EDS 212: Day 1, Lecture 1

Course intro, algebra refresher

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August 5^th, 2024

Welcome to EDS 212 - Essential Math in Environmental Data Science

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• Instructors: Ruth Oliver (rutholiver@ucsb.edu) & Sam Csik (scsik@ucsb.edu)
• Course assistant: Anna Pede
• Course hours: 10am - 4:30pm PST
• Location: NCEAS 1st Floor Classroom

Let’s take a look at the course syllabus together.

Course description

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• Units: 2
• Grading: Satisfactory/Unsatisfactory
• Description: Quantitative skills are critical when working with, understanding, analyzing and gleaning insights from environmental data. In the intensive EDS 212 course, students will refresh fundamental skills in math (algebra, uni- and multivariate functions, units and unit conversions), summary statistics and basic probability theory, derivative and differential equations, linear algebra, and reading, writing and evaluating logical operations.

Topics overview

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• Day 1: Course introduction & math basics refresher
• Day 2: Derivatives
• Day 3: Differential equations, intro to linear algebra
• Day 4: Linear algebra, summary statistics
• Day 5: Basic probability theory, Boolean algebra

Maybe you’re thinking…

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Why am I taking a math class?

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or,

Why do I have to know math when a computer will do it for me?

A quantitative brain warm-up

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MEDS students have diverse work & academic histories

This course (re)introduces math concepts and tools that:

• Focus on applications to environmental data science
• Are specifically relevant for MEDS projects, coursework
• Provide an entryway into building computational skills
• Refresh quantitative thinking skills generally
• Refresh essentials like units, conversions, notation, language

Math brain warm up

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• Algebra blitz
• Exponentials and logarithms
• Common units and unit conversions
• Functions
• Understanding graphs
• Interpreting equations

Math brain warm up

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• Algebra blitz
• Exponentials and logarithms
• Common units and unit conversions
• Functions
• Understanding graphs
• Interpreting equations

Algebra blitz

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You can get far with a few rules:

1. Order of operations
2. Equations are already solved (but sometimes we need them in a different format)
3. Do whatever you want but do the same thing to both sides

1. Order of operations (P-E-MD-AS)

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P - Parentheses

E - Exponents

M/D - Multipication / division

A/S - Addition / subtraction

1. Order of operations practice problems

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Simplify the following: \((12 - 2)/5 + 5(3+2)/6\)

Simplify the following: \(\frac{4-6}{2}(3+1)-\frac{1+2*4}{3}\)

Simplify the following: \(3x+4(8x-6x) -(2y-5)+\frac{2x(1-3)}{2}\)

1. \((12 - 2)/5 + 5(3+2)/6\)

• \((10/5) + 5 \times 5 / 6\)
• ANSWER: \(2 + 25/6\)

2. \(\frac{(4-6)}{(2)}(3+1)-\frac{(1+2*4)}{(3)}\)

• \(\frac{(-2)}{(2)}(4)-\frac{(9)}{(3)}\)
• \(-1 \times 4 - 3\)
• \(-4-3\)
• ANSWER: \(-7\)

3. \(3x+4(8x-6x) -(2y-5)+\frac{2x(1-3)}{2}\)

• \(3x+4 \times 2x - 2y + 5 + \frac{2x-6x}{2}\)
• \(3x+4 \times 2x - 2y + 5 + \frac{2x}{2}-\frac{6x}{2}\)
• \(3x+4 \times 2x - 2y + 5 + x - 3x\)
• \(11x - 2y + 5 - 2x\)
• ANSWER: \(9x - 2y + 5\)

1. Notation matters

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Simplify the following: \(6 \div 3(4+2)\)

What would be a harder-to-misinterpret way to write this?

ANSWER: 12

A harder to misinterpret way to write this: \((6 \div 3) \times (4+2)\)

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1. An important takeaway:

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Being readable & hard to incorrectly interpret is often as important as being technically “correct”

When designing things, it’s important to consider the different ways that users might misuse or misunderstand it - then build in safeguards to help them use it correctly. Clear communication and user-centered design is critical in environmental data science.

2. Equations are solved

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\[2x-5y+3.9=8x^2-100.7x\]

Provides solutions for the questions:

1. “What is the value of \(2x-5y+3.9\)?” and
2. “What is the value of \(8x^2-100.7x\)?”

2./3. It is often helpful to reorganize things

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In the equation on the previous slide (shown below), we might want to solve for y:

\[2x-5y+3.9=8x^2-100.7x\]

The one rule to rule them all:

You can do whatever you want to an equation, as long as you do the exact same thing to both sides. That includes ensuring that you are applying something entirely to each side.

3. We’re really just doing the same thing to both sides until we’re happy with the format

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Example:

Apply the same operation to each side of the following equation step-by-step to isolate \(x\) on one side. Write out all steps.

\[4x+8=5-2x\]

• Start with: \[4x+8=5-2x\]
• Add \(2x\) to each side: \(6x + 8 = 5\)
• Subtract \(8\) from both sides: \(6x = -3\)
• ANSWER: \(x = -\frac{1}{2}\)

3. We’re really just doing the same thing to both sides until we’re happy with the format

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Example:

Apply the same operation to each side of the following equation step-by-step to isolate \(a\) on one side. Write out all steps.

\[\frac{2(a+1)}{3a}+4=6\]

ENDING NOTE: We might encounter complex expressions for how values are related to each other. While the equation is already solved, it can be helpful to use some basic algebra to rearrange things to make our equations easier to understand.

• Start with: \[\frac{2(a+1)}{3a}+4=6\]
• Subtract 4 from both sides: \[\frac{2a+2}{3a}=2\]
• Multiply both sides by \(3a\): \[2a+2=6a\]
• Subtract \(2a\) from both sides: \[2=4a\]
• Divide each side by 4; ANSWER: \(a = \frac{1}{2}\)

Math brain warm up

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• Algebra blitz
• Exponentials and logarithms (this afternoon)
• Common units and unit conversions
• Functions
• Understanding graphs
• Interpreting equations

Exponents & how to do math with them

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\[x^n=x\times x\times\ x \times x...(n\space times)\]

Evaluate the following to find a value for \(y\):

1. \(y = 12-2^4\)

2. \(2y + 30=y+3^3\)

Sometimes things are related to each other in an exponential way, so we’re going to remind ourselves how they work using numbers.

1. \(y = 12-2^4\)

• \(y = 12 - (2 \times 2 \times 2 \times 2)\)
• \(y = 12 - 26\)
• ANSWER: \(y = -4\)

2. \(2y + 30=y+3^3\)

• \(2y + 30 = y + (3 \times 3 \times 3)\)
• \(2y + 30 = y + 27\)
• \(y + 30 = 27\)
• ANSWER: \(y = -3\)

Exponent rules

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• \(x^ax^b=x^{a+b}\) ; Example: \(x^5x^3=x^{5+3}=x^8\) ; Example: \(2^2 \times 2^2\) = \((2 \times 2) \times (2 \times 2)\) = \(2^4\)

• \(\frac{x^a}{x^b}=x^{a-b}\) ; Example: \(\frac{z^5}{z^3}=z^{5-3}=z^2\) ; Example: \(\frac{2^3}{2^2}\) = \(\frac{2 \times 2 \times 2}{2 \times 2}\) = \(2^1\)

• \(\frac{1}{x^a}=x^{-a}\) ; Example: \(b^{-4x}=\frac{1}{b^{4x}}\)

• \((x^a)^b=x^{ab}\) ; Example: \((2^3)^2=2^{3*2}=2^6=64\)

• \((\frac{x}{y})^a=\frac{x^a}{y^a}\) ; Example: \((\frac{y}{2^2})^2=\frac{y^2}{(2^2)^2}=\frac{y^2}{2^4}=\frac{y^2}{16}\)

• \((xy)^a=x^ay^a\) ; Example: \((3x)^2=3^2x^2\)

Exponent practice

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Simplify the following expressions using the rules of exponents:

1. \(3x^5x^8x^{-11}\)

2. \(\frac{-8x^6}{2x^4}+7x^2\)

3. \(\frac{3x}{x^5}-3.8x^4\frac{x^3}{x^6}+8.1x-11.2\)

1. \(3x^5x^8x^{-11}\)

• ANSWER: \(3x^2\)

2. \(\frac{-8x^6}{2x^4}+7x^2\)

• \(-4x^2 + 7x^2\)
• ANSWER: \(3x^2\)

3. \(\frac{3x}{x^5}-3.8x^4\frac{x^3}{x^6}+8.1x-11.2\)

• \(3x^{-4}-3.8x^4 \times x^{-3} + 8.1x - 11.2\)
• \(3x^{-4}-3.8x^1 + 8.1x - 11.2\)
• ANSWER: \(3x^{-4}+4.3x - 11.2\)

Multiplying expressions (FOIL)

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First, Outside, Inside, Last

Example:

\[(2x+5)(x-3)\]

\[= (2x \times x) (2x \times -3) (5 \times x) (5 \times -3)\]

\[= 2x^2 - 6x + 5x - 15\]

\[=2x^2-x-15\]

Math brain warm up

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• Algebra blitz
• Exponentials and logarithms
• Common units and unit conversions
• Functions
• Understanding graphs
• Interpreting equations

UNITS. UNITS. UNITS.

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Think about these statements, which all contain the same value of 4:

• There are four in the refrigerator.

• There are four burritos in the refrigerator.

• There are four roaches in the refrigerator.

• There are four million dollars in the refrigerator.

Units are critical in environmental data science

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We cannot responsibly work with data without knowing the units of each variable we’re working with.

That means we need to always familiarize ourselves with metadata, carefully check units and any unit conversions, and understand how units combine into the units of a dependent variable.

Dimensional analysis for unit conversions

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In dimensional analysis, we multiply initial units by a sequence of conversion factors to arrive at the final desired units.

For example, to convert \(100 \frac{g}{cm^3}\) into units of \(\frac{kg}{in^3}\), given that 1 cm³ = 0.061 in³.

\[100\frac{g}{cm^3}*\frac{1kg}{1000g}*\frac{1cm^3}{0.061in^3}=1.639\frac{kg}{in^3}\]

Unit conversion practice

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Practice dimensional analysis to perform the following conversions:

1. Convert \(8.1\frac{km}{s}\) to miles per hour, given that 1 km = 0.621 miles.

2. Convert a mass flux of \(3.2\frac{g}{min\cdot m^2}\) to \(\frac{mg}{s\cdot cm^2}\).

1. \(8.1\frac{km}{s} \times \frac{60s}{min} \times \frac{60min}{hr} \times \frac{0.621 miles}{1km} = 18,108.36 \frac{miles}{hr}\)

2. \(3.2\frac{g}{min\cdot m^2} \times \frac{1min}{60s} \times \frac{1000 mg}{g} \times \frac{m}{100cm} \times \frac{m}{100cm}\)

Math brain warm up

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• Algebra blitz
• Exponentials and logarithms
• Common units and unit conversions
• Functions
• Understanding graphs
• Interpreting equations

Functions

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Functions are mathematical expressions that tell us how input values are related to output values.

For example, \(y = 3x-5\) is a function that tells us the value of y at any value of x. In this scenario, we would probably say y is a function of x.

Could you also rewrite it and say x is a function of y? Here, with no knowledge of what’s an input and what’s an output, sure - but usually in environmental data science we specify the input variable(s), and the output variable(s) carefully. What follows is the expression in the format of: “[output variable(s)] is/are a function of [input variable(s)]”.

Thinking about inputs and outputs

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For the following combinations of related variables, which do you expect would be the input and the output in a function describing how they are related? Say your answer in a sentence, e.g. “Evapotranspiration is a function of air temperature.”

1. fuel (biomass) / slope / wildfire severity / windspeed / air temperature

2. wind speed / power generated by wind turbine

3. soil C:N ratio / bacterial biomass / soil water content / leaf litter decomposition rate

1. wildfire severity is a function of fuel and slope and windspeed and air temperature
2. power generated by wind turbine is a function of wind speed
3. leaf litter decomposition rate is a function of soil C:N ratio and bacterial biomass and soil water content

Function notation

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Single variable (univariate) function:

\(f(x) = [expression\space containing \space x]\)

Multivariate function:

\(g(a,T,z)=[expression\space containing \space a, T, \space and \space z]\)

Evaluating functions

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For continuous functions, we evaluate them by plugging in variable values.

Example:

Evaluate \(g(x,t)=2.4x+0.5t^2\) at \(x = 3\) and \(t = 10\)

\(g(3, 10) = 2.4(3) + 0.5(10^2)= 57.2\)

• \(g(3,10) = 2.4 \times 3 + 0.5 \times 10^2\)
• \(g(3,10) = 7.2 + 50 = 57.2\)