Names for extremely large numbers are kinda pointless. The purpose of a name is to communicate something, and if the meaning of the name is not widely known, the name doesn’t serve that purpose. It’s even counterproductive if two people have incompatible ideas what the word means. It’s much safer to simply use scientific notation for extremely large numbers.

Obscure or confusing

Everyone knows what a million is. Any larger than that and you may run into trouble. The next large number name, billion, means 10⁹ to some people and 10¹² to others.

When writing for those who take one billion to mean 10⁹, your audience may or may not know that one trillion is 10¹² according to that convention. You cannot count on people knowing the names of larger numbers: quadrillion, quintillion, sextillion, etc.

To support this last claim, let’s look at the frequency of large number names according to Google’s Ngram viewer. Presumably the less often a word is used, the less likely people are to recognize it.

Here’s a bar chart for the data from 2005, plotting on a logarithmic scale. The chart has a roughly linear slope, which means the frequency of the words drops exponentially. Million is used more than 24,000 times as often as septillion.

Here’s the raw data.

    |-------------+--------------|
    | Name        |    Frequency |
    |-------------+--------------| 
    | Million     | 0.0096086564 |
    | Billion     | 0.0030243298 |
    | Trillion    | 0.0002595139 |
    | Quadrillion | 0.0000074383 |
    | Quintillion | 0.0000016745 |
    | Sextillion  | 0.0000006676 |
    | Septillion  | 0.0000003970 |
    |-------------+--------------|

Latin prefixes for large numbers

There is a consistency to these names, though they are not well known. Using the convention that these names refer to powers of 1000, the pattern is

[Latin prefix for n] + llion = 1000ⁿ⁺¹

So for example, billion is 1000²⁺¹ because bi- is the Latin prefix for 2. A trillion is 1000³⁺¹ because tri- corresponds to 3, and so forth. A duodecillion is 1000¹³ because the Latin prefix duodec- corresponds to 12.

    |-------------------+---------|
    | Name              | Meaning |
    |-------------------+---------|
    | Billion           |   10^09 |
    | Trillion          |   10^12 |
    | Quadrillion       |   10^15 |
    | Quintillion       |   10^18 |
    | Sextillion        |   10^21 |
    | Septillion        |   10^24 |
    | Octillion         |   10^27 |
    | Nonillion         |   10^30 |
    | Decillion         |   10^33 |
    | Undecillion       |   10^36 |
    | Duodecillion      |   10^39 |
    | Tredecillion      |   10^42 |
    | Quattuordecillion |   10^45 |
    | Quindecillion     |   10^48 |
    | Sexdecillion      |   10^51 |
    | Septendecillion   |   10^54 |
    | Octodecillion     |   10^57 |
    | Novemdecillion    |   10^60 |
    | Vigintillion      |   10^63 |
    |-------------------+---------|

Code

Here’s the Mathematica code that was used to create the plot above.

    BarChart[ 
        {0.0096086564, 0.0030243298, 0.0002595139, 0.0000074383, 
         0.0000016745, 0.0000006676, 0.0000003970}, 
         ChartLabels -> {"million", "billion", "trillion", "quadrillion", 
                         "quintillion", "sextillion", "septillion"}, 
         ScalingFunctions -> "Log", 
         ChartStyle -> "DarkRainbow"
    ]

Which has more mobility on an empty chessboard: a rook, a king or a knight?

On an ordinary 8 by 8 chessboard, a rook can move to any one of 14 squares, no matter where it starts. A knight and a king both have 8 possible moves from the middle of the board, fewer moves from an edge.

If we make the board bigger or higher dimensional, what happens to the relative mobility of each piece? Assume we’re playing chess on a hypercube lattice, n points along each edge, in d dimensions. So standard chess corresponds to n = 8 and d = 2.

A rook can move to any square in a coordinate direction, so it can choose one of n-1 squares in each of d dimensions, for a total of (n-1)d possibilities.

A king can move a distance of 0, 1, or -1 in each coordinate. In d dimensions, this gives him 3^d – 1 options. (Minus one for the position he’s initially on: moving 0 in every coordinate isn’t a move!)

What about a knight? There are C(d, 2) ways to choose two coordinate directions. For each pair of directions, there are 8 possibilities: two ways to choose which direction to move two steps in, and then whether to move + or – in each direction. So there are 4d(d – 1) possibilities.

In short:

• A king’s mobility increases exponentially with dimension and is independent of the size of the board.
• A rook’s mobility increases linearly with dimension and linearly with the size of the board.
• A knight’s mobility increases quadratically with dimension and independent of the size of the board.

The rules above are not the only way to generalize chess rules to higher dimensions. Here’s an interesting alternative way to define a knight’s moves: a knight can move to any lattice point a distance √5 away. In dimensions up to 4, this corresponds to the definition above. But in dimension 5 and higher, there’s a new possibility: moving one step in each of five dimensions. In that case, the number of possible knight moves increases with dimension like d⁵.

Related post: A knight’s random walk in 3 or 4 dimensions

The sum of the squares of consecutive Fibonacci numbers is another Fibonacci number. Specifically we have

and so we have the following right triangle.

The hypotenuse will always be irrational because the only Fibonacci numbers that are squares are 1 and 144, and 144 is the 12th Fibonacci number.

It’s hard to find bounds on binomial coefficients that are both simple and accurate, but in 1990, E. T. H. Wang upper and lower bounds on the central coefficient that are both, valid for n at least 4.

Here are a few numerical results to give an idea of the accuracy of the bounds on a log scale. The first column is the argument n, The second is the log of the CBC (central binomial coefficient) minus the log of the lower bound. The third column is the log of the upper bound minus the log of the CBC.

    |---+--------+--------|
    | n |  lower |  upper |
    |---+--------+--------|
    | 1 | 0.0000 | 0.3465 |
    | 2 | 0.0588 | 0.2876 |
    | 3 | 0.0793 | 0.2672 |
    | 4 | 0.0896 | 0.2569 |
    | 5 | 0.0958 | 0.2507 |
    | 6 | 0.0999 | 0.2466 |
    | 7 | 0.1029 | 0.2436 |
    | 8 | 0.1051 | 0.2414 |
    | 9 | 0.1069 | 0.2396 |
    |---+--------+--------|

And here’s a plot of the same data taking n out further.

So the ratio of the upper bound to the CBC and the ratio of the CBC to the lower bound both quickly approach an asymptote, and the lower bound is better.

Quaternion multiplication is associative but not commutative. An earlier post looked at the average size of the commutator xy – yx as a measure of how far quaternion multiplication is from being commutative.

This post looks at an analogous question for octonions. Octonion mulitplication is neither commutative nor associative. So in this post we look at the associator of three octonions (xy)z – x(yz) as a measure of how far octonion multiplication is from being associative.

(A post from yesterday looked at how close octonion multiplication comes to being associative in an algebraic sense, looking at weak forms of associativity. This post looks at how close multiplication comes to being associative in an analytical sense, looking at norm distances rather than algebraic identities.)

We will use a simulation to look at the average norm of the octonion associator over the octionions of unit length, analogous to the earlier post that looked at the commutator of the quaternions. We developed code for octonion multiplication in the previous post and will reuse that code here. We also developed code for generating random unit-length octonions in the same post.

Here’s code to find the average norm of the associator and plot a histogram of its values.

    
    import matplotlib.pyplot as plt

    # omult is octonion multiplication. 
    # See previous post.
    def associator(x, y, z):
        return omult(omult(x, y), z) - 
               omult(x, omult(y, z))

    N = 100000
    s = 0
    h = np.zeros(N)
    for n in range(N):
        x = random_unit_octonion()
        y = random_unit_octonion()
        z = random_unit_octonion()    
        t = norm(associator(x, y, z))
        s += t
        h[n] = t

    print(s/N)
    plt.hist(h, bins=50)
    plt.show()

This gives an average of 1.095 and the histogram below.

Update: Greg Egan calculated the exact mean value for the norm of the associator to be 147456/(42875 π) ≈ 1.0947336. Here are the details.

A group is a set with a binary operation that

1. closed
2. associative,
3. has an identity, and
4. has inverses.

This post will look at each of these properties and what happens if you modify or remove it.

Magmas

Closed means that applying the binary operation to any two elements of the set yields another elements of the set. (I saw a book one time that had at the top of some chapter a quote something like “The product of two real numbers is a real number; we should be surprised if it were a head of cabbage.”) For example, the positive integers are closed under multiplication but not under division.

A set with a closed binary operation is called a magma.

Semigroups

Associative means that you can parenthesize an expression any way you want and always get the same result. It may be practically important how you parenthesize an expression. For example, matrix multiplication can far fewer operations if you carry out the multiplications in the right order. In programming, left folds and right folds (such as foldl and foldr in Haskell) may have very different efficiency, but they’re supposed to return the same result if you’re folding an associative operation.

A magma with an associative operation is called a semigroup.

Differential equation theory is concerned with analytic semigroups. If you advance the solution of an evolution equation by a positive time increment t and then by an increment s, you end up at the same place as if you’d advanced the solution by t + s.

Notice that the time increment is required to be positive. That’s because the initial conditions for an evolution equation might reside in a different function space than the solutions at any future time. That’s the case, for example, with the heat equation, where initial conditions can be rough but the solution smooths out immediately as you move forward in time.

Monoids

If you add the requirement that a semigroup has an identity, you get a monoid. Partial differential equation theory is more concerned with analytic semigroups than analytic monoids for the reason explored above: you might not have an identity.

Positive integers with multiplication form a monoid but not a group because they have an identity, 1, but positive integers other than 1 do not have a multiplicative inverse that is an integer.

A monoid in which every element has an inverse is a group.

Quasigroups

So far we’ve built up a progression of stronger and stronger requirements leading up to groups, starting with requiring that our operation is associative. What if we don’t have associativity, but we do have other properties?

Non-associative operations are less common. Perhaps the best known operation that is not associative is octonion multiplication.

There are four division algebras over the reals: the reals themselves, complex numbers, quaternions, and octonions. Real and complex numbers are both fields. Quaternions are not a field because multiplication is not commutative. Octonions are even further from being a field because multiplication is not even associative.

But division is possible for octonions, except for 0, and so non-zero octonions form a quasigroup, a magma with division.

Loops

A quasigroup that has an identity element is called a loop. Note that for quasigroups we said “division is possible,” not that elements have inverses. If elements do have inverses, then you can divide by an element by multiplying by its inverse. But you can define division by the ability to solve equations. We say you can divide b by a if ax = b and ya = b have unique solutions.

A loop with associativity is a group.

Alternative loops

Loops are not required to have an associative property, but they may satisfy a weak form of associativity.

An alternative loop satisfies x(xy) = (xx)y and x(yy) = (xy)y. Non-zero octonions form an alternative loop with respect to multiplication.

Moufang loops

A Moufang loop is a loop that satisfies these requirements:

1. z(x(zy)) = ((zx)z)y
2. x(z(yz)) = ((xz)y)z
3. (zx)(yz) = (z(xy))z
4. (zx)(yz) = z((xy)z)

These conditions imply the alternative loop conditions, so Moufang loops are alternative loops.

Moufang loop are still not required to be associative, but they’re getting closer. And as you might have anticipated, non-zero octonions with respect to multiplication form a Moufang loop.

Related posts

• How many ways can you parenthesize an expression?
• Functional folds and conjugate models
• Dot, cross, and quaternion products
• How far is xy from yx for quaternions?

Start with the sequence of positive integers:

1, 2, 3, 4, …

Now take partial sums, the nth term of the new series being the sum of the first n terms of the previous series. This gives us the triangular numbers, so called because they count the number of coins at each level of a triangular arrangement:

1, 3, 6, 10, …

If we repeat this again we get the tetrahedral numbers, the number of balls on each level of a tetrahedral stack of balls.

1, 4, 10, 20, …

We can repeat this process and general define T_{n, d}, the nth tetrahedral number of dimension d, recursively. We define T_{n, 1} = n and for d > 1,

This is just a formalization of the discussion above.

It turns out there’s a simple expression for tetrahedral number of all dimensions:

Here’s a quick proof by induction. The theorem clearly holds when n = 1 or d = 1. Now assume it hold for all n < m and d < m.

The last line follows from the binomial addition identity

It also turns out that T_{n, d} is the number of ways to select d things from a set of n with replacement.

Related posts:

• When is a triangle a square?
• Sums of consecutive powers