# The Adventurer’s Guide to Mathematopia: Part 1

Zhaorui Xu

And then I shall unfold to you
the scroll of the grand spectacle
of the most magnificent terrain of
Mathematopia!

## Full Map of Mathematopia

When looking at the landscape, you may have noticed that the whole territory lies within the drainage basin of the River of Proof. Indeed, without proofs, maths loses its essence of rigour and the whole system becomes untenable. As newcomers, one of the first lessons to learn is how to properly and rigorously construct proofs with mathematical symbols in order to justify your propositions. Some of the most basic but powerful weapons include proof by induction, proof by contradiction and proof by contraposition.

The whole civilisation is centred around the Temple of Logic. If you are (as the author herself is) into not only maths but also the boundary where maths meets philosophy, then you will no doubt be visiting this temple a lot! Neighbouring the temple you will find the Castles of Set Theory, which were first constructed by a great warrior from Germany called Georg Cantor. It was from here that cardinals, infinite cardinals and even different sizes of infinities, were given a definition. The later developed Zermelo-Frankel set theory is now widely regarded as the foundational system of mathematics. Also situated close to the Temple of Logic is the Village of Model Theory. The village is resided by logicians who study models of various mathematical structures including groups, fields and graphs. The region of mathematical logic looks peaceful and harmonious on the surface, but beware, there are dangers lurking: paradoxical ouroboros that are scattered in the grass, swamps of continuum that once trapped the aforementioned Cantor, and the rift of Gödel’s incompleteness theorems that place the entire civilisation on the brink of collapse…

Anyway, as a newcomer, the first few months of your apprenticeship will not usually take you too deep into the tangles at the foundations of maths, nor lead you too far into the wilderness. Instead you will be based at the training grounds on the outskirts to allow you to practice the basic skills that prepare you to become a real warrior. The following is the guide to the parts of the mathematical landscape that you will be exploring as a fresher (first year):

## Mathematical Analysis

This is the area where you encounter the notions of countability, infinity, infinitesimal, limits, convergence, continuity of functions, derivative, integral, and many more besides. Instances of the basic notions you will be introduced to are sequences, series, and their convergence. A sequence is an ordered list of numbers and a series the sum of all the terms in a sequence.

If you have studied A-level (high school) maths then you have already encountered some examples of sequences and series. Recall an infinite geometric series such as

1 + ½ + ¼ + ⅛ +… = 2.

This series is the sum of the sequence 1, ½, ¼, ⅛, …, 1/2n, …. and so on. The ideas of limit and convergence have in fact emerged from this simple equation: the sum of the terms converges to 2, and the limit of the sum 1+ ½ + …+ 1/2n is 2. Moreover, the sequence 1, ½, ¼, ⅛, …, 1/2n, ….converges to 0. You will soon be introduced to many more complicated sequences and series, and spend a lot of time carrying out tests for convergence and looking for limits. Once you become experienced in this task, you will have a good grasp of the idea of infinitesimals. Infinitesimals in maths are the extremely small quantities that are closer to zero than any real number, but are not zero themselves. And it is with this powerful weapon that functions are analysed. In fact, infinitesimals are the symbol of the region of analysis such is their foundational importance.

Armed with your knowledge of infinitesimals, you will begin to hunt for the continuity and differentiability of functions, scouring the Woods of Functions in search of derivatives and differentials, and conquering the hills of Riemann integrals and Lebesgue integration. New weapons will be unlocked along the way such as L’Hopital’s rule, Taylor series, power series; and treasure boxes will be discovered, such as the Mean Value Theorem and the Fundamental Theorems of Calculus.

‘A battle in vain!’ roars the Dragon of Exponential.

In the future, young warriors shall continue their adventure into Metric Spaces and onto the Complex ‘Plain’. The preliminary training of real analysis naturally provides you with an insight into the ‘distance’ between elements in a set, and based on this a metric space will be defined. The study of metric spaces will empower your exploration into functions with a complex variable.

## Calculus

The Land of Calculus was first explored by two outstanding adventurers, Issac Newton and Gottfried Wilhelm Leibniz, independently of one another. It soon became a prosperous land due to trading contacts with many other parts of the territory, especially the Land of Geometry. The discoveries and inventions of the Land of Calculus are taken by fearless sailors into the Ocean of Applied Maths, and introduced to Engineers, Physicists and Economists across the sea.

Legend has it that the Region of Analysis (at least as a discipline in modern mathematics) is a descendent of the Land of Calculus. The close ties between the regions can be seen through the use of infinitesimals, with many prophets arguing that calculus can be seen as the study of continuous change using the method of infinitesimals. The two main branches of calculus are differential calculus and integral calculus, which are linked together by the Fundamental Theorems of Calculus.

Apart from the basic techniques of differentiation and integration that young warriors have previously learned at school, new skills and tools including partial differentiation, double integrals, Jacobians, Taylor’s theorem and Lagrange’s multipliers form the focus of the primary exploration into this region.

As a final remark, it is important to remember that, neither the map, nor my guide here, covers all the wonderful landmarks or secret wonders of the landscapes of logic, analysis and calculus. What has been revealed is rather a silhouette and more surprises are left for you to discover…

In Part 2 of the Adventurer’s Guide to Mathematopia we will continue our journey to the east river bank and the Land of Algebra to foretell the breathtaking spectacles that await.

The full ‘Map of Mathematopia’ can be found here.

Part 2 of the ‘Adventurer’s Guide to Mathematopia’ is here.