Different flavors of Mathematics

 

mapOfmath

Many people encounter arithmetic and algebra in school but don’t know that there are other types of math. People that know there are other types of math often don’t know that there are dozens of types, even more than the big categories shown in the picture above. If you go on to university, then you also get calculus and possibly differential equations, a tiny sample. This week, Occupy Math looks at the different kinds of math there are. It’s interesting to note that mathematics, as a discipline, is arguably larger than science in the number of different topics it covers. If you would like a post explaining why math is not itself a science, please let Occupy Math know. We have continuous math, like the number line or geometry, we have discrete math that deals with whole numbers or collections of individual objects, and almost all kinds of math have grown, creating more and more general forms of the math known in ancient times.

Arithmetic

Arithmetic is the first part of math. Arithmetic was discovered by many different human civilizations. There is a whole field of study in looking at the way that different ancient civilizations represented numbers. The Arabic numerals that we use are one of many systems. They are far more convenient than Roman numerals. In particular, they have a zero which is incredibly useful. Arithmetic is usually developed to either help with business or to keep track of time — critical for things like planting crops.

It turns out that thinking about arithmetic leads you to discover the prime numbers and, when you start trying to figure out how they work, you get number theory. Number theory is the study of the properties of the whole numbers, and it starts off pretty abstract. In the end, though, secret codes, financial security, and even the blockchains that underlie crypto-currencies are the children of number theory.

group

Algebra

The first part of algebra may be familiar to you – the quadratic equation. The quadratic equation appears on Babylonian cuneiform tablets and it was used for the ballistics of cannons in Europe. It is a big deal. The algebra most people encounter is simplifying expressions in order to solve them. It turns out that algebra goes well beyond that.

Abstract algebra includes linear algebra. Linear algebra grew out of solving simultaneous linear equations, but it appears everywhere from computer graphics to mechanical engineering and is the “basic math” of higher dimensional space. The key objects in linear algebra are matrices — rectangular arrays of numbers that specify a transformation from one space to another. It is possible, for example, to construct a 2×2 matrix that, when applied with matrix multiplication, rotates the plane about its center. The normal plane is transformed into the rotated plane. It turns out that the transformations are part of a larger world.

When we move beyond linear algebra, we get group theory which generalizes the idea of symmetry. Examples of group theory in the real world include Rubik’s cube and space groups used to solve for the structure of protein when using X-ray crystallography. Group theory also helped us discover new elementary particles.

When we generalize normal arithmetic, with addition and multiplication, we find there are an infinite number of arithmetic systems called rings and fields. Fields underlie one of the other great branches of math (analysis, look below) and also show up extensively in cryptography and computer science. There are rings where multiplication does not commute: the result of multiplying is different if you multiply objects in different orders. A simple example of failure to commute is this. Suppose you put the letter ‘A’ at the center of the plane. If you rotate the plane a quarter turn and then slide it upward, the ‘A’ will be in a different place than if you instead slide the plane upward the same amount and then rotate.

poles

Analysis

Where number theory arises from studying whole numbers, analysis arises from studying all the numbers — the whole numbers and the ones between them. It turns out that even thinking about these numbers between the whole numbers leads to some very strange places. You can win a large prize, for example, if you solve the Riemann hypothesis, which is where the picture above comes from.

Analysis is a generalization of calculus, but it achieves that generalization by considering, in general, spaces that are continuous (as opposed to discrete, like the whole numbers). The field of point, set topology or general topology is the underlying language used in analysis. There are different types of analysis for different types of numbers. Real analysis uses real numbers, those that can be distances, while complex analysis uses the complex numbers. These are the numbers that include the number you are used to, but also contain the square roots of negative numbers. Complex numbers are useful for generating fractals.

Fano

Combinatorics

Occupy Math’s own field of math is combinatorics which includes graph theory, counting things, and even fields like design theory. In some ways combinatorics was the math left over when the other fields formed and it arose from puzzles and recreational math, like Sudoku, but also includes a lot of unsolved problems.

Combinatorics includes finite geometry. Look at the picture at the top of this section. It has seven points and seven lines (one of them is drawn like a circle). The only things that matter are the seven points and seven “lines” — the object is not drawn in plane, in spite of being drawn on paper. Imagine the diagram floating off the paper — this is the thing we are considering. Any two points define a unique line. Any pair of lines intersect in exactly one point. This finite geometry, the Fano plane, obeys many of the laws of Euclidean geometry in the infinite plane. It turns out that the Fano plane has applications.

Combinatorics forms a foundation of probability theory. The odds of rolling a seven on two dice is one-sixth. If you count the number of ways to roll a seven (1+6, 2+5, 3+4, 4+3, 5+2, 6+1) there are six of them and there are 6×6=36 ways two dice can roll. This means the odds are six out of thirty six, or one-sixth. Odds in card games or other games of chance are calculated by counting the number of ways to win and dividing by the number of things that can happen at all.

trefoil

Geometry and Topology

Geometry is another area many of Occupy Math’s readers will have encountered. This goes back as far as Euclid’s Elements with applications from architecture to navigation. Geometry turns out to have many forms beyond the flat or plane geometry of Euclid which are called Non-Euclidean geometry.

Geometry is also where topology started. Occupy Math has a recent post on some topics in topology. Topology investigates the possible shapes and structures of abstract spaces. Topology uses both algebra and analysis (the fields above) to this end and is one of the most intuition-heavy fields of mathematics. It includes the study of knots, like the trefoil knot shown above.

Alexander_horned_sphere

This picture shows the Alexander horned sphere. It is topologically equivalent to a normal sphere, something that is not at all clear from its picture. Topologists figure out what spaces are possible to construct and what their properties are. Topology has applications from understanding how DNA knots and folds to cosmology.

 

Foundations

The last of the big fields of math that Occupy Math will mention is foundations of mathematics. This area includes set theory. The area of mathematical foundations looks at the philosophy and logical origins of mathematics. A recent post discusses the way that all math grows from its assumptions and foundations are the formal study of that process.

All of the different fields of math overlap a little bit. There are some combinatorial problems that require complex analysis to solve. Occupy Math has solved graph theory problems for colleagues trying to understand a type of differential equation, and topology and number theory have subfields that use abstract algebra and analysis. Moving an idea from one field of math to another is both profitable and is its own field of math!

Occupy Math hopes you are better informed by knowing that there are many different types of math. Each of these fields also has its own culture and brand of intuition about the nature of mathematics. Math is a gigantic market of different ideas unified by the idea of logically proving your claims. Have a field of math you would like to know more about? Please comment or tweet!

I hope to see you here again,
Daniel Ashlock,
University of Guelph,
Department of Mathematics and Statistics

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