2 Algebra Review

2.1 Overview

Algebra, arthmetic, and probability are underlying frameworks for statistics. The following elements of algebra are particularly important:

Understanding symbols as variables, and what they can stand for
Factoring out common terms: $a x w + b x = x (a w + b)$
Factoring out negation of a series of added terms: $- a - b = - (a + b)$
Simplification of fractions
Addition, subtraction, multiplication, and division of fractions
Exponentiation with both fractional and whole number exponents
Re-writing exponentials of sums: $b^{u + v} = b^{u} \times b^{v}$
Logarithms
- log to the base $b$ of $x$ = $\log_{b} x$ is the number $y$ such that $b^{y} = x$
- $\log_{b} b = 1$
- $\log_{b} b^{x} = x \log_{b} b = x$
- $\log_{b} a^{x} = x \log_{b} a$
- $\log_{b} a^{- x} = - x \log_{b} a$
- $\log_{b} (x y) = \log_{b} x + \log_{b} y$
- $\log_{b} \frac{x}{y} = \log_{b} x - \log_{b} y$
- When $b = e = 2.71828 \dots$ , the base of the natural log, $\log_{e} (x)$ is often written as $\ln x$ or just $\log (x)$
- $\log e = \ln e = 1$
Anti-logarithms: anti-log to the base $b$ of $x$ is $b^{x}$
- The natural anti-logarithm is $e^{x}$ , often often written as $\exp (x)$
- Anti-log is the inverse function of log; it ‘undoes’ a log
Understanding functions in general, including $min (x, a)$ and $max (x, a)$
Understanding indicator variables such as $[x = 3]$ which can be thought of as true if $x = 3$ , false otherwise, or 1 if $x = 3$ , 0 otherwise
- $[x = 3] \times y$ is $y$ if $x = 3$ , 0 otherwise
- $[x = 3] \times [y = 2] = [x = 3 and y = 2]$
- $[x = 3] + 3 \times [y = 2] = 4$ if $x = 3$ and $y = 2$ , $3$ if $y = 2$ and $x \neq 3$
- $x \times max (x, 0) = x^{2} [x > 0]$
- $max (x, 0)$ or $w \times [x > 0]$ are algebraic ways of saying to ignore something if a condition is not met
Quadratic equations
Graphing equations Once you get to multiple regression, some elements of vectors/linear algebra are helpful, for example the vector or dot product, also called the inner product:
Let $x$ stand for a vector of quantities $x_{1}, x_{2}, \dots, x_{p}$ (e.g., the values of $p$ variables for an animal such as age, blood pressure, etc.)
Let $β$ stand for another vector of quantities $β_{1}, β_{2}, \dots, β_{p}$ (e.g., weights / regression coefficients / slopes)
Then $x β$ is shorthand for $β_{1} x_{1} + β_{2} x_{2} + \dots + β_{p} x_{p}$
$x β$ might represent a predicted value in multiple regression, and is known then as the linear predictor

2.2 Some Resources

2.3 Algebra Prequisites

Students should have a good command of college algebra. Specifically, you should have mastered variables, functions, grouped expressions, algebraic fractions, polynomials, exponents, logarithms, and uses of the mathematical constant $e$ . If you need to review algebra, there are many good study books and web sites. One good book is Forgotten Algebra, Second Edition by Barbara Bleau (Barrons Educational Series, Hauppauge, New York, 1994). You may skip the following Units in this book:

Topic	Unit Numbers
Division of polynomials	17
Factoring polynomials	18
Solving second degree equations-quadratic formula	22
Solving third-degree and higher equations	23
Solving systems of equations	26-27
Solving inequalities-second-degree	29
Right triangles	31

The topics to study, and corresponding unit numbers in Bleau’s book (second edition, may need to be updated for the third edition), are as follows.

Topic	Unit Numbers
Signed Numbers	1
Grouping Symbols and Simplifying Expressions	2
Solving First-Degree Equations	3
Removing Multiple Grouping Symbols; Solving 1st-Degree Eq.	4
Fraction Equations	5
Literal Equations	6
Applied Problems	7
Positive Integral Exponents	8
Negative Exponents	9
Division of Powers	10
Addition and Subtraction of Fractions	11
Multiplication and Division of Fractions	12
Fractional Exponents	13
Simplifying Expressions with Fractional Exponents	14
Additional Practice with Exponents	15
Multiplication of Monomials and Polynomials	16
Factoring Special Binomials	19-20
Factoring to Solve 2nd Degree Equations	21
Graphing Linear Equations in Two Variables	24
Graphing Quadratic Equations in Two Variables	25
Solving First-Degree Inequalities	28
Logarithms	30

Note that in statistics one most often uses logarithms to the base $e$ =2.71828…, and $\log x$ with no base indicated often means log to the base $e$ .

A more comprehensive but much more expensive book we also recommend is Intermediate Algebra by Alan Tussy and R. David Gustafson, Pacific Grove CA, Brooks/Cole, 1999.

# Algebra Review ## Overview Algebra, arthmetic, and probability are underlying frameworks for statistics. The following elements of algebra are particularly important: * Understanding symbols as variables, and what they can stand for * Factoring out common terms: $axw + bx = x(aw + b)$ * Factoring out negation of a series of added terms: $-a - b = - (a + b)$ * Simplification of fractions * Addition, subtraction, multiplication, and division of fractions * Exponentiation with both fractional and whole number exponents * Re-writing exponentials of sums: $b^{u + v} = b^{u}\times b^{v}$ * Logarithms + log to the base $b$ of $x$ = $\log_{b}x$ is the number $y$ such that $b^{y} = x$ + $\log_{b}b = 1$ + $\log_{b}b^{x} = x \log_{b}b = x$ + $\log_{b}a^{x} = x \log_{b}a$ + $\log_{b}a^{-x} = -x \log_{b}a$ + $\log_{b}(xy) = \log_{b}x + \log_{b}y$ + $\log_{b}\frac{x}{y} = \log_{b}x - \log_{b}y$ + When $b = e = 2.71828\ldots$, the base of the natural log, $\log_{e}(x)$ is often written as $\ln{x}$ or just $\log(x)$ + $\log e = \ln e = 1$ * Anti-logarithms: anti-log to the base $b$ of $x$ is $b^{x}$ + The natural anti-logarithm is $e^{x}$, often often written as $\exp(x)$ + Anti-log is the inverse function of log; it 'undoes' a log * Understanding functions in general, including $\min(x, a)$ and $\max(x, a)$ * Understanding indicator variables such as $[x=3]$ which can be thought of as true if $x=3$, false otherwise, or 1 if $x=3$, 0 otherwise + $[x=3]\times y$ is $y$ if $x=3$, 0 otherwise + $[x=3]\times[y=2] = [x=3 \,\textrm{and}\, y=2]$ + $[x=3] + 3\times [y=2] = 4$ if $x=3$ and $y=2$, $3$ if $y=2$ and $x\neq 3$ + $x\times \max(x, 0) = x^{2}[x>0]$ + $\max(x, 0)$ or $w \times [x>0]$ are algebraic ways of saying to ignore something if a condition is not met * Quadratic equations * Graphing equations Once you get to multiple regression, some elements of vectors/linear algebra are helpful, for example the vector or dot product, also called the inner product: * Let $x$ stand for a vector of quantities $x_{1}, x_{2}, \ldots, x_{p}$ (e.g., the values of $p$ variables for an animal such as age, blood pressure, etc.) * Let $\beta$ stand for another vector of quantities $\beta_{1}, \beta_{2}, \ldots, \beta_{p}$ (e.g., weights / regression coefficients / slopes) * Then $x\beta$ is shorthand for $\beta_{1}x_{1}+\beta_{2}x_{2} + \ldots + \beta_{p}x_{p}$ * $x\beta$ might represent a predicted value in multiple regression, and is known then as the <em>linear predictor</em> ## Some Resources * [tutorial.math.lamar.edu/pdf/Algebra_Cheat_Sheet.pdf](http://tutorial.math.lamar.edu/pdf/Algebra_Cheat_Sheet.pdf) * [www.khanacademy.org/math/algebra](https://www.khanacademy.org/math/algebra) * [www.purplemath.com/modules/index.htm](http://www.purplemath.com/modules/index.htm) ## Algebra Prequisites Students should have a good command of college algebra. Specifically, you should have mastered variables, functions, grouped expressions, algebraic fractions, polynomials, exponents, logarithms, and uses of the mathematical constant $e$. If you need to review algebra, there are many good study books and web sites. One good book is [Forgotten Algebra](http://www.amazon.com/Forgotten-Algebra-Barbara-Lee-Bleau/dp/0764120085/ref=pd_bbs_sr_1?ie=UTF8&s=books&qid=1201146512&sr=8-1), Second Edition by Barbara Bleau (Barrons Educational Series, Hauppauge, New York, 1994). You may skip the following Units in this book: | *Topic* | *Unit<br>Numbers* | |---------|-------------------| |Division of polynomials|17| |Factoring polynomials |18| |Solving second degree equations-quadratic formula|22| |Solving third-degree and higher equations|23| |Solving systems of equations|26-27| |Solving inequalities-second-degree|29| |Right triangles|31| The topics to study, and corresponding unit numbers in Bleau's book (second edition, may need to be updated for the third edition), are as follows. | *Topic* | *Unit<br>Numbers* | |---------|-------------------| |Signed Numbers|1| |Grouping Symbols and Simplifying Expressions|2| |Solving First-Degree Equations|3| |Removing Multiple Grouping Symbols; Solving 1st-Degree Eq.|4| |Fraction Equations|5| |Literal Equations|6| |Applied Problems|7| |Positive Integral Exponents|8| |Negative Exponents|9| |Division of Powers|10| |Addition and Subtraction of Fractions|11| |Multiplication and Division of Fractions|12| |Fractional Exponents|13| |Simplifying Expressions with Fractional Exponents|14| |Additional Practice with Exponents|15| |Multiplication of Monomials and Polynomials|16| |Factoring Special Binomials|19-20| |Factoring to Solve 2nd Degree Equations|21| |Graphing Linear Equations in Two Variables|24| |Graphing Quadratic Equations in Two Variables|25| |Solving First-Degree Inequalities|28| |Logarithms|30| Note that in statistics one most often uses logarithms to the base $e$ =2.71828..., and $\log x$ with no base indicated often means log to the base $e$. A more comprehensive but much more expensive book we also recommend is _Intermediate Algebra_ by Alan Tussy and R. David Gustafson, Pacific Grove CA, Brooks/Cole, 1999.