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

Math Review “Don’t worry about your difficulties in Mathematics, I can assure you that mine are still greater” Albert Einstein

NASANational Space Grant Network “The greatest sin committed by formal education is the continued and relentless extermination of the natural pleasure of learning; and the greatest sin committed by "progressive" education is the delusion that such pleasure need not be accompanied by a certain measure of pain”. Sydney Harris, Thoughts at Large

Math ReviewThursday-Monday, June 3-7 2003 • Introduction • Symbols • Operations • Central Tendencies • Linear Algebra • Correlation/Regression Analysis • System of Equations: Linear/Quadratic • Applied Calculus

Math Review #1Thursday, June 3 2003 http://www.columbia.edu/itc/sipa/envp/louchouarn/courses/mpa.htm • Introduction • Symbols • Operations • Central Tendencies • Linear Algebra

E = mc2 • F = ma Basic Math Review Symbols We need symbols to simplify expressions and develop abstract arguments • N = ike  particularly for quantitative analysis.

A Better Math Curriculum 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 number “Sex Drives” Or the “Gender Wars”

What’s in a number Really, shouldn’t we be serious?

What’s in a number “Sex Drives” vs.

What’s in a number “Bullshit detector”

Basic Math Review Operations Summation: S (sigma) Sxi = x1 + x2 + x3 + …+ xn Where: n, represents the sample size x represents the variable, and xi represents the value of the ith observation S(xi+yi)2 = (x1+y1)2 + (x2+y2)2+ (x3+y3)2 + …+ (xn+yn)2

x(a/b) = bth root of (xa) = x(-a) = x(a - b) = Basic Math Review Operations Powers (Exponents): xaxb = x(a + b) xaya = (xy)a (xa)b = x(ab)

also: Basic Math Review Operations Logarithms (base 10): logb(x) = yif and onlyifby = x Warning: logb(x)logb(y)  logb(xy) logb(b) = 1 logb(1) = 0 logb(xy) = logb(x) + logb(y) logb(x/y) = logb(x) - logb(y) logb(xn) = nlogb(x)

Basic Math Review Operations Logarithms (natural log): ln(x) = yif and onlyifey = x ln(e) = 1 ln(1) = 0 ln(xy) = ln(x) + ln(y) ln(x/y) = ln(x) - ln(y) ln(xy) = yln(x) ln(ex) = xln(e) = x  1 = x eln(x) = x

Basic Math Review Operations Solve for x: ln(ea) = bx Solve for y using common logarithms (base 10): y = 175 Solve for y using common logarithms (base 10): y = 175 - 127

Basic Math Review Central Tendencies The most commonly used descriptive statistics are measures of central tendency The sample mean (: pronounced “x bar”) is: Where Sxi represents the sum of all values in the sample and n represents the sample size

Basic Math Review Central Tendencies Let’s assume we have a student population (n = 47) But what happens if we have an outlier (skewed distribution )?

Basic Math Review Central Tendencies Let’s assume we have a real student population (n = 47)

Basic Math Review Central Tendencies • Mean: arithmetic average • Median: middle value of a set of values • Mode: the data value that occurs most often

Friday • Populations Lecture 1 • Math Review #2: Linear Algebra - Correlation/Regression Analysis • Don’t forget the website AND the math sheets! Monday • Lamont orientation (LDEO Exec. Director and DEES Chair) • Math Review #3: System of Equations: Linear/Quadratic - Applied Calculus http://www.columbia.edu/itc/sipa/envp/louchouarn/courses/mpa.htm

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