1. a × 10n Topic 10 Scientific Notation 2 3 4 5 6
Definition: Scientific Notation, also known as Standard Form is a way of writing numbers
that are too large or too small to be written in the conventional way. Writing in
scientific notation allows us to eliminate zeros in front of a very small decimal or
behind a very large number. This very concise way of writing numbers is often
used by scientists, mathematicians and engineers, who find that this way of writing
numbers is more convenient than the usual decimal form of the number.
In scientific notation all numbers are written like this:
a × 10n
Where a is any real number called the coefficient and it is usually a decimal with a
value between 1 and 10 ( 1 < a < 10) , and the exponent 10n will always have a base of
10 and n is always an integer.
If a = 1 or a = 10 then we usually omit the a and write the number as 10n.
1. How to convert large numbers into Scientific Notation.
We start by looking at simple numbers that are all powers of ten; these numbers are the building
blocks of the decimal number system and are easily converted into Scientific Notation as can be
seen in the table below.
Name
Number
Scientific
Notation
one 1 10
ten
hundred
100 2
thousand
1,000 3
ten thousand
10,000 4
hundred thousand
100,000 5
million
1,000,000 6
For example the number ten thousand is 10,000 = 10 x 10 x 10 x 10 = 104
The number one is a little unusual as it can be written in a number of ways, for example the
number one can be written as 20 , 40 etc but we used the form 1 = 100 to be consistent with the
other numbers so that they all use a base of ten.
It is easy to convert any of these powers of ten into Scientific Notation – all you need to do is to
count the number of zeros that follow after the initial 1, so 10,000,000,000,000 = 1013
Page | 1

2. (a) 100,000,000,000 (a) 30,000,000,000 (b) 4 hundred million =
Topic 10 Scientific Notation
Example 1: Convert the following numbers into Scientific Notation.
(a) 100,000,000,000
Solution(a): 100,000,000,000 =
(b) A hundred million
(c) Ten trillion
(1 followed by 11 zeros)
1011
Solution(b): A hundred million =
100,000,000 =
108
(1 followed by 8 zeros)
Solution(c): Ten trillion =
10,000,000,000,000 =
1013
(1 followed by 13 zeros)
Next we look at numbers that start with a single digit and then have trailing zeros.
Example 2: Convert the following numbers into Scientific Notation.
(a) 30,000,000,000
Solution(a): 30,000,000,000 = =
(b) 4 hundred million
3 x 10,000,000,000
3 x 1010
(c) 500,000,000,000,000
(1 followed by 10 zeros)
Solution(b): 4 hundred million =
400,000,000
4 x 100,000,000 =
Solution(c): 500,000,000,000,000 =
4 x 108
5 x 100,000,000,000,000
5 x 1014
(1 followed by 8 zeros)
(1 followed by 14 zeros)
Lastly we look at how to convert any large number into a x 10n its Scientific Notation.
The process involves moving the decimal point to the left until the new value of the number is
between 1 and 10 this new value will be our coefficient a, while the number of places that we
move the decimal point will be our integer power n.
Example 3: Convert the following numbers into Scientific Notation.
(a) 87,000
(d) 150,000
(g) 16,400
(j) 54
(m) 23.5 thousand
(b) 16,000,000
(e) 2,400,000
(h) 115,000
(k)
(n) 156 billion
(c) 200,000,000
(f) 8,210,000,000
(i) 1,234
(l) 32 million
(0) million
Page | 2

3. 107 Topic 10 Scientific Notation Solution (a):
87,000 is written as 87,000.
Decimal point
We move the decimal point to its new location and remove the trailing
zeros so that it is now 8.7 this is the value of the coefficient a = 8.7
Also since the decimal point moved 4 places to the left the power of n = 4
So 87,000 =
8.7 × 104
Solution (b):
Solution (c):
16,000,000. =
200,000,000 = 16 2
107
108
move the decimal point 7 places to the left.
move the decimal point 8 places to the left.
Solution (d):
Solution (e):
150,000.
2,400,000. = 15 24
105
106
move the decimal point 5 places to the left.
move the decimal point 6 places to the left.
Solution (f):
8,210,000,000 = 8 21
109
move the decimal point 9 places to the left.
Solution (h):
Solution (i):
115,000.
1,234. =
1 15
1 234
105
103
move the decimal point 5 places to the left.
move the decimal point 3 places to the left.
Solution (j):
54. =
5.4
101
move the decimal point 1 places to the left.
Solution (k): =
move the decimal point 2 places to the left.
Solution (l):
32 million =
32,000,000. =
3.2
107
move the decimal point 7 places to the left.
Solution (m):
23.5 thousand =
23,500. =
2.35
104
move the decimal point 4 places to the left.
Solution (n):
156 billion =
156,000,000,000. =
1.56
1011
move the decimal point 11 places to the left.
Solution (l):
23.45 million=
23,450,000. =
2.345
107
move the decimal point 7 places to the left.
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4. Topic 10 Scientific Notation
Exercise 1: 1.
Convert the following numbers into Scientific Notation.
For example 10,000 would be written in Scientific Notation as 104.
(a) 100,000,000,000,000
(b) one thousand million
(c) one hundred trillion 2.
Convert the following numbers into Scientific Notation.
For example 80,000 would be written in Scientific Notation as 8 x 104.
(a) 900,000,000,000
(b) 40 thousand million
(c) 8,000,000,000,000 3.
Write down each of the following numbers in standard form.
For example would be written in standard form as 3.4 x 104.
(a) 324
(e) 356,000,000
(i) 8,600
(m)
(b) 32
(f)
(j)
(n)
(c) 3,890,000
(g)
(k)
(o)
(d) 120
(h) 67,000
(l) 84
(p) 4.
Write down each of the following numbers in standard form.
For example 22,100,000 would be written in standard form as 2.21 x 107.
(a) 1 ,903
(d)
(g)
(b) 346, 000,000
(e)
(h) 5,120,000
(c) 28 ,000,000,000,000,000
(f) 281,000
(i) 5.
For each of the following numbers
(i) Write it out in figures.
(ii) Write it in scientific notation.
(a) 20 thousand
(d) 150 thousand
(g) 2.3 billion
(b) 16 million
(e) million
(h) 9 thousand million
(c) 200 million
(f) million
(i) thousand
(j) sixteen thousand five hundred
(k) one million two hundred and fifty thousand
Page | 4

5. Decimal point moved 2 points to the right
Topic 10 Scientific Notation
2. How to convert Scientific Notation into large numbers
When a number is written in its Scientific Notation form a x 10n where n is a positive integer
then it can be converted into a normal number.
The method that is used is to move the decimal point in the coefficient a, n places to the right.
Example 1: Convert the number x 102 into an ordinary number.
Solution:
We take the term x 102 and start with its coefficient and move its
decimal point 2 places to the right.
becomes 123.4
Decimal point moved 2 points to the right
So x 102 = 123.4
Example 2: Convert the number x 103 into an ordinary number.
Solution:
We take the term x 103 and start with its coefficient and move its
decimal point 3 places to the right.
becomes 1234.
Decimal point moved 3 points to the right
So x 103 = 1234
Example 3: Convert the number x 107 into an ordinary number.
Solution:
We take the term x 107 and start with its coefficient and move its
decimal point 3 places to the right.
becomes 12,340,000.
Decimal point moved 7 points to the right
So x 107 = 12,340,000
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6. 4.5 x 103 103 108 Topic 10 Scientific Notation
Example 4: Write down each of the following numbers as a normal number
(a) 4.5 x 103
(d) 4 x 106
(g) 5 x 102
(b) x 103
(e) x 102
(h) x 107
(c) x 104
(f) x 104
(i) x 101
Solution (a):
Solution (b):
4.5 x 103
x 103 =
4,500
3,872.2
move the decimal point 3 places to the right.
Solution (c):
Solution (d):
3.87 x =
4 x =
38,70 0
4,000,000
move the decimal point 4 places to the right.
move the decimal point 6 places to the right.
Solution (e):
Solution (f):
2.999 x 102
2.999 x 104 =
299.9
29,990
move the decimal point 2 places to the right.
move the decimal point 4 places to the right.
Solution (g):
5 x 102 =
500
move the decimal point 2 places to the right.
Solution (h):
1.83 x =
18,300,000
move the decimal point 7 places to the right.
Solution (i):
3.7 x 101 = 37
move the decimal point 1 places to the right.
Exercise 2: 1.
Write down each of the following numbers as a normal number
For example 3.87 x 105 would be written as 387,000.
(a)
102
(b)
101
(c)
104
(d) 5 8
(g)
107
103
(e)
(h)
103
105
(f) 2
(i)
100
107
(j) 5
106
(k) 4 13
104
(l)
1010
(m) 3 45
(p) 6 34
106
101
(n)
(q)
103
102
(o)
(r)
108
103
Page | 6

7. 0.0 0 0 01 has 5 zeros to the left of the number 1 so 0.00001 = 10-5
Topic 10 Scientific Notation
3. How to convert small numbers into Scientific Notation.
We start by looking at simple numbers that are all powers of ten; these numbers are the building
blocks of the decimal number system and are easily converted into Scientific Notation as can be
seen in the table below.
Name
Decimal
Form
Fractional
Scientific
Notation
one 1 10
tenth
0.1 -1
hundredth
0.01 -2
thousandth
0.001 -3
ten thousandth -4
hundred thousandth -5
millionth -6
For example the number ten thousandth is =
= 10- 4
The number one is a little unusual as it can be written in a number of ways, for example the
number one can be written as 20 , 40 etc but we used the form 1 = 100 to be consistent with the
other numbers so that they all use a base of ten.
It is easy to convert any of these decimals into powers of ten, you just count the number of zeros
to the left of the first non zero number.
has 5 zeros to the left of the number 1 so = 10-5
For example
For example
has 4 zeros to the left of the number 2 so = 2 x 10-4
has 3 zeros to the left of the number 3 so = 3.24 x 10-3
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For example

8. Topic 10 Scientific Notation
Example 1: Convert the following numbers into Scientific Notation.
(a)
(d)
(g)
(j)
(b)
(e)
(h)
(k)
(c) 0 19
(f)
(i)
(l)
Solution(a): =
4.3 x 10-2
Since there are 2 zeros at the start.
Solution(b): =
3.57 x 10-4
Since there are 4 zeros at the start.
Solution(c): 0 19 =
1.9 x 10-1
Since there is 1 zero at the start.
Solution(d): =
9.8 x 10-3
Since there are 3 zeros at the start.
Solution(e): =
7.04 x 10-5
Since there are 5 zeros at the start.
Solution(f): 0 06 =
6 x 10-2
Since there are 2 zeros at the start.
Solution(g): =
8 x 10-7
Since there are 7 zeros at the start.
Solution(h): =
5.22 x 10-1
Since there is 1 zero at the start.
Solution(i): =
2.007 x 10-5
Since there are 5 zeros at the start.
Solution(j): =
7.8 x 10-9
Since there are 9 zeros at the start.
Solution(k): =
Solution(l): =
1.4 x 10-3
4.5 x 10-4
Since there are 3 zeros at the start.
Since there are 4 zeros at the start.
Exercise 3: Convert the following decimal numbers into Scientific Notation.
(a)
(d) 0 8
(g)
(j)
(m)
(b)
(e)
(h)
(k)
(n)
(c)
(f)
(i)
(l)
(o)
Page | 8

9. Topic 10 Scientific Notation
4. How to convert Scientific Notation into small numbers
In these situations we will be given a number written in scientific notation a x 10n where n will
be a negative integer. There are three possible strategies for doing this conversion and an
example of each is given below.
Example 1: Change 3.1 x into an ordinary number.
Solution:
Our first strategy is to convert the power 10n to a decimal and then multiply the
coefficient a and the converted power together to obtain the result.
3.1 x 10- 4 =
3.1 x
This method is seldom used as it is not concise and can easily lead to very messy
calculations especially if n is a large integer.
In our second strategy we start with the coefficient a and move the decimal point n
places to the left filling in with zeros when necessary.So in this situation we take
the coefficient a = 3.1 and move the decimal point 4 places to the left to get the
result So
3.1 x 10- 4 =
In our third strategy we remove the decimal point from the coefficient a = 3.1 to get
31 and then adding 4 zeros to the left of a to get the result
This is a very concise method as it involves a few steps that usually can be
performed at once to quickly give the result.
Example 2: Write each of the following as an ordinary number :
(a) 4 3
(d) 7 65
10-3
10-7
(b) 2 1
(e) 3 11
10-5
10-1
(c) 6 41
(f) 1 253
10-2
10-8
Solution(a): 4 3
Solution(b): 2
10-3
10-5 =
0.0043
0.0000
3 zeros to the left of 43
5 zeros to the left of 2
Solution(c): 6 41
Solution(d): 7 65
Solution(e): 3 11
10-2
10-7
10-1 =
0.311
2 zeros to the left of 641
7 zeros to the left of 765
1 zeros to the left of 311
Solution(f): =
8 zeros to the left of 1253
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10. Topic 10 Scientific Notation
Exercise 4: Write each of the following as an ordinary number.
For example 7 12
10-3 would be written as 1.
(a) 2 6
10-3
(b) 2
10-6
(c) 2 4
10-2
(d) 5 55
10-7
(e) 7 06
10-3
(f)
10-2
(g) 8
10-6
(h) 9
10-5
(i)
10-5
(j)
10-2
(k) 6 53
10-1
(l) 3
10-10
(m) 9 167
10-2
(n) 9
10-6
(o)
10-1
(p) 3 23
10-10
(q)
10-6
(r)
10-4
Page | 10

11. Topic 10 Scientific Notation 5. Problems involving Scientific Notation
There are many situations that involve using numbers written in Scientific Notation in
calculations. This can happen when we use very large or very small numbers in formula such as
those used by physicist, chemists as well as engineers.
The main strategy for performing multiplications that involve Scientific Notation is to follow
these 4 steps.
Step 1: Separate the terms into coefficients and powers of 10
Step 2: Multiply the coefficients together and the powers of ten together as separate
calculations.
Step 3: Convert the product of the coefficients into Scientific Notation if needed.
Step 4: Combine the individual parts to obtain the final result.
Here are some examples of this process in action.
Solution:
(7.2 x 103 ) x (2.5 x 105 ) = =
( 7.2 x 2.5) x ( 103 x 105 )
18 x 108
1.8 x 101 x 108
1.8 x 109
Step 1
Step 2
Step 3
Step 4
Example 2: Calculate (7.51x 1020 ) x (9 x ) giving your answer in Scientific Notation.
Solution:
(7.51 x 1020 ) x (9 x ) = =
( 7.51 x 9) x ( 1020 x )
67.59 x 10-6
6.759 x 101 x 10-6
6.759 x 10-5
Step 1
Step 2
Step 3
Step 4
Solution:
(6.32 x 10-2) x ( 3.2 x 10-13)= =
( 7.51 x 3.2) x ( 10-2 x )
x 10-15
x 101 x 10-15
x 10-14
Step 1
Step 2
Step 3
Step 4
Example 4: Calculate (1.5 x 108) x ( 2.6 x 10-6) giving your answer in Scientific Notation.
Solution:
(1.5 x 108) x ( 2.6 x 10-6) =
( 1.5 x 2.6) x ( 108 x 10-6 )
3.45 x 102
Step 1
Step 2
Page | 11

12. Topic 10 Scientific Notation
The main strategy for performing a division that involve Scientific Notation is to follow these 4
steps.
Step 1: Separate the terms into coefficients and powers of 10
Step 2: Divide the coefficients together and the powers of ten together as separate calculations.
Step 3: Convert the division of the coefficients into Scientific Notation if needed.
Step 4: Combine the individual parts to obtain the final result.
Here are some examples of this process in action.
Example 5: Calculate
giving your answer in Scientific Notation.
Solution: =
Step 1 =
0.125 x 10-12
1.25 x 10-1 x 10-12
1.25 x 10-13
giving your answer in Scientific Notation.
Step 2
Step 3
Step 4
Example 6: Calculate
Solution: =
Step 1
0.25 x 10-3-(-7) =
0.25 x 104
2.5 x 10-1 x 104
2.5 x 10-3
Step 2
Step 3
Step 4
Example 7: Calculate (7.2 x 103 )
(2.5 x 105 ) giving your answer in Scientific Notation.
Solution:
(7.2 x 103 )
(2.5 x 105 ) = =
Step 1 =
2.88 x 10-2
Step 2
Page | 12

13. Topic 10 Scientific Notation
Another common situation is to use Scientific Notation in formula and in multiple calculations.
Example 8: There are approximately 5.9 x miles in one light year. How many miles are
there in 2,500 light years? Write your answer in scientific notation.
Solution:
Number of miles = =
( 2,500 ) x (5.9 x 1012 )
( 2.5 x 103 ) x (5.9 x 1012 ) =
( 2.5 x 5.9 ) x (103 x 1012 )
14.75 x 1015
1.475 x 101 x 1015
1.475 x 1016
Step 1
Step 2
Step 3
Step 4
Example 9: The number of atoms in 12 grams of carbon is 6.12 x Find the number of
atoms in 1 gram of carbon. Write your answer in scientific notation.
Solution:
Number of atoms = = =
5.1 x 10-1 x
5.1 x
Example 10: The amount of energy that is equivalent to a mass of m kg is given by the equation
E = mc2 where m is the mass in kg and c = 3 x 109 m/sec is the speed of light.
(a) How much energy is contained in a single proton that weighs
m = kg?
(b) 1 gram of hydrogen contains 6.02 x 1023 protons, how much energy is there in
1 gram of hydrogen?
Solution(a): We express the mass in Scientific notation.
m = = x 10-27 E =
mc2
(1.672 x ) x ( 3 x 109 )2
(1.672 x ) x ( 3 x 109 ) x ( 3 x 109 ) =
(1.672 x 3 x 3 ) x (10-27 x 109 x 109)
x 10-9
x 101 x 10-9
x Joules
Step 1
Step 2
Step 3
Step 4
Solution(b): E =
( 6.02 x 1023 ) x ( x 10-8 ) =
( 6.02 x ) x (1023 x 10-8 )
x Joules
9,058,889,600,000,000 Joules
Step 1
Step 2
Page | 13

14. computer take to do 8 million calculations?
Topic 10 Scientific Notation
Exercise 5: 1. 2.
(a) The distance between the planet Earth and the sun is a approximately miles.
Write this distance in Scientific Notation.
(b) Jupiter's closest satellite is called Amalthea and is approximately miles from
the centre of the planet. Write this distance in Scientific Notation.
What is written in standard form? A B
C 3,015
D 30,150 3. 4. 5.
An electrons mass is about kg.
Write this number in Scientific Notation.
The Earths mass is about kg
Calculate the following, giving your answer in standard form.
(a) 42 million times 56 million.
(b) x
(c) ( 1.2 x 105 ) x ( 2.5 x 109 )
(d) ( 4 x 10-3 )
( 8 x 108 )
(e) ( 2.9 x 103 ) x ( 5.9 x 103 )
(g)
(f) 1
(h) 6.
There are an estimated 1022 stars in the visible universe. It has been suggested by a scientist
that 0.03% of these stars will have earth like planets around them.
Which of the following calculations will give you the number of earth like planets in the
visible universe? A
B. 3
1020
C. 3
1018
D. 3
1026 7. 8. 9.
There are approximately 5.9 x miles in one light year.
How many miles are there in 3,500 light years? Write your answer in scientific notation.
The mass of one proton is approximately 1.7 x gram.
Find the mass of 4 million protons. Write your answer in scientific notation.
A certain computer does one calculation in 6 x 10-7 second. How long does this
computer take to do 8 million calculations?
10. The number of atoms in 12 grams of carbon is 6.12 x Find the number of atoms in 18
grams of carbon. Write your answer in scientific notation.
11. The amount of energy that is equivalent to a mass of m kg is given by the equation
E = mc2 where m is the mass in kg and c = 3 x 109 m/sec is the speed of light.
How much energy is contained in an object that weighs a million Kg
Page | 14

15. 1.(q) 0.000 001 566 Topic 10 Scientific Notation Solutions
Exercise 1: 1.(a) 1014
(b) 109
(c) 1014
2.(a) 9 x 10 11
(b) 4 x 10 10
(c) 8 x 1012
3.(a) 3.24 x 102
(b) 3.2 x 101
(c) 3.89 x 106
(d) 1.2 x 102
3.(e) 3.56 x 10 8
(f) 1.9 x 10 10
(g) 8.9 x 10 10
(h) 6.7 x 104
3.(i) 8.6 x 103
(j) 2.3 x 103
(k) 4.25 x 105
(h) 8.4 x 101
3.(m) 6.7 x 10 7
(j) 4.1 x 10 4
(k) 5 x 10 9
(h) x 103
4.(a) x 103
(b) 3.46 x 108
(c) 2.8 x 1016
(d) 6.34 x 101
4.(e) x 10 2
(f) 2.81 x 10 5
(g) x 10 1
(h) 5.12 x 106
4.(i) x 102
5.(a) 20,000 = 2 x 104
(b) 16,000,000 = 1.6 x 107
5.(c) 200,000,000 = 2 x 10 8
(d) 150,000 = 1.5 x 105
5.(e) 2,400,000 = 2.4 x 106
5.(i) 8,670 = 6.67 x 103
(f) 8,210,000 = 8.21 x 106
(h) 9,000,000,000 = 9 x 109
(j) 16,500= 1.65 x 104
5.(g) 2,300,000,000 = 2.3 x 10 9
5.(k) 1,250,000 = 1.25 x 10 6
Exercise 2: 1.(a) 240
1.(e) 6,040
1.(i) 12,400,000
1.(m) 3,450,000
1.(q) 121.7
(b) 36.1
(f) 2
(j) 5,000,000
(n) 1,903
(r) 4,007
(c) 70,030
(g) 2356
(k) 41,300
(o) 346,450,000
(d) 58,000,000
(h) 2,356
(l) 84,320,000,000
(p) 63.4
Exercise 3: (a) x 10-3
(e) 8.96 x 10-8
(i) 5.28 x 10-1
(m) 7 x 10-7
Exercise 4: 1.(a)
1.(e)
1.(i)
1.(m)
(b)
(f)
(j)
(n)
3.04 x 10-3
6.45 x 10-7
2.07 x 10-5
5.67 x 10-9
(b)
(f)
(j)
(n)
(c) 5 x 10-7
(g) 8.3 x 10-4
(k) 9.8 x 10-9
(k) 9 x 10-5
(c) 0.024
(g)
(k) 0.653
(o)
(d) 8 x 10-1
(h) 8 x 10-10
(l) 1.9 x 10-5
(d)
(h)
(l)
(p)
1.(q)
Exercise 5: 1.(a) 9 x 107
(r)
1.(b) 1.1 x 105 2. 3. 4. C
9.1 x 10-31
x 1024
5.(a) x 1015
5.(e) x 107
(b) x 1012
(f) 2.5 x 10-8
(c) 3 x 1014
(g) 2 x 108
(d) 5 x 10-12
(h) 9 x 10-7
6.C
x 1016
x 100 sec
x 10-18
x 1023
11. 9 x 1024
Page | 15