the
last digit 5 and round it to the next position. As it’s greater or equal to 5, I am rounding up
and so will increase the 4 by 1 giving me 0.00000765.
Now let’s get that 7 to the left of the decimal point. How many times will I need to move
the decimal place to the right to put the 7 to the left of it. Counting along, I make it 6
jumps. Here’s that number with the decimal place moved to the right 6 places, with 3
significant figures..
7.65
Now I need to let the reader know what the conversion factor will be to put this number
back to a tiny decimal. I counted 6 jumps to right to move the decimal place, so will put
that as the power of my exponent remembering to add a – sign to show we are decreasing
the magnitude.
7.65⋅10
− 6
Bringing it all together
As with all good superhero stories, I should end this explanation of scientific notation with
a final grand battle. This battle is between the evil of dividing really big numbers, and the
heroics of scientific notation in making the calculation really easy.
Our evil numbers are
592000000 and 2812000000
and we are going to divide the smaller one by the bigger one.
I’m going to keep the 4 significant figures shown for the calculation so I don’t risk losing
any detail, but there’s an awful lot of zeros. I even remember some calculators that
26

couldn’t handle such large numbers, so I can’t even cheat. Lets use scientific notation to
make these numbers easier to work with.
592000000 can be written with one decimal to read 592000000.0. I want to move the
decimal point so that only the 5 is on it’s left. That takes, by my count, 8 jumps. Keeping
to 4 significant figures the new number is
5.920
but I have to add back in the conversion factor. My decimal place jumped 8 places to the
left, meaning that I need to multiply by an exponent with a power of 8. That power should
be positive as I will want to make the number bigger again. So my exponent is
5.920⋅10
8
Repeating for 2812000000, let me first write it with 1 decimal place so I can see the decimal
point.
2812000000.0
Now I want to jump the decimal place so it is just to the right of the 2. How many jumps?
I make it 9. After ditching the trailing non-significant zeros, the number now reads.
2.812
Now add back in my conversion factor. I counted 9 jumps, and the conversion factor will
need to be positive to make the number bigger again. So we end up with
2.812⋅10
9
Now lets do our division. Notice how I keep the number and it’s exponent of 10 together
using brackets. This is a great way of reminding us that everything in the the brackets on
the left, is being divided by everything in the brackets on the right.
(5.920⋅10
8
)÷(2.812⋅10
9
)
27

Because of the nature of exponents, we can first calculate the value of the exponents. We
know that, when multiplying or dividing, all we have to do is add or subtract respectively.
In this case it’s division so 10
-8
/ 10
-9
gives us 10
-1
. We’ve dealt with that, so can move it to
the end of the expression ready to convert our number back again.
(5.920)÷(2.812)⋅10
−1
Now lets deal with dividing our nice small numbers.
5.920÷2.812=2.105
I’ve kept the answer to 4 significant figures and can substitute this back in to our
expression.
2.105⋅10
− 1
If you like, we can expand this out to give the full decimal.
0.2105
What have we achieved
So looking at the last section, where we brought everything together, what have we
achieved by using scientific notation.
We can take very large numbers or very small numbers and convert them in to values that
we can easily work with.
We can easily convert different units to like units to make calculations easy and
meaningful.
We can use exponents to avoid nightmare multiplication or subtraction of very large or
very small numbers by adding or subtracting the powers of the exponent.
Most of all, though we have learned the scientists always look for the most simple
explanation, and the easiest way of doing or explaining things.
28

Bibliography
(1) Wikipedia Entry - Babylonian Numerals ~ https://en.wikipedia.org/wiki/Babylonian_numerals
obtained 2nd
March 2017.
29

Explanation of scientific notation

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