We may be celebrating Pi Day here at io9, but we would be irrational to deny that there’s more to mathematical interestingness than simply dividing an object’s circumference by its diameter. Here are seven numbers we love as much as pi.

#### 1. 1

1 may be the loneliest number, but it’s the littlest number that could — the first non-zero integer that displays remarkable properties of self-reliance. Aside from being the first whole number, it is its own square, cube, and factorial. It’s also very stubborn; when you raise 1 to any power — even a number as high as a googolplex (1 followed by 10 to the 100th power, or 10^(10^100)) — you still get 1. It's the first and second number in the Fibonacci sequence. It is neither a composite number, nor a prime number (mathematicians rejected this idea because it complicates fundamental theorems of arithmetic). It is, however, a unit (like -1). And it’s the only positive number that’s divisible by exactly one positive number.

#### 2. *i*

Any number that doesn’t actually exist, but is still useful, has to be considered cool. Also called the **imaginary unit**, *i* is the square root of -1 (*i*^{2} = -1). This number cannot exist because no number multiplied by itself can equal a negative number.

At first, **imaginary numbers** were considered useless (an imaginary number is a number that, when squared, gives a negative result; e.g. *5i* = -25). But by the Enlightenment Era, thinkers began to demonstrate its value in math and geometry, including Leonhard Euler, Carl Gauss, and Caspar Wessel (who used it when working with complex planes). They’re useful in that they can be used to find the square root of a real negative number.

Today, *i* is used in signal processing, control theory, electromagnetism, fluid dynamics, quantum mechanics, cartography, and vibration analysis. The figure *j* is often substituted in these fields, which is used to represent the electric field current. The imaginary number also appears in several formulas, including the Euler Identity.

As an aside, Isaac Asimov’s short story “The Imaginary” (1942) featured the eccentric psychologist Tan Porus who explained the behavior of a mysterious species of squid by using imaginary numbers in the equations which describe its psychology.

#### 3. Graham's Number

Simply put, this is the largest useful (i.e. non-arbitrary) number known to mathematicians. But it’s an *astoundingly* large number. Named after Ronald Graham, it’s the upper bound to a certain question that involves Ramsey Theory (a branch of math that studies the conditions under which order must appear). Consequently, it’s the biggest number used for a serious mathematical proof.

This number’s “root” arises from the extreme addition, multiplication, and powering of threes. It’s subsequently a very big power of three, and the number itself is considerably larger than a googolplex. In fact, Graham’s number is so mindboggingly huge that it cannot be expressed using conventional notation of powers, and even powers of powers. It’s so large, that if all the material in the universe were turned into pen and ink it would not be enough to write the number down. Consequently, mathematicians use a special notation devised by Donald Knuth to express it.

It’s so big that it’s physically impossible for our brains to comprehend. AI theorist Eliezer Yudkowsky put it this way:

Graham's number is far beyond my ability to grasp. I can describe it, but I cannot properly appreciate it...My sense of awe when I first encountered this number was beyond words. It was the sense of looking upon something so much larger than the world inside my head that my conception of the Universe was shattered and rebuilt to fit. All theologians should face a number like that, so they can properly appreciate what they invoke by talking about the "infinite" intelligence of God.

Interestingly, if not ironically, the lower bound to the Ramsey problem that gave birth to that number — rather than the upper bound — is probably six. *Note: A reader alerted me to this **study**, which suggests a lower bound raised to 11, and then to 13.*

#### 4. 0

The number 0 is totally taken for granted, which, when considering that it represents nothing, is somewhat understandable. But it does serve some important functions, including as an empty place-value in our decimal number system. How else, for example, could we express the year 1906 in the decimal system without it?

Sure, the universe starts to melt when you try to divide by it, but 0 can serve some important roles in equations, including those that involve addition, multiplication, and subtraction. Numbers can also be raised by the power 0, which will always produce the value of 1. And if you raise 0 to power of anything, you still get 0. But, if try to do 0^0, math goes all squirrely again and the answer becomes basically anything (an “indeterminate form”).

Lastly, the sum of 0 numbers is 0, but the product of 0 numbers is 1. And 0 is neither positive, nor negative. It’s not a prime number, and it’s not a unit — but it is an even number.

#### 5. *e*

Yes, there’s a number called ‘*e*’, but it’s also known as **Euler’s Number**. Like pi, it’s an important mathematical constant, an irrational number that goes like this: 2.71828182845904523536...

Named after Leonhard Euler (1707-1783), it’s the base of John Napier’s Natural Logarithms — the logarithm to the base *e*, where *e* is an irrational and non-algebraic number (what’s called a **transcendental constant**, much like pi). Some people refer to it as the natural base. Euler devised the following formula to calculate e:

e= 1+ 1/1 + 1/2 + 1/(2 x 3) + 1/(2 x 3 x 4) + 1/(2 x 3 x 4 x 5) + . . . (alternately: 1 + 1/1 + 1/2! + 1/3! + 1/4! + 1/5!)

Mathematicians have calculated *e* to over a trillion digits of accuracy.

Euler's interest in *e* came about when calculating continuously compounded interest on a sum of money. And in fact, the limit for compounding interest can be expressed by the constant *e*. So, if you invest $1 at an interest rate of 100% per year, and the interest is compounded continuously, you will have $2.71828 (or so) at the end of the year.

*e* also shows up in probability theory and the Bernoulli trials process (which is helpful for calculating things like probabilities in gambling). Other applications include derangements (the so-called hat-check problem), asymptotics (when describing limiting behavior, a useful concept in computer science), and calculus.

#### 6. Tau

Tau is simply **2pi**, or the constant that is equal to the ratio of a circle’s circumference to its radius. Thus, tau is written out like 6.283185...

Tau is the 19th letter of the Greek alphabet and was chosen as the symbol for 2pi by Michael Hartl, a physicist, mathematician, and author of "The Tau Manifesto," along with Peter Harremoës, a Danish information theorist (who knew math could get so political?).

Tau is considered by some to be more useful than pi for measuring circles because mathematicians tend to use radians instead of degrees. According to Kevin Houston from the University of Leeds, the most compelling argument for tau is that it is a much more natural number to use in the fields of math involving circles, like geometry, trigonometry and even advanced calculus.

What this means, of course, is that Tau Day should be celebrated on June 28 (6/28).

#### 7. Phi (φ)

Also called the **Golden Number**, Phi (rhymes with "fly") is an important mathematical figure that’s written out as 1.6180339887...

Unlike pi, which is a transcendental number, phi is the solution to a quadratic equation. But like pi, phi is a ratio that’s defined by geometric construction. Two quantities fit within the golden ratio if the ratio of the sum of the quantities to the larger quantity is equal to the ratio of the larger quantity to the smaller one. Because of its unique properties, phi is used in math, art, and architecture. The Greeks discovered it as the dividing line in the extreme and mean ratio, and for Renaissance artists it represented the Divine Proportion.

Phi also has interesting equivalent ratios when the number one is introduced, like φ:1 is equal to φ+1:φ, or 1:φ-1. Also, two successive fibonacci numbers, when divided, produce a number close to phi. The further through the series, the more accurate (or detailed) phi becomes.

*Special thanks to Calvin Dvorsky for helping me with this article!*

*Top image: Sashkin/Shutterstock.*

## DISCUSSION

Any number that doesn’t actually exist, but is still useful, has to be considered cool.No number actually exists, zero and pi included. They're concepts. Math is the study of things that are made out of rules. "Imaginary" numbers are just one set of objects that follow a slightly different set of rules from "real" numbers.

You want to define the imaginary numbers, if you've already got the reals defined? Here's one way: Define a "complex number" as an ordered pair of real numbers. Define addition of complex numbers, (a1,a2) + (b1,b2) = (a1+b1,a2+b2), and multiplication, (a1,a2) * (b1,b2) = (a1*b1-a2*b2,a2*b1+a1*b2), then subtraction and division as their inverses per usual (with additive identity (0,0) and multiplicative identity (1,0)). Now an imaginary number is just a complex number of the form (0,x), and i=(0,1).

3. Graham's Numberio9 covered this number a while back. A link might be in order, for the curious.

How else, for example, could we express the year 1906 without it?MXMVI, as some other reader has probably already said. :)

φ:1 is equal to 1:1+φWhat? No. φ:1 is equal to φ+1:φ, or 1:φ-1. In your version, not only are the ratios different, one is obviously >1 and the other is obviously <1.