Quantitative Reasoning Week 2 PDF

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Lecture Notes Week 2 2021...

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Quantitative Reasoning Week 2: What is so hard about doing maths?   

Boring: repetition of seemingly pointless exercises. Culture: tend to reinforce maths as hard and not fun. Strongly hierarchical: difficult to progress is you have not mastered earlier ideas.

The more you know how to do, the better:  

We can only hold about 7 different concepts in our mind at once. o Think about driving a car, playing a musical instrument, computer game, etc. The more automated a process becomes, o The faster we can use it. o The faster we can learn about other processes that use it.

Algebraic operations (+, -, X, divide or /): 

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Addition and multiplication o 2+3=5 o 2x3=6 Their inverses, subtraction and division o 2-3=-1 o 2/3= 0/666… Repeated multiplication: raising to a power. o 23=2x2x2=8 o 102=10x10= 100

The laws of algebra for the natural number (1, 2, 3): 

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Commutative law (for + and x) o A+b=b+a o Axb=bxa Associative law (for + and x) o (a+b)+c=a+(b+c) (2+3)+4= 2+(3x4) o (axb)xc= ax(bxc) (3x(2+3) Distributive law o Ax(b+c)= axb+axc

The number line: Magnitude or absolute value:  

The distance from the origin, so |4|=4 and |-4|=4 Always positive or 0

Greater than or less than:   

Greater than means in the direction if increasing numbers from Less than means in the direction of decreasing numbers from It has nothing to do with magnitude, e.g., -6 negative number Multiplying/dividing with an even number of negatives -> positive number o Ax(-b)= -(axb) o (-a)xb= -(axb) Rules for adding and subtracting fractions: o To add or subtract fractions, you must convert them to a common denominator.

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Rules for multiplying and dividing fractions. o Multiplying: multiply numerators and denominators

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Dividing: invert the divided fraction and multiply

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Whole numbers can be written as fractions over 1, i.e.

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Rules for exponents

Representing fractions as percentages o A percentage is the numerator of a fractions out of 100. o 2/5=40/100=40% o 1/3=33.3/1--=33/3% o 2=200/100=200%

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Be kind: use brackets. o “+” out the front -> keep the same sign. o “- “out the front -> flip each sign. o 1+(2-3)= o 1-(2-3)= o 2+3x(2-3)= o 2-3x(2-3)=

Units:     

I will be there in 5. I got done doing 62. It is 28 today, but it will be 31 tomorrow. I want a coffee, but I have only got two-fifty on me. Measuring a quantity results in a number on an agreed scale. o That scale reflects a choice of units (definite predetermined amounts). o Standardising the scale allows comparison.

Units for quantities: 

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In Australia, we use Systeme International (SI) units. o Lengths in metres (m) o Mass in kilograms (kg). o Time in seconds (s). o Temperature in Kelvin (K). o Amount of substance in moles (mol). o Luminous intensity in candelas (cd). Everything else can be written as some combination of these. o E.g., speed= distance/time, so its units are m/s (or ms-1). Common combinations are given their own name. o E.g., pressure has units of kgm-1s-2, otherwise called pascals. o Other better-known examples: joules, watts, volts, hertz, newtons. Rates of change o Acceleration= rate of change of velocity/time, so units are… o Power= rate of change of energy/time, so units are… Other formulas o Energy= ½ x mass x velocity2, so units are… o Energy= mass x acceleration due to gravity x height so units are… o Diffusivity= distance2/ time, so units are… o Units of force (=mass x acceleration) o Units of frequency (=number per second) o Units of radians (=arclength/radius). There are number of non-SI units accepted for use alongside SI units, e.g., o Angle in degrees, minutes, and seconds (e.g., latitude and longitude) o Area in hectares o Volume in litres o Time in minutes, hours, days, years o Temperature in degrees Celsius. Americans use their own customary units (related to British Imperial), e.g, o Length in inches, feet, yards, and miles (and area in acres)

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Mass in pounds Temperature in degrees Fahrenheit (F) (note that zero is different in this scale).

Converting between units:   

What is the world record for the women’s 4x100m freestyle? How fast is that (average speed) in m/s? How fast is that (average speed) in km/h?

Finding relations via units: 

My mower can handle mowing a lawn at about 2m2/s. how many minutes would is take me to more my parents ¼-acre lawn?

The decimal system: Thousands (1000) 10 3

Hundreds (100) 10 2

Tens (10)

Ones (1)

10 1

10 0

Tenths (1/10) 10-1

Hundredths (1/100) 10-2

All rationales have either finite or repeating decimal expansions:

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How do we conveniently express these values? o Radius of an atomic nucleus is about 0.0000000000001m o Size of our galaxy is about 100000000000000000000m.

(Normalised) Scientific notation: 

Avogadro’s number N o Based on the number of atoms in 12g of Carbon-12  N=60221407600000000000000 o We use comma separation to make it clearer.  N=602,214,076,000,000,000,000,000

Thousandth s (1/1000) 10-3

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Surely, we can make writing this number down easier.

The carbon atoms on a graphene layer (folded to make Carbon Nanotubes) are 0.00000000000142m apart. o 10=1=1/10=0.1, 10-2=1/ (10x10) =0.01, 10-3=1/ (10x10x10)=0.001 o 0.0000000000142-1.42x0.0000000001- 1.42x10-9. A number is in normalizes scientific notation if it is in the form. Where: o R is a real number (called the significand) with 1< r < 10 (r can equal 1, but not 10); o M is an integer (called the exponent). Process 1: o Start with the original number, e.g., 142.14. o Hop the decimal place toward the first non-zero digit ( lead digit) in the number.  For each hop to the left, multiply by 10.  For each hop to the right, divide by 10 (i.e., multiple by 10-1). o Keep going until the decimal point is to the right of the lead digit.  142.14- 14.214 x10- 1.4214x 102.  0.000314- 0.00312 x10-1- 0.0312 x 10-2- 0.312 x 10-3- 3.12 x 10-4.

Process 2: o The power of 10 whose lead digit is in the same position as the original number will be 10m.  100=10(2) for 142.14, 0.0001=10(4) for 0.000314. o Write the digits of the number, putting a decimal after the lead digit- this will be r.  142.14= 1.421x10(2), 0.000314= 3.14x 10(4). Examples: o Diameter of the earth (12742000m). o Radius of an atomic nucleus (0.000000000000001m) o Age of the universe in seconds o Average mass of a blood cell (0.000000000027g).  Notes: on a calculator/computer: 6.02x 10(23) = 6.02E23 and 1.42x10(10) = 1.42E-10.  Engineering notation uses exponents in multiples of three, i.e., 602x 10(21) or 142x10 (12).

SI prefixes and illions: 

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We use SI prefixes to represent large and small numbers, e.g., o Separation of carbon atoms was 1.42x10(9)m= 1.42nm. o Australians consume about 10MWh per capita per annum. We typically stick to the powers-of-3. o Common exceptions: cm, dB, hPa, cI o Random examples: MW, GByte, fs, nm. With m, km, kg, yr. o It is common to use million, billion, trillion etc, rather than M, G, T, etc.

Expressing tiny fractions and huge numbers:

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Separation of carbon atoms in a carbon nanotube. o 0.00000000142m o 1.42x10(9)m (calculator says 1.42e-9). o 1.42nm (1 nanometre=10(9) m. Age of the universe o 13800000000 years. o 13.8 billion years (0r 13.8 Tera years- correct, but non-standard). o 13.8x 10(9) years= 1.38 x10 (10) years (calculator says 1.38e10). o 1.38x10(10) x (365.25) x (24) x (60) x (60) = 4.35x10(17) s.

Orders of magnitude:  

When values vary across a broad scale, it is helpful to think in terms of the exponent, which we call an order of magnitude. If the significand= 1, the exponent is the logarithm (m=log(100m).

Converting between units:   

What is the height of the Eiffel tower in microns? What is the age of the universe in hours? What is the speed of light in parsecs/hour?

Relations using units: How long would it take to burn off the calories from a Mars Bar?  

A) running, if I burn 80 kj/min; or Swimming, if I burn 60 kj/min?

Switching between mol and g? 

We need a 0.1 mol/l solution of 1-octanol (130.23 g/mol). How many grams of 1-octanol will there be in 1cm (3) of this solution? o The molecular weight of a compound gives the number of grams per mol. It is the sum of the molecular weight of all its atoms.

Linear chemistry problems:

A patient has a serum potassium level of 4mmol/l.   

How many mmol are present in a 20ml sample? How many mg is present in the 20ml sample? What is the difference between % w/w and % w/v? o % w/w is the percentage weight in the total weight (grams in 100g). o % w/v is the percentage weight in the total volume (grams in 100ml).  A mouthwash contains 0.1% w/v chlorhexidine gluconate. How much chlorhexidine gluconate is contained in 250ml of the mouthwash?  What weight of miconazole is required to make 40g of a cream containing 2% w/w of the drug?

Switching between mol and g: 

Apomorphine (267.322 g/mol) can be administered to dogs to induce vomiting after ingestion of toxic substances. The maximum recommended dose is 0.04mg per kg of bodyweight. If apomorphine comes in a 3.6 mmol/l solution, and Monty weighs 10kg, how many ml of solution should the vet inject?...