Lecture 0. Some Basic Maths for Actuarial Students

We will regularly need to employ certain calculations. In MATH10951 the context might vary but the maths varies much less. These notes are more of a background check on prequisties. We cover

Power, the exponential, logarithms, the (natural) logarithm.
Arithmetic and Geometric progressions.

Powers and the Exponential

Often we will care about $a^t$ where $t$ is an amount of time and, for $a=(1+r)>0$ , $r$ is interpreted as a rate of interest.

Fact: multiplication becomes addition,

$a^{s} \cdot a^t = a^{s+t}.$

Fact: powers become multiplication:

$(a^{s} )^t = a^{s\cdot t}.$

We care about powers for a “special” number $e\approx 2.7182$ .

Fact:

$\frac{d e^{x}}{dx} = e^x \qquad \text{and}\qquad \int e^x dx =e^x$

Since we care about rates of change (i.e. differentiation and integration) and interest (i.e. powers), you can imagine the above fact is very handy.

Warning! maths joke!

All the mathematical functions go to a party, $\sin x$ , $\cos x$ , the cantor function,…, and $e^x$ who hangs around the corner of the room on his own. One of the functions, the cantor function, asks him why he does not socialize with the other functions. To which $e^x$ replies “I keep trying to integrate but every time I try I just end up back with myself”.

Okay you can probably tell I didn’t go to a lot of parties at Uni… so let’s swiftly move on…

Fact: $e^x$ has a nice expansion

$e^x= 1 + \frac{x}{1!} + \frac{x^2}{2!} + \frac{x^3}{3!} +...$

Fact: $e^x$ is what happens when you make lots of small multiplications: as $n\rightarrow \infty$

$\left( 1+ \frac{x}{n}\right)^n \rightarrow e^{x}.$

(Here read “ $x\rightarrow y$ “ as “ $x$ goes to $y$ ”.)

Fact: $e^x$ grows really fast (when compared with polynomials): as $x\rightarrow \infty$

$\frac{e^x}{x^n} \rightarrow 0.$

Also you can Google “plot e^x and x^10″ and expand the $y$ axis.

Logarithms

If you give me $1$ and I agree to give you a yearly rate of interest $r$ (I.e. your money multiplies by $a=(1+r)$ every year) and I give you back $y$ pounds.

Q. How long has the money been invested for?

A. We look for the number $t$ such that $a^t=y$ . The solution to this logarithm the logarithm $y$ with base $a$ , $t=\log_a y$ .

We used logarithm to convert units of money to units of time and the exponential to convert units of time to units of money. In otherwords the logarithm is the inverse of the power function $x\mapsto a^x$ .

Definition: Given $a$ and $y$ the solution, $t$ , to the equation $a^t=y$ is called the logarithm of $y$ to the base $a$ and is written $\log_a y .$ In otherwords

$y=a^t \qquad \text{if and only if}\qquad t=\log_a y.$

The next facts, follow from the above statements about exponentials.

Fact:

$\log_a (x\cdot y) = \log_a (x) + \log_a y.$

Fact:

$\log_a (x^t) = t\log_a (x).$

Fact: Moving between bases:

$\log_a x = \log_b (x) \cdot \log_a(b)$

To see this check the following

$x= a^{\log_a(x)} = a^{\log_a(b)\cdot \log_b(x)} = b^{\log_b x}=x.$

Definition: the Natural logarithm is the logarithm with base $e$ and is written $\log x$ .

(sometimes people write $\ln$ for the natural logarithm but not mathematicians)

Fact:

$\frac{d \log (x)}{dx} = \frac{1}{x}\qquad \text{and}\qquad \int \log (x) dx =x \log x- x.$

Fact: $\log (1-x)$ has a nice expansion, for $|x|<1$ $\log(1-x)= 1 - \frac{x}{1} - \frac{x^2}{2} - \frac{x^3}{3} -...$

Arithmetic and Geometric Series

A arithmetic progression is of the form

Fact:

$\sum_{n=1}^N n = \frac{1}{2}N(N+1)$

Picture proof given in lectures. (Hint: the sum on the left is set of points laid out in a triangle, which is half a square)

(Carl Fredrick Gauss famously figured this out when he was $8$ ).

A geometric progression is of the form

Fact: For $a\neq 1$

$\sum_{n=1}^N a^n = \frac{a^{N+1}-1}{a-1}$

Picture proof given in lectures. (Hint: let $S_N$ be the sum on the right. What is $aS_N - S_N$ ? Notice lots of things cancel out…)

Another Maths Joke (sorry!) An infinite number of mathematicians walk into a bar, the first orders a pint. So the barman pours a pint. The second orders a half pint. So the barman pours a half pint. The third orders a quarter pint. The barman stops, says “wait a minute!”, takes the beer back, and pours two pints.

Okay I don’t get down the pub much either…so that’s probably enough for now…

Powers and the Exponential

Logarithms

Arithmetic and Geometric Series

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