“Don’t worry about your difficulties in Mathematics, I can assure you that mine are still...

““Don’t worry about your difficulties in Don’t worry about your difficulties in Mathematics,Mathematics,

I can assure you that mine are still I can assure you that mine are still greater”greater”Albert EinsteinAlbert Einstein

Math ReviewMath Review

NASANASANational Space Grant NetworkNational Space Grant Network

““The greatest sin committed by formal education The greatest sin committed by formal education is the continued and relentless extermination of is the continued and relentless extermination of the natural pleasure of learning; and the greatest the natural pleasure of learning; and the greatest sin committed by "progressive" education is the sin committed by "progressive" education is the delusion that such pleasure need not be delusion that such pleasure need not be accompanied by a certain measure of pain”.accompanied by a certain measure of pain”.

Sydney HarrisSydney Harris,, Thoughts at Large Thoughts at Large

A)A) IntroductionIntroductiona.a. SymbolsSymbolsb.b. OperationsOperationsc.c. Central TendenciesCentral Tendencies

B)B) Linear AlgebraLinear AlgebraC)C) Correlation/Regression AnalysisCorrelation/Regression AnalysisD)D) System of Equations: Linear/QuadraticSystem of Equations: Linear/QuadraticE)E) Applied CalculusApplied Calculus

Math ReviewMath ReviewThursday-Monday, June 3-7 2003Thursday-Monday, June 3-7 2003

A)A) IntroductionIntroductiona.a. SymbolsSymbolsb.b. OperationsOperationsc.c. Central TendenciesCentral Tendencies

B)B) Linear AlgebraLinear Algebra

Math Review #1Math Review #1Thursday, June 3 2003Thursday, June 3 2003

http://www.columbia.edu/itc/sipa/envp/louchouarn/courses/mpa.htm

















a. SymbolsBasic Math ReviewBasic Math Review

We need symbols to simplify expressions and develop abstract arguments

E = mc2

particularly for quantitative analysis.

F = ma

QuickTime™ and aCinepak decompressor

are needed to see this picture.N = ike

A Better Math CurriculumA Better Math CurriculumDr. Tom Davis (B.Sc. Math at Caltech, Ph.D. in Math at Stanford, post-doc in electrical engineering at Stanford)Dr. Tom Davis (B.Sc. Math at Caltech, Ph.D. in Math at Stanford, post-doc in electrical engineering at Stanford)

The problem today: “math is generally taught by and aimed at mathematicians”

(In many universities, engineering Depts teach their own math courses since students are unable to solve engineering (applied) problems with the tools they learn from the math Dept!)

Basic Math:- Finances (household…)- Problem-solving skills!!!- Basic numeracy- Estimation (including probability & statistics)- Visualization, and- Bullshit detector!!!

What’s in a numberWhat’s in a number“Sex Drives”

Or the“Gender Wars”

What’s in a numberWhat’s in a numberReally, shouldn’t we be serious?


vs.

Average driving distance of men's and women's tours (PGA and LPGA)

200

210

220

230

240

250

260

270

280

290

300

1990 1992 1994 1996 1998 2000 2002 2004

Year

Average driving distance (Ft)

Men

Ladies


vs.


y = 1.0082x - 36.144

R2 = 0.9392

210

215

220

225

230

235

240

245

250

255

260

255 260 265 270 275 280 285 290

Men's driving distance (Ft)

Women's driving distance (Ft)


210

215

220

225

230

235

240

245

250

255

260

255 260 265 270 275 280 285 290

Men's driving distance (Ft)

Women's driving distance (Ft)

What’s in a numberWhat’s in a number“Bullshit detector”

Tour de France

y = 0.173x - 305.41

R2 = 0.8601

20

25

30

35

40

45

1915 1925 1935 1945 1955 1965 1975 1985 1995 2005Year

Average Speed (km/hr)

Tour de France

50

70

90

110

130

150

170

190

210

230

250

1915 1925 1935 1945 1955 1965 1975 1985 1995 2005Year

Total Time (hr)

What’s in a numberWhat’s in a number“Bullshit detector”

Tour de France

2020

2520

3020

3520

4020

4520

5020

5520

6020

1915 1925 1935 1945 1955 1965 1975 1985 1995 2005Year

Distance (km)

Tour de France

3000

3500

4000

4500

5000

5500

6000

20 30 40Average Speed (km/hr)

Distance (km)

Summation: (sigma)xi = x1 + x2 + x3 + … + xn

Where:n, represents the sample sizex represents the variable, andxi represents the value of the ith observation

b. OperationsBasic Math ReviewBasic Math Review

€

x =x i∑

n=

1

nx i∑

xi+yi)2 = x1+y1)2 + x2+y2)2 + x3+y3)2 + … + xn+yn)2


€

1

xaxx(-a)(-a) = =

Powers (Exponents):Powers (Exponents):xxaa xxbb = = xx(a + b)(a + b)

xxaayyaa = ( = (xyxy))aa

((xxaa))bb = = xx(ab)(ab)

€

xabxx(a/b)(a/b) = b = bthth root of ( root of (xxaa) = ) =

€

xa

xbxx(a - b)(a - b) = =


Logarithms (base 10):Logarithms (base 10):loglogbb((xx) = ) = yy if and onlyif and only ifif bbyy = = xx

loglogbb(1) = 0(1) = 0

loglogbb(b) = 1 (b) = 1

loglogbb((xxyy) = log) = logbb((xx) + log) + logbb((yy))

loglogbb(x/y) = log(x/y) = logbb((xx) - log) - logbb((yy))

loglogbb((xxnn) = nlog) = nlogbb((xx))

€

xy = balso: also: Warning:

loglogbb((x)x)loglogbb((yy) ) log logbb((xxyy))

€

logb(x)

logb(y)≠ logb(

x

y)


Logarithms (natural log):Logarithms (natural log):lnln ((xx) = ) = yy if and onlyif and only ifif eeyy = = xx

ln(1) = 0ln(1) = 0

ln(ln(ee) = 1 ) = 1

ln(ln(xxyy) = ln() = ln(xx) + ln() + ln(yy))

ln(x/y) = ln(ln(x/y) = ln(xx) - ln() - ln(yy))

ln(ln(xxyy) = ) = yyln(ln(xx))

ln(ln(eexx) = ) = xxln(ln(ee) = ) = x x 1 = 1 = xx

eeln(x)ln(x) = = xx


a)a) Solve for Solve for xx: ln(: ln(eeaa) = b) = bxx

b)b) Solve for Solve for yy using common logarithms (base 10): using common logarithms (base 10):y = 175

c)c) Solve for Solve for yy using common logarithms (base 10): using common logarithms (base 10):y = 175 - 127

c. Central TendenciesBasic Math ReviewBasic Math Review

€

x =x i∑

n=

1

nx i∑

The most commonly used descriptive statistics are The most commonly used descriptive statistics are measures of central tendencymeasures of central tendencyThe The sample meansample mean (: pronounced “ (: pronounced “xx bar”) is: bar”) is:

Where Where xxii represents the sum of all values in the sample represents the sum of all values in the sample and and n n represents the sample sizerepresents the sample size


Let’s assume we have a student population (Let’s assume we have a student population (nn = 47) = 47)

But what happens if we have an outlier (skewed But what happens if we have an outlier (skewed distribution )?distribution )?

Frequency Distribution

0

2

4

6

8

10

12

14

22 23 24 25 26 27 28 29 30 31 32 33 34

Age

Frequency


Let’s assume we have a real student population (Let’s assume we have a real student population (nn = = 47)47)

Frequency Distribution

0

1

2

3

4

5

6

7

8

9

10

20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50

Age

Frequency


Mean: arithmetic averageMean: arithmetic average Median: middle value of a set of valuesMedian: middle value of a set of values Mode: the data value that occurs most oftenMode: the data value that occurs most often

FridayFriday Populations Lecture 1Populations Lecture 1 Math Review #2: Linear Algebra - Math Review #2: Linear Algebra -

Correlation/Regression AnalysisCorrelation/Regression Analysis Don’t forget the website AND the math sheets!Don’t forget the website AND the math sheets!


MondayMonday Lamont orientation (LDEO Exec. Director and DEES Lamont orientation (LDEO Exec. Director and DEES

Chair)Chair) Math Review #3: System of Equations: Math Review #3: System of Equations:

Linear/Quadratic - Applied CalculusLinear/Quadratic - Applied Calculus

















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