Transcript BMET 4350

BMET 4350
Lecture 1
Notations & Conventions
Electrical Units

Electrical engineers and
technologists have their
cryptic signs and symbols,
just as the medical
profession does.

Letters are used in
electronics to represent
quantities and units.

The units and symbols are
defined by the SI system.
Magnetic Units

Letters are also used to represent magnetic
quantities and units in the SI system.
Metric Prefixes
Metric prefixes are symbols that represent the
powers of ten used in Engineering notation.
BMET 4350
Math Refresher
Algebra Review

Algebra is a system for representing
numbers by letters and then performing
operations with them.
• That is, juggling letters according to certain
rules.
Properties of Equality

We define three basic properties as
follows:
• a = a (reflexive property)
• If a = b then b = a (symmetric property)
• If a = b and b = c, then a = c. (transitive)
Commutative Rules

Addition:
• A+B=B+A
• The order of addition is unimportant!
• 10 + 20 = 30 = 20 + 10

Multiplication:
• AB=BA
• The order of multiplication is unimportant!
• 10  20 = 200 = 20  10
Associative Rule

Addition:
•
(A + B) + C = A + (B + C)
• Typically, we add two numbers at a time!
• Again, the order of adding is unimportant!
•

(5 + 10) + 20 = 35 = 5 + (10 + 20)
Multiplication:
•
(A  B)  C = A  (B  C)
• Typically, we multiply two numbers at a time!
• Again, the order of multiplication is unimportant!
•
(5  10)  20 = 1000 = 5  (10  20)
Distributive Rule

Multiplication:
• A  (B + C) = A  B + A  C
• Multiplication distributes over addition.
• 5  (10 + 20) = 5  10 + 5  20 = 150
Identities

Additivie Identity:
•
The additive identity is the number that when added to an
initial number does not change the value of the initial
number.
•

The additive identity is 0.
•
Note 0 can take on many values:
• (3-3) = 0
Multiplicitive Identity
•
The multiplicitive identity is the number that when multiplied
to an initial number does not change the value of the initial
number.
•
The additive identity is 1.
•
Note 1 can take on many values:
• 3/3 = 1
Addition & Subtraction of
Fractions

The lowest common denominator
method is employed.
•
A C A D B C AD  BC
 


B D BD BD
BD
•
1 1 1 4 3 1 43 7
     

3 4 3 4 3 4
12
12
Division

To represent the division operation of
algebra, we can write:
• B = C/A
• 8 = 32/4

Division becomes a little more
complicated when dealing with fractions,
because we have to distinguish between
•
A
C AC
 A 
B / C 
B
B
•
A/ B A 1
A
  
C
B C BC
and

When a fraction appears in the
denominator,
• invert the fraction
• multiply the inverted fraction by the
numerator.
•
2
4 2 4 8
 2 

3 / 4
3
3
3

When a fraction appears in the
numerator,
• the fraction in the numerator is multiplied by
the reciprocal of the denominator.
• The reciprocal of a number is simply one divided
by the number
• The reciprocal of T is 1/T.
•
3 / 4  3  1 
2
4 2
3 1 3

4 2 8
Exponential Numbers

In science, we often encounter numbers
with an awkward surplus of zeros:
• a megohm is 1,000,000 ohms
• a microampere is 0.000001 amps.

To prevent writing so many zeros, mathematical tricks are
used.
•
•
Very large or very small numbers are written as exponents
or powers of 10.
•
•
•
100 is 102 (10  10)
1000 is 103 (10  10  10)
10000 is 104 (10  10  10  10)
So
•
•
•
102 can be used instead of 100,
103 can be used instead of 1000,
104 can be used instead of 10000.
•
•
200 = 2  102
5,000,000 = 5  106

Similarly
• 0.1 = 10-1
• 0.01 = 10-2
• 0.001 = 10-3
• 5.5  10-6 = 0.0000055
Multiplication & Exponents

A convenience of writing numbers is
exponential form:
• Allows multiplication of numbers by adding
their exponents
• 2  102  3  103 = 2  3  102+3 = 6  105
• Allows division of numbers by subtracting the
exponents
•
3 10 3
3
3 2
1


10

1
.
5

10
 15
2
2
2 10
Scientific Notation


Scientific notation is a method of
expressing numbers.
A quantity is expressed as a number
between 1 and 10, and a power of ten.
Example:
5000 would be expressed as 5 x 103 in
Scientific notation.
Powers of Ten


The power of ten is expressed as an
exponent of the base 10.
Exponent indicates the number of places
that the decimal point is moved to the
right (positive exponent) or left (negative
exponent).
Engineering Notation
Engineering notation is similar to
Scientific notation, except that
engineering notation can have from 1 to
3 digits to the left of the decimal place,
and the powers of 10 are multiples of 3.
Scientific notation vs
Engineering notation
Consider the number: 23,000
In Scientific notation it would be expressed as:
2.3 x 104
In Engineering notation it would be expressed
as:
23 x 103
Example of Metric Prefix
Consider the quantity 0.025 amperes, it could be expressed
as 25 x 10-3 A in Engineering notation, or using the metric
prefix as 25 mA.
Logarithms

There are two types of logarithms generally
used in science:
•
The common logarithm
•
The natural logarithm
• log = log10
• ln = loge
•
•
e = 2.718
e appears often in nature
• radioactive decay
• time charge and discharge
• statistical analysis

The logarithm of a number is simply the
exponent placed on 10 or e to get that number.
•
The logarithm, or log, of 100 is 2.
•
Mathematically,
• 102 = 100
• log10100 = 2
•
•
The subscript 10 is the base
• The number that the exponent is placed.
Base 10 is so common that the previous expression can
be written as:
• log 100 = 2
Operations with Logarithms



log (x  y) = log x + log y
log (x/y) = log x – log y
log xn = n  log x
Miscellaneous Mathematical
Symbols
•
•>
• >>
•<
• <<
•
•
approximately equals (10.0001  10)
greater than (10 > 2)
much greater than (1000 >>2)
less than (2 < 10)
much less than (2 << 1000)
change in (A2 – A1 where A2 > A1)
infinity, or very, very large