DR ABDUL WAKEEL UNIVERSITY OF AGRICULTURE FAISALABAD PUNJAB PAKISTAN

1. Applied Mathematics
for Soil Science
Lecture 2: Scientific Notions
Dr. Abdul Wakeel
Associate Professor
University of Agriculture Faisalabad

2. Scientific Notions
Scientific notation is a way of writing very large or very small
numbers.
A number is written in scientific notation when a number between
1 and 10 is multiplied by a power of 10.
For example, 650,000,000 can be written in scientific notation as
6.5 ✕ 108.
0.000000065 can be written as in scientific notion 6.5 ✕ 10-8
The number is written in two parts:
1. Just the digits, with the decimal point placed after the first
digit
2. × 10 to a power that puts the decimal point where it should
be. (i.e. it shows how many places to move the decimal point)
Scientific Notation was developed in order to easily represent
numbers that are either very large or very small.
104 10 x 10 x 10 x 10 10000
103 10 x 10 x 10 1000
102 10 x 10 100
101 10 10
100 1
10-1 0.1 0.1
10-2 0.1 x 0.1 0.001
10-3 0.1 x 0.1 x 0.1 0.001
10-4 0.1 x 0.1 x 0.1 x 0.1 0.0001

3. Scientific Notions
Examples 1: Convert 1600,000 g into scientific notions
Solution:
1600000 g = b x 10n
1600000 g = 1.6 x 106 g
Example 2: Convert 0.00000000000003 cm into scientific notion
Solution:
0.00000000000003 cm = b x 10n
0.00000000000003 cm = 3 x 10-14 cm

4. Exponential Notions
These are more generally used in
daily life as well as in scientific
documents.
Examples:
Square meter = m2
Square centimeter = cm2
Cubic meter =m3
Exponents are also used while
writing the dominator of fraction.
Example: the unit of speed is
km/h, which can be written as km
h-1 and this is the preferable form
in scientific documents. Other
examples include t ha-1, mg hg-1

5. Exponential Notions
Exponents are the shortened way of writing repeated multiplications
Example: 2 x 2 x 2 x 2 x 2 x 2 =26 =64
Example: 3 x 3 x 3 x 3 x 3 x 3 =36 =729
In exponential units the base of 10 is not needed, base of any digit can
be used very easily
Question 1: What is the numerical value of 55?
Question 2: Which one is bigger , 5-4 or 5-3?

6. Addition and Subtraction of Exponential
Numbers
Addition
(b × cn) + (d × cn) = (b + d) × cn
Question 1: What is the sum of (2 × 106) and (3 × 106)?
Solution: 2 × 106= 2000000
3 × 106= 3000000
2000000+ 3000000= 5000000
Or
(2 × 106) + (3 × 106)= (2 +3) × 106) =5 × 106 = 5000000

7. Addition and Subtraction of Exponential
Numbers
Subtraction
(b × cn) - (d × cn) = (b - d) × cn
Question 2: Subtract (2 × 106) from (3 × 106)
Solution: 3 × 106= 3000000
2 × 106= 2000000
3000000- 2000000= 1000000
Or
(3 × 106) - (2 × 106)= (3-2) × 106) =1 × 106 = 1000000

8. Addition and Subtraction of Exponential
Numbers
Subtraction
(b × cn) - (d × cn) = (b - d) × cn
Question 3: Subtract (2 × 105) from (3 × 106)
Solution: 3 × 106= 3000000
2 × 105= 200000
3000000 - 200000= 2800000
Or
3 × 106= 2000000
2 × 105 0.2 x 106
(3 × 106) - (0.2 × 106)= (3-0.2) × 106) =2.8 × 106 = 2800000

9. Multiplication and Division of Exponential
Numbers
Multiplication
(b × cn) x (d × cm) = (b x d) × cn+m
Question 4: What is the answer of (2 × 105) x (3 × 106)
Solution: 2 × 105= 200000
3 × 106= 3000000
200000 x 3000000= 600000000000
Or
(2 × 105) x (3 × 106) = 3 x2 x105+6 = 6 x 1011= 600000000000

10. Multiplication and Division of Exponential
Numbers
Division
(b × cn) ÷ (d × cm) = (b ÷ d) × cn-m
Question 5: What is the answer of (4 × 106) ÷ (2 × 105)
Solution: 4 × 106= 4000000
2 × 105= 200000
4000000 ÷ 200000= 20
Or
(4 × 106) ÷ (2 × 105) = 4 ÷ 2 x 106-5 = 2 x 10 =20

11. Exponential Numbers
Example
53 + 2-2
Solution
First solve the exponents:
53 = 5 × 5 × 5 = 125
2-2 = 1/2 × 1/2 = 1/4 = 0.25
Then add the two results:
53 + 2-2 = 125 + 0.25 = 125.25

12. Raising Powers
• Sometimes we need to raise an exponential
number to a power [102]3. This can be solved
by following the multiplication principles of
exponential notions.
Solve (52)5
= 52×5
= 510

13. Logarithms
• Logarithms are the exponents to which a base
must be raised to get a particular numerical
value. Two bases are commonly used, base 10
and the natural logarithm which uses base
e(e~ 2.718).
• We use log10 or log when we want to indicate
that we are using base 10 and loge or in when
we want to indicate that we are using natural
logarithm.
• Logarithms are not used very much any more
to do simple calculation however, they play a
very important role in mathematically
describing biological soil processes.

14. Logarithms
Base 10 Logarithms
The log of 100 = 2 and
the log of 0.1 = -1.
That is because 102 = 100
and 10-1 = 0.1.
Value (A) Exponential Notation (B) Logarithms (A)
10,000
1000
100
10
1
0.1
0.01
0.001
0.0001
104
103
102
101
100
10-1
10-2
10-3
10-4
4
3
2
1
0
-1
-2
-3
-4
Number that are not a power of 10 need to be looked up in the tables of common
logarithms or calculated using a calculator. For example, log 2 = 0.3010 and log 5 =0.6989.
Numbers less than one have negative logarithms. So, log 0.03= -1.5228 and log 0.4= -0.3979.
A number less than zero cannot be assigned a logarithm.

15. Mathematical Operations Involving
Logarithms Multiplication log (nm) = 𝑙𝑜𝑔𝑛 + 𝑙𝑜𝑔𝑚
Division log n/m = 𝑙𝑜𝑔𝑛 − 𝑙𝑜𝑔𝑚
Raising to a power log Cn= 𝑛 𝑙𝑜𝑔 C
Question:
What is log ( 6 × 2 )?
Solution: log ( 6 × 2 )
= log 6 + log 2
log 6 = 0.7781
log 2 = 0.3010
=0.7781 + 0.3010 = 1.0791 log (6× 2 ) = log 12 = 1.0791
or

16. Question:
What is log 28?
Solution
log 28 = 8 × log 2
log 2 = 0.3010
8× 0.3010 = 2.4082
0r
log28 = 𝑙𝑜𝑔256=2.4082
Question: What is log (1000/10)?
Solution
log(1000/10) = log 1000 – log 10
= 3-1
= 2
Or
log 100 = 2
Mathematical Operations Involving
Logarithms

17. Questions
1. 1000- 43
2. 54 ÷ 62
3. 54 × 6-2
4. What is the numerical value of 38?
5. What is the numerical value of 52?
6. Which is larger, 2-5 or 5-2?
7. What is (1 × 105) + (2 × 105)?
8. What is 82 + 42?
9. What is (2 × 63) + ( 4 × 63)?
10. What is 45 – 35?
11. What is 79 - 76?
12. What is ( 5 × 42) – ( 3 × × 42)?
13. What is 104 × 105?
14. What is ( 0.1 × 25) × ( 5 × 25)?
15. What is 102 ÷ 103?
16. What is log 1000?