1. Basic Math Techniques .
Chapter 13

2. Exponents and Scientific Notation
Exponents are used to indicate multiple multiplication
2 x 2 x 2 x 2 = 16
24 = 16
Here 2 is the base number and 4 is the exponent
103 = 10 x 10 x 10 = 1000
10-3 = 1/10 x 1/10 x 1/10 = 1/1000 = .001

3. Rules for Exponents
To multiply two numbers with exponents where the base is the same, add the exponents
Am x An = Am+n
53 x 54 = 57
10-3 x 104 = 101
Divide two numbers with same base
Subtract exponents
Am/Ab = A(m-b)

4. Continued
To multiply an exponent number to a higher power, multiply the numbers
(Am)n = Amxn
(23)3 = 23x3 = 29
(103)-4 = 10-12
To multiply or divide different base numbers with exponents, convert to no exponent number and multiply or divide
32 x 24 = 9 x 16 = 144

5. Continued
To add and subtract exponents, convert to numbers and add and subtract
= = 73
Any number raised to the zero power is 1
430 = 1

6. Exponents where the base is 10
For numbers greater than 1
Exponent represents number of places after the number and before the decimal point
Exponent is positive
Larger the positive exponent, larger the number
For numbers less than 1
Exponent represents the number of places to the right of the decimal

7. Continued The exponent is negative
The larger the negative exponent, the smaller the number
May use “order of magnitude” to related exponents
102 is two orders of magnitude less than 104
108 is three orders of magnitude greater than 105

8. Scientific Notation
Uses exponents to simplify handling large and small numbers
= 6.66 x
= 4.32 x 1011
Comprised of coefficient and 10x
Convert 3450 to scientific notation
Convert to scientific notation

9. Rules for Scientific Notation
1. All numbers big and large are expressed as numbers between 1 and 10 multiplied by 10 to a power.
2. The number of places that the decimal must be moved from its original position to the position that makes the number between 1 and 10, is the exponent number.
3. The sign of the exponent is determined by whether the number is greater or less than 1.  If the number is less than 1, the exponent will be negative.  If the number is greater than 1 the exponent will be positive.
EXAMPLE:  2,300,000 is 2.3 x 106
Convert 3450 to scientific notation
Convert to scientific notation

10. Calculations with Scientific notation
Multiplication
Multiply coefficients and add exponents
Division
Divide coefficients and subtract exponents
Addition/subtraction
Convert to regular notation and add/subtract
OR use a calculator (lets practice)

11. Logarithms Common Logarithms or log or log10
Log of a number is the power to which 10 must be raised to give that number
100 = 102 so the log is 2
0.001 is 10-3 so the log is -3
5 = 10? = ? Use log table or hit log on calc
What is the log of 48?
Log 10 = 1 and log 100 = 2 so 48 is between one and two. Actually

12. Antilogarithms Is the number corresponding to the logarithm
If the log of 5 is 0.699, then the antilog of is 5
On a calculator use 2nd log or inv log to get this number

13. Natural Logarithms Have a base of 2.7183, called e
When e is the base, the log is called a natural log or ln or loge
Tables of calculators may be used to find these values
Log10 is most common in Bio labs

14. Applications
What is the pH of an acid whose H+ concentration is 1 x 10-3?
pH = -log[H+] = -log[1 x 10-3] = 3
What is the H+ concentration of an acid having a pH of 4?
[H+] = 10-pH = 10-4 = 1 x 10-4

15. Units of Measurement Measurement is a process of counting
Length is a property that is measured.
1.55 is a value of length
Cm is a unit of length
Unit of measurement is a precisely defined amount of a property, such as length or mass.
A standard is a physical embodiment of a unit
Cm, mm, oC, ml are units of measurement.
Rulers, graduate cylinders thermometers are standards
Units are not effected by environment. However, standards may be effected.

16. Systems of measurement
Most labs use the metric system
English system not used much
In the metric system, units are related to each other
Centi = 1/100 so centimeter is a meter/100
Kilo = 1000 so a km = 1000 meters
Common metric units include grams, meters, and liters

17. Metric Table giga mega kilo unit = l, g, s, m centi milli micro nano g
109
106
103
100
10-2
10-3
10-6
10-9 9 6 3 -2 -3 -6 -9

18. Rules for metric conversion
The above table and the following rules will help you to convert one SI unit into another.
1.    To convert any unit to another unit, the decimal place must be moved either to the right of the left. 2.    If the unit that you wish to convert to is to the right of the original unit, move the decimal to the right. 3.    If the unit to be converted to is to the left of the original unit, move the decimal to the left. 4.    The number of places that the decimal should be moved is equal to the difference between the exponent numbers.

19. Equations and relationships
Equations are a way to describe relationships using mathematical symbols
=, <, or > all indicate relationships
Ex: V = I R
Where V = voltage, I is current, and R is resistance
What can you tell me about the relationship of V, I, and R
We have a system where V = 12, I = 3 and R = 4.
What happens if R increase to five?
V may inc and I remain the same.
I may decrease and V remain the same. -

20. Continued Equations for Temperature oF = 1.8(oC) +32
1.8 and 32 are constants because they are always present and always the same value
oF and oC are variables, can vary
Given one variable, the other may be calculated

21. Units and Mathematical Operations
A = L x W (area = length x width)
What is the area of a room measuring 12 ft in length by 10 feet in width?
A = 12 ft x 10 ft = 120 ft x ft or 120 ft2
Multiplying and dividing of units follows the same rules as do exponents
40 cm3/20cm2 = 40 cm(3-2)/20 = 2cm