dr. hugh blanton entc 3331. dr. blanton - entc 3331 - math review 2

Dr. Hugh Blanton

ENTC 3331

Dr. Blanton - ENTC 3331 - Math Review 2

Measurement UnitsMeasurement Units

• The System of International Units (SI units) was adopted in 1960.• The use of older systems still persists,

but it is always possible to convert non-standard measurements to SI units.

SI (International Standard) Base UnitsSI (International Standard) Base Units

• meter (m) = about a yard• kilogram (kg) = about 2.2 lbs• liter (l) = about a quart• liter (l) = 1000 mL

Fundamental UnitsFundamental Units

Physical Property Unit (abbreviation)

length meter (m)

mass kilogram (kg)

time second (s)

electric current ampere (A)

temperature kelvin (K)

number of atoms or molecules mol (mol)

light Intensity candela (cd)

Seven Fundamental physical phenomena.

Unit ConversionsUnit Conversions

•When converting physical values between one system of units and another, it is useful to think of the conversion factor as a mathematical equation.

• In solving such equations, one must only multiply or divide both sides of the equation by the same factor to keep the equation consistent.

ft 23ft 3

yd 1ft 23

Quantityyd 3

yd 1ft 23

•Example: 23 feet = ? yards

yd 7.6ft 3

yd 1ft 23

•Example: 5 yd2 = ? ft2

ft 3 yd 5 2

ft 3 yd 5

ft 9 yd 5 2

22 ft 45

ft 9 yd 5

UnitsUnits

• Fundamental Units• The SI system recognizes that there are

only a few truly fundamental physical properties that need basic (and arbitrary) units of measure, and that all other units can be derived from them.

• Derived Units• The funadmental units are used as the

basis of numerous derived SI units.• Note that derived SI units are

sometimes named after famous physicists.

Derived UnitsDerived UnitsMeasureme

ntUnit Unit Description

force newton (N) Force required to accelerate a mass of 1 kg at 1 m/s2

pressure pascal (Pa) Pressure that exerts a force of 1 newton per m2 of surface area

frequency hertz (Hz) Number of cycles of periodic activity per second

energy joule (J) Energy expended in moving a resistive force of 1 newton over 1 m.

power watt (W) Rate of energy expenditure of 1 joule per second.

electrical charge

coulomb (C)

Charge that passes a point in an electrical circuit if 1 ampere of current flows for 1 second.

electrical resistance

ohm Ratio of the voltage divided by the current in an electrical circuit.

Unit Multiplication FactorsUnit Multiplication Factors

• An additional letter that denotes a multiplying factor may prefix fundamental or derived units.• The more common multiplying factors

increase or decrease the unit by powers of ten.

Unit Multiplication FactorsUnit Multiplication Factors

• An additional letter that denotes a multiplying factor may prefix fundamental or derived units.• The more common

multiplying factors increase or decrease the unit by powers of ten.

tera (T) 1012

giga (G) 109

mega (M) 106

kilo (k) 103

hecto (h) 102

deca (da) 10

deci (d) 10-1

centi (c) 10-2

milli (m) 10-3

micro 10-6

nano (n) 10-9

pico (p) 10-12

femto 10-15

Powers of Ten (big)Powers of Ten (big)

•101 = 10•103 = 1000 (thousand)•106 = 1,000,000 (million)•109 = 1,000,000,000 (billion)

Powers of Ten (small)Powers of Ten (small)

•100 = 1•10-3 = 0.001 (thousandth)•10-6 = 0.000001 (millionth)•10-9 = 0.000000001 (billionth)

Scientific NotationScientific Notation

• 7,000,000,000

• = 7 billion

• = 7 109

• 7,000,000

• = 7 million

• = 7 106

Scientific NotationScientific Notation

• 7,240,000

• = 7.24 million

• = 7.24 106

3 significant digits

Very Large QuantitiesVery Large Quantities

• 7,240,000 = 7.24 106

6 decimal places

Very Small QuantitiesVery Small Quantities

• 0.0000123 = 1.23 10-5

5 decimal places

Engineering NotationEngineering Notation

• Exponents = 3, 6, 9, 12, . . .• Instead of 5.32 107

• we write• 53.2 106

• Decimal part got bigger

Exponent got

smaller

Adding and SubtractingAdding and Subtracting

•Exponents must be the same!•(1.2 106) + (2.3 105)

•change to•(1.2 106) + (0.23 106)

•= 1.43 106

MultiplyingMultiplying

•Exponents Add•(3.1 106)(2.0 102)

•= 6.2 108

DividingDividing

•Exponents Subtract

•(3.8 106)•(2.0 102)

•= 1.9 104

Adding FractionsAdding Fractions

•You can only add like to like• Same Denominators

Different DenominatorsDifferent Denominators

•Make them the same• find a common denominator

•The product of all denominators is always a common denominator• But not always the least common denominator

Finding the LCDFinding the LCD

•Example:

Factor the DenominatorsFactor the Denominators

15 3 5 12 2 2 3

Assemble LCDAssemble LCD

15 3 5 12 2 2 3

2 2 3 5 60

Build up Denominators to LCDBuild up Denominators to LCD

Add NumeratorsAdd Numerators

And Reduce if Needed

Rational ExpressionsRational Expressions

•Example:

2 12 2

Factor the DenominatorsFactor the Denominators

x x x2 1 1 1 ( )( )

x xx x

2 2 11 1

( )( )

Assemble LCDAssemble LCD

( )( )x x 1 1

( )( )( )x x x 1 1 1

Build up Fractions to LCDBuild up Fractions to LCD

1 1( )( ) ( )( )

LCD x x x ( )( )( )1 1 1

x ( )1

x ( )1)( x( )1

x( )1FACTORED

Add NumeratorsAdd Numerators

( )( ) ( )

( )( )( )

x x x x

1 1 2 1

x x x x

2 22 1 2 2

( )( )( )

Simplify NumeratorSimplify Numerator

( )( )( )

( )( ) ( )

( )( )( )

x x x x

1 1 2 1

RadicalsRadicals

xRadicand

Radical

MeaningMeaning

if and only if

ExampleExample

because

An AmbiguityAn Ambiguity

because•but it’s also true that. . .

It’s also true thatIt’s also true that

( ) 5 252

•So why not say

25 5 •?

Two Answers?Two Answers?

•Roots with an even index always have both a positive and a negative root

•Because squaring either a negative or a positive gives the same result

Principal RootPrincipal Root

•To avoid confusion we define the principal root to be the positive root, so:

25 5 5 (not )

The Negative RootThe Negative Root

•If we want the negative root we use a minus sign:

Negative RadicandsNegative Radicands

•Do Not Confuse

25•With 25 •!!!

25 •Does not exist

Negative RadicandsNegative Radicands

•You cannot take an even root of a negative number

•Because you cannot square any number and get a negative result

Odd Roots of Negative RadicandsOdd Roots of Negative Radicands

•You can take odd roots of negative numbers:

2 2 2 8

3 because

( )( )( )

Some Square Root IdentitiesSome Square Root Identities

•for all non-negative x

•for all x

A Common ErrorA Common Error

a b a b •for example, you cannot say

3 4 72 2 (WRONG!)•What is the correct result?

First Evaluate InsideFirst Evaluate Inside

ProductsProducts

•You can “split up” a radical when it contains a product (not a sum!):

ab a b•(as long as a and b are non-negative)

ExampleExample

400 16 25

4 5 20

Perfect SquaresPerfect Squares

•Perfect squares are numbers that have whole number square roots: 4, 9, 16, 25, 36, 49, 64, etc.

•All other numbers have irrational roots

NumbersNumbers

• Natural Numbers: 1, 2, 3, . . .

• Whole Numbers: 0, 1, 2, 3, . . .

• Integers: . . . , -2, -1, 0, 1, 2, . . .

NumbersNumbers

• Rational Numbers

• a/b (a,b integers, b not zero)

• Irrational Numbers Cannot be a ratio of integers Decimals never repeat or end. (decimals of rationals do)

Rational and IrrationalRational and Irrational

45454545.011

41421356.12

Rational(Terminates)

Rational(Repeats)

Irrational

NumbersNumbers

• Real Numbers Rationals + Irrationals All points on number line All signed distances

The Number Line

Imaginary NumbersImaginary Numbers

•Square root of a negative number•We Define:

1 i Math, Physics

1 j Engineering, Electronics

Properties of jProperties of j

2 1j By Definition

3j j Because j 3= j 2j = (-1)j

4 1j Because j 4= j 2j 2 = (-1)(-1)

5j j Because j 5= j 4j = (1)j

Expressing Square Roots of Negative NumbersExpressing Square Roots of Negative Numbers

4 ( 1)4

4 2 2j j

Expressing Square Roots of Negative NumbersExpressing Square Roots of Negative Numbers

3 ( 1)3

Complex NumbersComplex Numbers

•Real Part + Imaginary Part•Example:

Real Part = 6

Complex NumbersComplex Numbers

•Real Part + Imaginary Part•Example:

Imaginary Part = 2

Adding and Subtracting Complex NumbersAdding and Subtracting Complex Numbers

•Likes stay with likes• Re + Re = Re• Im + Im = Im

•Just collecting like terms

•Example:

(6 2 ) (2 3 )j j

•Example:

(6 2 ) (2 3 )j j

•Example:

(6 2 ) (2 3 )j j

•Remember that j 2 = -1

(6 2 )(2 3 )j j 212 18 4 6j j j

)1(61412 j

DividingDividing

•Complex Conjugate• Reverse sign of imaginary part

6 2 jConjugate of

6 2 jis

DividingDividing

• Write as fraction• Multiply numerator and denominator by

the complex conjugate of denominator• Multiply and simplify

DividingDividing

(6 2 ) (2 3 )j j

(6 2 )

(2 3 )

DividingDividing

(6 2 )

(2 3 )

12 18 4 6

4 6 6 9

DividingDividing

12 18 4 6

4 6 6 9

12 22 6

DividingDividing

12 22 6

6 22 6 22

13 13 13

Graphing Complex NumbersGraphing Complex Numbers

•Real part is x-coordinate

•Im. part is y-coordinate

Graphing Complex NumbersGraphing Complex Numbers

•Example: 3 + 2j (3, 2)

Polar FormPolar Form

•Example: 3 + 2j 3.633.4°

•Re + j Im rrej

2 2Re Imr

1 Imtan

Re cosr

Im sinr

Trigonometric FormTrigonometric Form

•r (cos + j sin )•Start with Re + j Im

•Substitute•Re = r cos •Im = r sin

Trigonometric FormTrigonometric Form

•Start with Re + j Im

•Substitute

•r cos + j r sin

•r (cos + j sin )

sincos je j

Euler’s Identity

sincos jrre j

Complex ArithmeticComplex Arithmetic

•Addition & Subtraction• Easiest in rectangular form

•Multiplication & Division• Easiest in polar form

Multiplication in Polar FormMultiplication in Polar Form

•(r11) (r22)

•= r1r2 (1+2)

Division in Polar FormDivision in Polar Form

•(r11) / (r22)

•= r1 / r2 (1-2)

Vectors & ScalersVectors & Scalers

• There is a fundamental distinction between two types of quantity:• Scalers and• Vectors

• Scalers possess a magnitude, whereas vectors have both magnitude and direction.

• Properties such as mass and temperature clearly have no directionality and are examples of scalers.

• A complete description of force would be impossible without specifying both the magnitude and direction of the quantity.

VectorsVectors

•Represent magnitude and direction•Example: Displacement

• “go 2 miles East”

Vector QuantitiesVector Quantities

•Force•Velocity•Magnetic Field

Vector NotationVector Notation

•Vector: Bold or arrow over

•Scalar: Italic, no arrow

Numerical DescriptionNumerical Description

•Polar Form• Magnitude and angle

•Rectangular Form• x- and y-components

A vector can be represented in:

V = (r, )V =(53, 65°)

V = rV = 5365°

Rectangular FormRectangular Form

Vx=V cos

Vy=V sin

Rectangular to PolarRectangular to Polar

2 2x yV V V

1tan y

Vector AdditionVector Addition

•Resultant vector•Not the sum of the magnitudes•Vectors add head-to-tail

Vector Addition ExampleVector Addition Example

•Go 3 miles East,•then 4 Miles North

R = 5 miles at 53°

Adding Nonperpendicular VectorsAdding Nonperpendicular Vectors

•x-components add to givex-component of resultant•y-components add to givey-component of resultant

Rx = Ax + Bx Ry = Ay + By

R = A + B

Trigonometric FunctionsTrigonometric Functions

•Right Triangles Only!

hypotenuse

adjacent

Trigonometric FunctionsTrigonometric Functions

hypotenuse

opposite

Similar TrianglesSimilar Triangles

Same Angle

Same Ratiosof Sides

•Ratios of sides depend ONLY on •So the ratio is a function of

Ratios of SidesRatios of Sides

•Six Possible

hypotenuse

adjacent

sin = opp hyp

cos = adj hyp

tan = opp adj

Ratios of SidesRatios of Sides

sin = opp hyp

cos = adj hyp

tan = opp adj

csc = hyp opp

sec = hyp adj

cot = adj opp

The Main 3 Trig FunctionsThe Main 3 Trig Functions

sin = opp hyp

cos = adj hyp

tan = opp adj

S O H C A H T O A

Solving TrianglesSolving Triangles

•Find all 3 sides and 3 angles•Need: 1 side plus 2 more items

• Only one more thing if it is given that one angle is 90°

Right TrianglesRight Triangles

•Need 2 sides•OR

•1 side and 1 angle

Tool KitTool Kit

•The Trig functions• (sin, cos, tan)

•The inverse Trig functions• (sin-1, cos -1, tan -1)

•The Pythagorean Theorem•Sum of angles is 180°

The Trig FunctionsThe Trig Functions

•Find a side•Given 1 side and 1 angle

sinopp

cosadj

tanopp

The Inverse Trig FunctionsThe Inverse Trig Functions

•Find an angle•Given 2 sides

The Pythagorean TheoremThe Pythagorean Theorem

•Find a side•Given 2 sides

Angles add to 180°Angles add to 180°

•Find an angle•Given the other angle

Vector MultiplicatonVector Multiplicaton

• Three types vector multiplication:• Simple multiplication• Dot Product

• Always yields a scaler answer.

• Cross Product• Always gives a vector result.

Dot ProductDot Product

ABABBA cos

22332 222 A

ˆˆˆ 3z3y2x

dr. hugh blanton entc 3331. dr. blanton - entc 3331 - math review 2

units fundamental units

blanton entc

math review

system of units

measurement units

funadmental units

hugh blanton entc

arbitrary units of measure

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