The **Turkish currency** is called the ** lira**. This, of course, is an anagram of the

**Georgian currency**, the

**. I have yet to stop mixing these words up. As far as I can tell, most Anglophones use**

*lari**lira*as both singular and plural, which they also do for the lari. To my ear this smacks of

**Orientalism**, and I prefer to use regular English plurals:

*liras*and

*laris*.

The current lira regime was introduced in 2005, when the **“new lira”** replaced the **“old lira”**. The “old lira” had become **grossly inflated**, and was exchanged for “new liras” at a rate of **1,000,000:1**. At the time of its demise, the old lira was being issued in denominations as high as twenty million.

The **Turkish lira sign**, like the **Georgian lari sign**, was introduced quite recently (2012). It looks like a capital *L* (with two lines across, of course) bent into the shape of half an **anchor**. This was intended to show that the currency is a “safe harbor”. I think they should add a crow’s nest at the top to show that the currency is “looking ahead”, and maybe some other **nautical symbolism**.

Again as with Georgia, one can’t help but wonder why a country intent on **joining the EU** and presumably adopting the euro would try to better establish its own currency.

My favorite piece of Turkish money is the **Turkish 10 lira bill**, which is unusual in featuring a mathematician and a **complex mathematical formula**. The mathematician is named **Cahit Arf**, and the formula is called the **Arf invariant**. I don’t really understand what this is, but I do know that it has something to with * quadratic forms over fields of characteristic 2*. I’ll try to explain what this means as best as I can, in this order: field, characteristic, quadratic form.

Consider the integers, i.e. the set {…,-2,-1,0,1,2,…}, under the operation of addition. Note the following properties:

- The integers are
*closed*under addition. That is, given any two integers, their sum is also an integer. - Addition is
*associative*, meaning that for any three integers a,b,c, (a+b)+c=a+(b+c). - There is an
*identity element*, 0, such that for any integer a, a+0=a. (This turns out to be unique.) - Every integer a has an
*inverse element*, -a, such that a+-a=0. (Inverses also turn out to be unique.)

In general, we say that a set forms a *group* under an operation if that operation satisfies these criteria. A group that satisfies a+b=b+a is called a *commutative group*. Not all groups are commutative, although the integers are. More examples:

- The even integers form a group under addition.
- The odd integers do
*not*form a group under addition, since they are not closed thereunder. For example, 5+3 is 8, which is not an odd integer. - The positive fractions form a group under multiplication: the identity element is 1, and the inverse of a is 1/a.
- The positive fractions do
*not*form a group under addition: besides not being closed under it, the set lacks an additive identity and additive inverses.

These examples have all involved *infinite* sets, but groups can be finite as well. Compositions of transformations of geometric figures often form groups (notably, in particular, the ** Rubik’s Cube** under various moves).

Importantly, the integers do not form a group under multiplication. However, we can “close” the integers under multiplication by adding a multiplicative identity (1) and multiplicative inverses (1/a for a). This gives us the *rational numbers, *i.e. the set of all fractions. (We could equivalently obtain this by “closing” the positive fractions under addition.)

The group is a generalization of the integers under addition. Similarly, the generalization of the rationals under addition and multiplication is the ** field**. A field is a set F upon which are defined two operations + and * such that F forms a commutative group under both + and * with the further requirement that * is

*distributive*over +: a*(b+c)=(a*b)+(a*c). Other common examples of fields are the real numbers and the complex numbers, both of which are extensions of the rationals. Fields, like groups, can also be finite.

Certain fields have the property that if you add “1” to itself enough times, you’ll end up with “0”, though of course these are not 1 and 0 as we normally know them, but rather the multiplicative and additive inverses, respectively. The ** characteristic** of a field is the number of 1s it takes to get 0; if this cannot be done (as with any field a non-mathematician would recognize), we say the field is of characteristic 0. Every finite field is of positive characteristic, since with finitely many elements you have to eventually “loop back around”, but the converse is not true: there are infinite fields of nonzero characteristic.

Switching gears, a * quadratic form* is merely a polynomial wherein the sum of the exponents of the variables of each term add up to exactly 2. An example of this is (for variables x and y) x^2+xy+y^2. The exponents of the variables of the first and third terms are obviously 2, while the exponents of the variables of the middle term add to 2 because xy=x^1*y^1 and 1+1=2.

** Exercise for the reader: **Explain why a typical

*quadratic function*ax^2+bx+c is not a quadratic form (unless b,c=0).

I don’t really understand why quadratic forms are important or interesting, though one of the recently-awarded ** Fields Medals** was given to

**Manjul Bhargava**for work dealing with them.

**Finally**, we’re in a position to explain (at a gravely basic level) what the Arf invariant is. Prior to Arf’s work, there were certain facts that could be proved about quadratic forms over fields of every characteristic but 2. Arf developed the Arf invariant to prove those facts (or maybe disprove them, I don’t know) for the remaining case. I don’t know why fields of characteristic 2 should be so different from other fields, but apparently they are.

Now, you might be thinking “Hey, wow, neat, a circulating bill with math on it. I bet that’s never been done before.” **Wrong**, it has been done before. From 1990 until the introduction of the **euro** in 2002, the **German 10 mark bill** featured the legendary mathematician **Carl Friedrich Gauss** along with some kind of statistics graph called a **Gaussian**** distribution **or **normal distribution**. I don’t know anything about stats, so I can’t explain it.

* Exercise for the reader:* In the comments section, explain in basic (but still sufficiently detailed) terms what a “Gaussian distribution” is. If someone has already done this, try to give an even better explanation.

*Note on terminology:** *In some languages, a field is called a ** body** (French:

*corps*; German:

*Körper*; Turkish:

*cisim*).

*Field*is used in Russian (Поле,

*polye*) and Georgian (ველი,

*veli*). Both

*field*and

*body*are used in Italian (

*campo*and

*corpo*), though the latter refers to what in English is called a

*skew field****. Groups, as far as I know, are the same in every language. This is usually some variant of

*group*(

*gruppe*, группа, etc.) but not always: the Georgian word for

*group*is the truly terrible ჯგუფი,

*jgupi*.

***A skew field is like a field except that its multiplication operation is not required to be commutative. It turns out, in a result known as *Wedderburn’s Little Theorem*, that all proper skew fields are infinite. Another proof of this theorem was among the mathematical publications of the **Unabomber**.

The origin of quadratic forms lies in the need to have a good distance function. The most basic of these is of course x²+y²+z² which gives the distance squared between the point (x,y,z) and the origin (0,0,0) and it’s called the Euclidean distance.

In relativity theory we have e.g. x²+y²+z²-c²t². This is the distance between to events (an event is a place and a time, c is the speed of light).

Gauss’s doctoral thesis was in fact the first comprehensive work about quadratic forms and it’s still very useful (e.g. http://en.wikipedia.org/wiki/Quadratic_reciprocity).

A Gaussian distribution explains what a graph of measurements that depend on several factors looks like. E.g. the length of people: most people will be round average height (the bulge in the middle), very few people are extremely tall or extremely tiny (the drops of the curve at both ends).

The location of the top of the curve is determined by the average or mean height. The sharpness of the peak (i.e. how fast it drops to both sides) is determined by what is called the standard deviation. It is a measurement of the average distance between the individual measurements and the mean.

Excellent remarks! In fact it was directly from Gauss’s Disquisitiones Arithmeticae that Bhargava took his start. Is it a coincidence that Gauss should come up in this post in two unrelated ways?