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Math150Mathematics for Natural Sciences

Z. Makukula P. Ittmann

UKZN, Pietermaritzburg

Semester 1, 2012

Makukula, Ittmann (UKZN PMB) Math150 2012 1 / 26

Introduction

Lecturers & Lectures

Lecturers:

Ms. Z. Makukula (Room F10) (Course co-ordinator)Mr. P. Ittmann (Room F35)

Lectures (Venue: MSB DSLT):

Monday 07h45 - 08h30Thursday 10h30 - 12h10Friday 08h40 - 09h25


Introduction

Tutorials

You will be allocated to one of the following tutorial times - pleasecheck the course website and/or the notice board for details

Tuesday 14h10 - 16h45 (Venues: MSB G17, MSB Geog L1, MSBPhys L2, New Science Seminar 1, New Science Seminar 2)

Friday 10h30 - 13h05 (Venue: OMB Temple)

Friday 14h10 - 16h45 (Venues: MSB G17, MSB Geog L1, NewScience Seminar 1)


Introduction

Assessment

Three class tests (Venue: TBA)

Test 1 01/03/2012Test 2 29/03/2012Test 3 10/05/2012

DP requirements

To be granted DP, the following requirements must be met:

A class mark of atleast 35%Attendance at atleast 80% of tutorials and lectures


Introduction

Aims of the course

Equip you with the necessary mathematical tools required by scientists

The study of practical examples to aid understanding and interest


Basics

Sets

Sets are fundamental building blocks in mathematics

A set is simply a collection of objects

Example

The following are all sets:

{MATH150,Peter, 983}{x : x is a positive multiple of 3} = {3, 6, 9, . . .}


Basics

Sets (cont.)

The objects in a set are called the elements of the set

If X is a set and x is an element of X , then we write x ∈ X . If not,then we write x /∈ X

Example

Let

X = {x : x is a positive multiple of 3 and less than 12}= {3, 6, 9}.

Then, 6 ∈ X and 5 /∈ X


Basics

Important Sets

The Natural Numbers, N

This set consists of all positive whole numbers. Thus,

N = {1, 2, 3, . . .}.

If n is a natural number, we write n ∈ N


Basics

Important Sets (cont.)

The Integers, Z

This set consists of all whole numbers (positive, negative and zero). Thus,

Z = {. . . ,−3,−2,−1, 0, 1, 2, 3, . . .}= {0,±1,±2,±3, . . .}.


Basics


The Rational Numbers, Q

This set consists of all numbers which can be expressed as a ratio of twointegers where the denominator is non-zero. That is, the set of allfractions. Thus,

Q =

{x

y: x , y are integers and y 6= 0

}.


Basics


The following are all elements of Q: −3, 2−6

, 132, 0

Note that 10is not an element of Q. Nor is

10=∞!


Basics


The Real Numbers, R

This set contains all numbers with a decimal notation � repeating, or not.

The following relationship holds:

R contains Q which contains Z which contains NExamples of real numbers which are not elements of Q are

√2, π and

e = 2.7182818284 . . .


Basics

Properties of the Rationals

Every rational number can be represented as either a repeating, or aterminating decimal

Example

Some rational numbers expressed as decimal numbers:

1

9= 0.11111 . . . ,

8

2= 4.0,

22

7= 3.142857142857 . . .

Conversely, every repeating, or terminating decimal, is a rationalnumber


Basics

Properties of the Rationals

Example

Let x = 0.532453245324 . . .. Then, x is a repeating decimal number. Notethe following:

10000x = 5324.53245324 . . .

Thus,9999x = 5324

and hence,

x =5324

9999.

We note the following: 227= 3.142857142857 . . . which is not equal to

π = 3.141592653589 . . .


Basics

Fractions vs. Decimals

In the previous slide, we convert repeating decimals to fractions

The converse (that is, converting fractions to decimals) is achieved bystandard division

Example

Knowledge of the decimal representation of common fractions and realnumbers is useful:

1

3= 0.333 . . . > 0.32

π = 3.14159 < 3.142857142857 . . . =22

7

The ability to execute such comparisons without calculators is vitallyimportant!


Basics

Percentages

We are all familiar with percentages in day-to-day life

In fact, percentages are simply ratios (that is, fractions)

Any number can be represented as a percentage by converting it to afraction of 100

Example

2 =200

100= 200%

0.3 =30

100= 30%

−0.01 = − 1

100= −1%


Basics

Parts per 100

A percentage is a representation of a number as a fraction of 100

Alternatively, a percentage could represent �parts per 100� (e.g.,concentration of a solution)

Example

If the atmospheric concentration by volume of CO2 is 12% then we can saythat the atmospheric concentration by volume of CO2 is 12 parts per 100


Basics

Parts per Notation

It is often the case that we deal with quantities so small that % orpart per 100 notation is unsuitable

We then use the following:

Parts per million (ppm), or parts per 106

Parts per billion (ppb), or parts per 109

Example

The percentage 0.0001% is 0.0001 parts per 100. However, 0.0001% is 1part per 106 (1 ppm) and 1000 parts per 109 (1000 ppb)


Basics

Parts per Notation (cont.)

Parts per notation is dimensionless quantity � no unit of measureapplies

This is because parts per notation represent ratios of quantities

Switching between di�erent parts per notations needs to be practised.


Basics

Parts per Notation (cont.)

Example

Consider the dimensionless quantity 125 ppm. This equals125

1000000= 0.000125. What is this in parts per 100 (%)? And in parts per

billion?Answer: Well, 0.000125 = 125000

1000000000= 0.0125

100. So, 125 ppm is 125000 ppb

and 0.0125%.


Basics

Dependent and Independent Variables

Consider the equationy = 3x − 2

For a given value of x ,there exists a unique valueof y

We say that y depends on

x

We call y a dependent

variable and x anindependent variable

Note that the opposite isalso true in this case (butnot always!)

-3 -2 -1 1 2 3x

-10

-5

5

y

Figure: Plot of y = 3x − 2


Basics

Dependent and Independent Variables (cont.)

Now consider the equationd = 5t2

In this case, t is anindependent variable andd is a dependent variable

Note that the reverse isnot true in this case!

-3 -2 -1 1 2 3t

10

20

30

40

d

Figure: Plot of d = 5t2


Basics

Dependent and Independent Variables (cont.)

In the case of physical experiments, dependent and independent

variables have other meanings

Suppose the equation d = 5t2 described the distance (in metres) arock travels after falling t seconds in an experiment we are running

We have control over t, since we can choose when to measure d (i.e.,we choose the value of t)

We have no control over d , since the distance a rock falls is governedby physical laws

In the context of this physical experiment, t is a dependent variable

and d is an independent variable


Basics

Functions

Revisit the equation y = 3x − 2

As discussed, y depends on x since every value of x yields a uniquevalue of y

We call y a function of x if this occurs

By this de�nition, d is a function of t in the case of the equationd = 5t2, but t is not a function of d

Can you say why?


Basics

Units

In the above physical experiment the equation d = 5t2 described thedistance (in metres) a rock travels after falling t seconds

If we changed the unit of measure of distance to centimetres, or theunit of measure of time to minutes, the equation would changedramatically


Basics

Units (cont.)

Example

If we changed the unit ofmeasure of time from secondsto minutes, then the equationwould change to the following:

d = 5(60t)2

= 18000t2-0.04 -0.02 0.02 0.04

t

10

20

30

40

d

Figure: Plot of d = 18000t2.Note the change of units on thet-axis


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