Many of my students tends to believe that associativity of a given operation is always true.
We have seen in the previous post that addition is associative so that we always have . This actually hold true for all integers (positives and negatives).
But clearly you would agree that subtraction is not associative. .
This has obviously to do with the way subtraction is define. One way to think of it is to write as the additive inverse (or opposite) of , such that: . The subtraction is then defined as . The rule is that we can replace directly by .
It follows that the additive inverse of is as . Therefore we chose to write the additive inverse of as .
Let us now have a look at which we write so that clearly subtraction is not associative
I recently found a book left on the AMRC coffee table “Fundamental Concepts of Mathematics” by R.L. Goodstein. Inside his book Goodstein propose to use mathematical induction to show that addition is commutative.
We start by defining addition
We first need to define the addition of two number as satisfying the following property: (this is associativity with under addition). Let us refer this property as .
Showing associativity for all integers
We can first show that associativity is always true. We write the property .
The proof by induction is:
1) we know to be true for as it give
2) let us assume it exists an integers satisfying so that we can write .
It follows that:
since is true for
We now see that if satisfies then the next integer satisfies . Since is true for it is true for all positive integers.
Showing that addition with is commutative
From here we can show commutativity with . We refer to this property as : for all (positive) integers.
The proof by induction is the following:
1) we know that is true for , as we simply have
2) let us assume that it exists an integer for which is true. Then we can write .
We now see that if satisfy then the next integer satisfy . Since is true for it is true for all positive integers.
Showing that addition is commutative for all integers
Next we can show the property : commutativity with any integer .
The proof by induction is:
1) we know is true for as it gives which simply is which we now know to be true.
2) let us assume that is true for some integer then we can write .
It follows that
since is true for
So that we see that if satisfy then the next integer satisfy . Therefore is true for all integers.
In “My Best Mathematical and Logic Puzzles” Martin Gardner’s first problem is titled “The Returning Explorer” in reference of the following riddle.
“”An explorer walks one mile due south, turns and walks one mile due east, turns again and walks one mile due north. He finds himself back where he started. He then shoot [or see (depending of the riddle you wish to tell)] a bear. What color is the bear ?””
The problem proposed by M. Gardner is then to find all points on the globe from which you can walk one mile south, east and north and finally end up at your starting point.
There is an infinite number of points which satisfies this property. Can you find them and give the latitude at which you need to start your journey ?
The answer is the following. Writing the earth radius and the latitude ( is integer), all latitude on which you can start are given by:
Which at the first order in leads to
Now, starting from any other latitude , you would not be able to “complete a triangle” by walking south for miles, east for miles and back north for miles. But how far away from the starting point would you be exactly ?
Well if you were to walk back west from the finish to the starting point, you would have to walk for a distance
Assuming we have
However, this is not the shortest distance. The fact is that the shortest path between two points and on a sphere is along the circle of radius passing by and . Can we find out what the shortest path is ?
I also had the chance to meet with Professor Pak Wing Fok. In addition of sharing a car for our regular trips between philadelphia and Newyark, both of us are sharing similar research interests in bio-physics and stochastic processes.
So far I had the change to give two presentation of my work:
———————————– ———————————– ———————————–
Department of Mathematical Sciences, University of Delaware :
Analytical Approaches for simple models of Gene Expression
———————————– ———————————– ———————————
Department of Physics, Virginia Tech, Society of Physics Students :
Birth and Death on Lineland or a purely geometrical model of creation and annihilation of particles
The works of Amanda Goodall is focused on the correlations between “manager’s expertise” and “success”. If we admit that “good management leads to better performance”, we surely should investigate what does define good management ?
Amanda Goodall will prove to you (with data to back it up) that “Experts, not managers, make the best leaders”. In her book Amanda shows that the performance of universities improves when led by presidents who are outstanding scholars. See here. Amanda explains why it is important for experts to be guided by expert leaders. Amanda raises the same statement in other sectors as well. One should mention that more and more hospitals are lead by managers with no expertise in medicine. In this field (but others too) it is of capital importance for managers to understand (best experience) what challenges are facing doctors and nurses everyday.
We will follow up on a related subject : How stress or well being affect your productivity. This leads to the work of Andrew Oswald (warwick university).