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.

•

•

BD

BCAD

D

C

B

B

D

D

B

A

D

C

B

A

12

7

12

34

4

1

3

3

4

4

3

1

4

1

3

1

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

• and

•

B

AC

B

CA

CB

A

/

BC

A

CB

A

C

BA

1/

When a fraction appears in the denominator, • invert the fraction

• multiply the inverted fraction by the numerator.

• 3

8

3

42

3

42

4/3

2

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.

•

8

3

24

13

2

1

4

3

2

4/3

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

• 15105.110

2

3

102

103 1232

3

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

• log = log10

• The natural logarithm

• 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.

• 102 = 100

• Mathematically,

• 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