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# Introduction

In Ireland at least, we first encounter fractions at age 6-8. At this age, because of our maturity, while we might be capable of some conceptional understanding, by and large we are doing things by rote and, for example, multiplying fractions is just something that we do without ever questioning why fractions multiply together like that. This piece is aimed at second and third level students who want to understand why the ‘calculus’ of fractions is like it is.

Mathematicians can in a rigorous way, write down what a fraction is… this piece is pitched somewhere in between these constructions — perhaps seen in an undergraduate mathematics degree — and the presentation of fractions presented in primary school. It is closer in spirit to a rigorous approach but makes no claims at absolute rigour (indeed it will make no attempt at rigour in places). The facts are real number axioms.

# Defining Fractions

We will define fractions in terms of integers and multiplication.

To get the integers we first define the natural numbers.

### Definition 1: Natural Numbers

The set of natural numbers is the set of counting numbers

$\mathbb{N}=0,1,2,3,4,\dots$,

together with the operations of addition (+) and multiplication $\times$.

TL;DR: The margin of error gets smaller with the proportion: the given margin of error is most relevant the closer to 50% the support.

Warning: I am taking the polls to be of the form

Will you vote for party X: yes or no?

There is more than a little confusion about the margin of error in political polling amongst the Irish political commentators.

Polls frequently come with margins of error such as “3%” and if a small party polls less than this some people comment that the real support could be zero.

In this piece, I will present a new way of interpreting low poll numbers, show how it is derived, further explain where the approximate 3% figure comes from and show what the calculation should be for mid-ranking proportions.

The main point is that none of these margins of error are accurate for small proportions.

For those who don’t like the mathematics of it all I will explain in a softer way why polling at 1-2% is very unlikely when your true support is 0.5%.

The reality of the situation is that a far greater proportion than 5.2% of Leaving Cert students sitting Higher Level maths are presenting work that should confer less than 40% on their exam and those in the system know that the marking scheme receives a thorough massaging in order to get 94.8% of this cohort through.

A number of students are simply not allowed fail because failing 8-20+% of those sitting would be a political and logistical nightmare.

The final result of this fear of failing students, is the dumbing down of Higher Level maths and this is good for nobody.

I feel a solution to this is to do away with the pass/fail regime for maths: keep LC points for grades above 40% but mark the papers properly. This would effectively do away with the requirement for a pass in maths for third level courses, save for those such as engineering and science who want students with certain grades.

This will allow students take on Higher Level maths in good faith with the knowledge that if they do get less than 40%, it will not be a total disaster and they will still be able to attend a third level institution.

There is still a cut off in that 35% would confer 0 points and 40% would confer 70 points but this cut-off already exists (in theory!) but it is the failing not the lack of points that is somehow forcing the Department of Education to pass these students who really should be failing.

We should keep the bonus points for higher level maths because we should want our pupils to have good maths skills for the smart economy. However, at the moment, the stick of failing is proving to have more impact than the carrot of the 25 extra points.

## Introduction

This is just a short note to provide an alternative way of proving and using De Moivre’s Theorem. It is inspired by the fact that the geometric multiplication of complex numbers appeared on the Leaving Cert Project Maths paper (even though it isn’t on the syllabus — lol). It assumes familiarity with the basic properties of the complex numbers.

## Complex Numbers

Arguably, the complex numbers arose as a way to find the roots of all polynomial functions. A polynomial function is a function that is a sum of powers of $x$. For example, $q(x)=x^2-x-6$ is a polynomial. The highest non-zero power of a polynomial is called it’s degree. Ordinarily at LC level we consider polynomials where the multiples of $x$ — the coefficients — are real numbers, but a lot of the theory holds when the coefficients are complex numbers (note that the Conjugate Root Theorem only holds when the coefficients are real). Here we won’t say anything about the coefficients and just call them numbers.

### Definition

Let $a_n,\,a_{n-1},\,\dots,\,a_1,\,a_0$ be numbers such that $a_n\neq 0$. Then

$p(x)=a_nx^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0$,

is a polynomial of degree $n$.

In many instances, the first thing we want to know about a polynomial is what are its roots. The roots of a polynomial are the inputs $x$ such that the output $p(x)=0$.

With the new Project Maths programme being developed as we speak, diligent students might like to know which proofs are examinable under the new syllabus so they know which to look at.

It can be difficult to sift through the syllabi at projectmaths.ie but I have gone through them and here are the proofs required.

Applied maths could be defined as the use of mathematics in studying natural phenomena. The branch of applied maths studied at Leaving Cert level is Newtonian mechanics. Mechanics is the study of systems under the action of forces. Newtonian mechanics is concerned with systems that can be adequately described by Newton’s Laws of Motion. Systems that aren’t adequately described by Newtonian mechanics include systems with speeds approaching the speed of light, systems of extremely small particles and systems with a large number of particles. Hence Leaving Cert Applied Maths is the study of simple macro-systems that have moderate speeds.

Applied maths is essentially a further study of the mathematics of chapters 6 through 13 in Real World Physics; following which more specific and involved questions than those of Leaving Cert Physics may be posed and answered. The emphasis in applied maths is more on problem-solving than anything and reflecting this, the need for rote-learning is almost non-existent. The course content itself could be presented on one A4 sheet. The skills required for the course include:
• Capacity for interpretation and visualisation
• Ability for strategic problem solving
• Competency in mathematics

The statistics over the last number of years are that typically over 90% of students taking applied maths do higher level, and of these about a quarter achieve an A, over half achieve an A or B and nearly two-thirds achieve an honour. 90% of students pass. So with strong maths and good, effective application to the task at hand, good grades are there for the taking.

Some LC Applied Math Notes and some Solutions to selected problems. Section 6.2 deals with the Fundamental Theorem of Calculus and Section 6.4 has a little integration. See Chapter 10 of these LC Maths Lecture Notes for some more on integration.

This is a short note covering pensions as done in LC HL Project Maths. I do not know how pensions are calculated “in the real world”.

## Fixed Number of Payouts

Suppose you want a pension that will pay you €20,000 per year for 25 years after retirement. How much should you have in your pension fund on retirement in order to have this? Suppose further that money can be invested at 3% per annum.

### Method 1

Suppose we need the pension fund to contain €X on retirement. Let $P(t)$ be the amount of money in the pension fund after $t$ years and suppose the pension fund is invested at 3%. Well, in the first year we need:

$p(0)=X$,

but then take out €20,000:

$p(1)=X-20,000$

We then accrue interest on this (capital + interest = $(1+i)$) — but then withdraw €20,000 at the end of the first year so:

$p(2)=X(1+i)-20,000(1+i)-20,000$.

Now this accrues interest but €20,000 is withdrawn:

$p(3)=X(1+i)^2-20,000(1+i)^2-20,000(1+i)-20,000$

$\vdots$

$p(25)=X(1+i)^{24}-20,000[(1+i)^{24}+(1+i)^{23}+\cdots+(1+i)+1]$.

## The Average

The average or the mean of a finite set of numbers is, well, the average. For example, the average of the numbers $\{2,3,4,4,5,7,11,12\}$ is given by:

$\text{average}=\frac{2+3+4+4+5+7+11+12}{8}=\frac{48}{8}=6.$

When we have some real-valued variable (a variable with real number values), for example the heights of the students in a class, that we know all about — i.e. we have the data or statistics of the variable — we can define it’s average or mean.

### Definition

Let $x$ be a real-valued variable with data $\{x_1,x_2,\dots,x_n\}$. The average or mean of $x$, denoted by $\bar{x}$ is defined by:

$\bar{x}=\frac{x_1+x_2+\cdots+x_n}{n}=\frac{\sum_{i=1}^nx_i}{n}.$

In Leaving Cert Maths we are often asked to differentiate from first principles. This means that we must use the definition of the derivative — which was defined by Newton/ Leibniz — the principles underpinning this definition are these first principles. You can follow the argument at the start of Chapter 8 of these notes:

to see where this definition comes from, namely:

$f'(x)\equiv \frac{dy}{dx}=\lim_{h\rightarrow 0}\frac{f(x+h)-f(x)}{h}.$ (*)

Suppose that we want to invest €$P$ over $t$ years with an interest rate of $i$ (if the interest rate is 4%, then $i=0.04$).

Initially we have €$P$ – called the principal. After 1 year however, we will have the principal plus the interest: which is $i$ of $P$. Hence the value after one year will be:

$P+Pi=P(1+i)$.

So to get the value of the investment you multiply the previous years value by $1+i$. So after three years, for example, the investment has value:

$P(1+i)(1+i)(1+i)=P(1+i)^3$,

and we can generalise to any number of years.

The final value of an investment of a principal $P$ after $t$ years at an interest rate of $i$, $F$, is given by:

$F=P(1+i)^t$.

p. 30 of the tables – Compound Interest