Fun with Numbers! Trivia Quiz

Answer: Probability theory

While the concepts that underlie the mathematical study of probability are much the same as you would use in day-to-day life, the language used by statisticians is certainly not. In probability theory, the set of possible outcomes of a trial is known as the 'sample space', with any subset of the sample space being termed an 'event'. Every event must have a probability assigned to it.

For example, if you were rolling a normal cubic dice, your sample space would be the set {1, 2, 3, 4, 5, 6}. If the event in which you were interested was 'rolling an even number', the subset for your event would be {2, 4, 6} and it would have a probability of one half.

Answer: Combinatorics

Combinatorics is one of the most intuitive areas of pure mathematics, and combinatoric problems can generally be understood without any advanced mathematical training. While there are several different areas of combinatorics (such as combinatorial optimization and extremal combinatorics), the most common area used by the amateur mathematician is known as 'enumerative combinatorics', better known as counting.

A simple combinatoric problem would be something like "In a class of 10 students there are six girls and four boys. How many different ways are there to split the class into two teams of five, each containing three girls and two boys?". Answers on a postcard...

Answer: Calculus

Differential calculus is concerned with the rate of change of a function, which is often referred to as the 'gradient' when considering a straight line. Many functions are smooth continuous curves rather than straight lines, and differential calculus allows us to calculate the instantaneous gradient at any point along the curve. On the other hand, integral calculus is concerned with calculating the area under or between curves.

Both branches of calculus are linked together by the inventively named 'Fundamental Theorem of Calculus', which (simply put) states that the operations of differentiation and integration are the inverse of one another. This means that if you take a function, differentiate it, then integrate it, you'll get back to your original function.

Answer: Group theory

A 'group', G, is a set of elements (often numbers, though can be other objects), combined with an operation, '~', that obeys the following four conditions:

Closure - If a and b are elements of G, then a~b is also in G.

Associativity - If a, b and c are in G, then (a~b)~c must be the same as a~(b~c).

Identity element - There must be an element, e, in G such that a~e = e~a = a for every a in G.

Inverse element - For every element a in G, there is an element â in G such that a~â = â~a = e.

An example of a group is the set of integers with the operation addition.

Answer: Topology

The types of deformation that are allowed in topology are stretching, crumpling and bending, with tearing and glueing not possible. The reason that a coffee cup is the same as a donut is that it is a solid object with only one hole in it (known formally in mathematics as a 'torus'). If you squish the coffee cup from above, you end up with a disc with a handle. If you then distribute the material in the disc evenly around the handle, smoothing the edge along the way, you end up with a donut. If you don't believe me, try searching 'coffee cup donut topology video' to watch the transformation happen in real time.

Answer: Game theory

Game theory is a branch of applied mathematics that considers structured interactions (known as 'games'), and the discovery of the optimal strategy for playing those games. It is necessary for the game to have a formal incentive for each player, such as 'you will get $500 if X event happens', in order to mathematically work out the best strategy.

A classic example of game theory is the so-called 'Prisoner's Dilemma'. In this 'game', two criminals have been arrested for a crime that they may have committed. Each is offered a deal for a reduced sentence if they snitch on their partner. However, if they both snitch, the sentence they each receive will be longer than if neither of them had said anything in the first place. What should they do?

Answer: Number theory

Number theory considers the properties of the whole numbers and functions related to them. It is one of the oldest areas of mathematics to still be studied academically today, with a history stretching back to ancient Mesopotamia (almost two millennia BCE).

While ancient mathematicians such as Pythagoras, Diophantus and Euclid were all interested in number theory, the creation of modern number theory is credited to Fermat, Euler and Gauss in the 17th and 18th centuries.

Answer: Knot theory

Knot theory is pretty much exactly what it says on the tin - the study of mathematical knots. Mathematical knots are almost the same as the knots that we encounter in day-to-day life, with the added complication that they are made from a single piece of material and thus form a continuous loop.

The simplest nontrivial knot (i.e. a knot that cannot be undone into a single loop) is known as the trefoil knot, while a series of knots joined together is called a link. Knot theorists often depict the knots they are studying in 2D, with overlaps between knots drawn such that they can tell which segment is in front and which is behind.

Answer: Linear algebra

Vector spaces share many aspects of their definition, such as associativity and an identity element, with the groups of group theory. However, vector spaces must be collections of vectors, and the operations that can be applied to them are known as 'vector addition' and 'scalar multiplication'.

To illustrate, vector addition in 3D Euclidean space (the space in which we all live) would look like this:

a = (1, 3, 5)
b = (2, -1, 0)

a+b = (3, 2, 5)

Where each element of a is added to its corresponding element in b. Scalar multiplication is where the entire vector is multiplied by a constant:

2a = 2(1, 3, 5) = (2, 6, 10)

Answer: Fractal geometry

A fractal can be generated using a set of instructions that are repeated over and over again, which gives the appearance that the fractal appears exactly the same no matter how far you zoom in. An example of such a set of instructions is the construction of a Koch curve, one of the earliest fractal curves to be defined:

- Start with a straight line of length 3 cm

- Remove the middle 1 cm and replace it with two diagonal 1 cm lines to form an equilateral triangle with no base

- Repeat the step above on each of the four 1 cm line segments now available, removing and replacing the central section each time

- Keep on repeating!

Three Koch curves placed together in an equilateral triangle form a shape known as the Koch snowflake.