Seven thousand seven hundred and seventy-seven.

There are 14 syllables while the original number has only 4 digits. This makes it very tedious to read numbers in English. Thus, I want to invent a better way to read numbers!

Single-digit numbers

First, let’s improve the first few numbers in English. They are

(I do not want to read 0 as /əʊ/ because it confuses with the letter O.) Here are several problems:

• Some words have more than one syllable or end with a consonant, this makes it hard to pronounce them fast in continuous speech.
• What do 10, 11, and 12 do here? They are not single-digit numbers and should not appear here.
• Some of the words have strange spelling.
• two has the same pronunciation as too, and four has the same pronunciation as for, which may be confusing sometimes.

I want to improve them as follows:

There are some advantages:

• Most of the words have similar pronunciation as the original words, this makes it easy to remember them. The only exceptions are sou and sah because there too many single-syllable English words that start with /s/, and I have to avoid the existing words.
• Each of the words contains only one simple syllable, which makes it easy to pronounce fast.
• The spelling pretty much follows the pronunciation and avoids large discrepencies among accents.
• The written length of the words are all the same (three letters), which makes it easier to sight-read.
• The words are pretty distinguishable by their pronunciations so that they do not confuse when being said in a noisy environment.

Positive integers smaller than 1000

In English, the suffix -ty is used to get multiples of ten. This suffix will sound too similar to tee (3) in our improved words for digits. Therefore, I want to propose a new suffix -ta, pronounced as /tə/. The multiples of ten are then

For numbers smaller than 100 and not multiples of 10, they are just the sum of the corresponding multiples of 10 and single-digit numbers. The two parts hyphenizes to represent the number. The numbers 11 etc. are then

Now we can read all numbers smaller than 100.

For numbers starting from 100, English uses the word hundred, which is very long, and it is almost always followed by a useless word and. A better way to express numbers starting from 100 is just how we express those from 10. I want to propose a new suffix -ha, pronounced as /hə/, to represent multiples of hundred. The multiples of hundred are then

Hyphenization gives other numbers below 1000:

Sometimes we also benefits from just reading the digits. If the speaker know for sure that the audience will not confuse whether the number is a categorical data or a cardinal data, then the speaker can just read the digits. This makes the delivery of information about numbers really fast considering that now every digit only has one syllable.

Integers starting from 1000

For numbers larger than 1000, just use how English deal with them, but use thou, mill, bill, etc. to replace the multi-syllable words thousand, million, billion, etc. Another way to do this is just reading digits.

(Actually, I do not think using thou, mill, etc. is a good idea because they coincide with existing English words and are hard to remember (for foreigners), but I cannot come up with a better idea.)

Decimals

For decimals, just use how English deal with them. Use dot instead of point to tell the decimal point because it is easier to pronounce. Also, I propose that we read dot as /dɒ/ instead of /dɒt/ to pronounce it faster.

Negative numbers

For negative numbers, add ne (from the word negative), pronounced as /neɪ/, before the number.

Fractions

English use ordinal numerals to represent fractions, which is very non-intuitive and sometimes ambiguous. There are various ways to represent fractions in English that do not involve ordinal numerals. For example, a half can be represented as

• 1 in 2,
• 1 (divided) by 2,
• 1 over 2,
• 1 to 2 (only used for ratios),
• 1 slash 2,

etc., but none of them is a good way because they may be ambiguous. We need to invent a new word. I propose using the word ci, from the second syllable in the word reciprocal, pronounced as /sɪ/.

Scientific notation

How do you read $1.234\times10^{-8}$?

One point two three four times ten to the power of negative eight.

That is very long! I propose that we write the number as 1.234e-8, and just pronounce how it looks (pronounce the e here as /iː/).

Ordinal numerals

English uses the suffixes -th, -st, -nd, and -rd to represent ordinal numerals, which is actually a disaster…

• Do you like $(n+1)$st or $(n+1)$th? (People surely have personal preferences over these two according to the answers, and there is no standard.)
• You may know Platform Nine and Three-Quarters if you are a Harry Potter fan, but what is the ordinal of $9\frac34$?
• Because there are multiple suffixes, there is a disaster for non-native speakers to write ordinal numerals. You may Google “21th” to see how many people have made the mistake.
• How to ask a question w.r.t. an unknown ordinal numeral? “The how many-th president is Barack Obama?” (Most the answers in this link cannot generalize to other questions like “How many-th most beautiful person is Kat in the world?” or even weirder questions like “You love Kat the how many-th most in the world?”)

Therefore, we need to have a consistent way to turn any cardinal numeral into an ordinal numeral, and we need to invent a way to ask questions about ordinal numerals.

I then want to use the word ra (abbreviation for rank, although this is an existing English meaning the hawk-headed sun god), pronounced as /rɑː/. Put the word ra before a number to represent the ordinal numeral.

To ask a question about an ordinal numeral, just convert the structure “ra + num. + sth” into “ra what + sth”, and form the rest of the sentence to get a question. If there is a definite article before ra, omit it in the question.

Example sentences:

• The ra 21 century is the century of biology.
• Kat wants a ra tue child.
• The ra $n$ number in the Fibonacci sequence is $F_n$.
• Students board Hogwarts Express at the ra $9\frac34$ platform on the ra wah day in September.
• The ra ne wah element in an array is defined to be its last element. Generally, the ra $-k$ element in an array is defined to be its ra $n-k$ element, where $n$ is the length of the array.
• In an arithmetic progression, the difference between the ra $n+1$ term and the ra $n$ term is the same for any $n$.
• Ra what president is Barack Obama?
• Ra what most beautiful person is Kat in the world?
• Ra what most do you love Kat in the world? / Ra what most in the world do you love Kat?

Although I invented a set of new rules for ordinal numerals, I do not delete the word first in English. The word first now means the frontmost thing in spatial or chronological order. The difference between being the first and being the ra wah appears when the things in order are uncountable or does not start from number 1. For example, we can say the first element in the interval $[0,1]$ is $0$ if we order the elements from the small to large, but there is no ra wah element in it. For another example, because indices start from 0 in programming languages, the first element in an array is the ra zoh element instead of the ra wah element.

The word first can also be used in the following circumstances:

• In the structure first… next… then… last… or other similar structures;
• In collocations like in the first place, first of all, etc.

Sometimes, we may use firstly instead of first when it is used as an adverb (the same thing applys to other ordinal numerals). I would like to just replace firstly, secondly, etc. with ra wah, ra tue, etc.

Numerical prefixes

Here is another mess with English: numerical prefixes. Just look at how many systems of numerical prefixes we have in English.

However, I do not have motivations to fix the numerical prefixes… After all, people will invent their own. Also, there is an advantage of using numerical prefixes different from normal numerals: they avoid ambiguity in oral speech.

The spacetime of higher dimensions that we mention here refers to a Galileo universe with $\iota$ time dimensions and $\chi$ space dimensions. It has affine structure, and we manually define a coordinate system on it. Galilean transformations include uniform-velocity motion (note that velocities are matrices (see below)), spacetime translation, and space rotation. The principle of Galilean relativity still holds.

The universe is $\left(\iota+\chi\right)$-dimensional, manually separated into an $\iota$-dimensional subspace and a $\chi$-dimensional subspace, where the former is called the time space, and the latter is called the space space. World points (events) are described by the combination of time coordinates and space coordinates.

Note that, when we find the derivative of a function w.r.t. time, we need to distinguish the total partial derivative and the partial partial derivative. The former regards all variables as functions of time, while the latter one does not. We denote the total partial derivative of function $F$ w.r.t. the $j$th time coordinate as $\partial_jF$, and the partial partial derivative as $\frac{\partial F}{\partial t_j}$.

To avoid confusion, there is an example. For example, the total partial derivative of $F\!\left(q,t\right)$ w.r.t. $t_j$ is $\partial_jF=\frac{\partial F}{\partial q}\partial_jq+\frac{\partial F}{\partial t_j}$. We can see that the total partial derivative has an extra term in addition to the partial partial derivative, which originates from the change of other independent variables of the function due to the change of time.

A system with $s$ DOF needs $s$ multivariable functions $q_k\!\left(t\right)$ to describe, where $k$ is the subscript, and $t\in\mathbb R^\iota$. The numbers $q_k$ are called generalized coordinates. Generalized coordinates are mappings from the time space to the space space.

Then, the generalized velocities become matrices, whose each component represents each generalized coordinate changes w.r.t. each component of time coordinates. Written explicitly, it is $\partial_jq_k$. It can be regarded as the Jacobian matrix of generalized coordinates. We may think that the generalized velocities span a space called the velocity space.

Just like traditional universe with one time dimension, we define the action $\mathcal S$ as the volume integral of the function $\mathcal L$ in the time space, where the Lagrangian $\mathcal L$ is a scalar function defined on the space-velocity-time phase space. Written explicitly, it is $\mathcal S\coloneqq\int_M\mathcal L\\,\mathrm dV_t$, where $M\subseteq\mathbb R^\iota$ is a region in the time space, and $\mathrm dV_t$ is the volume element in the time space.

Now, Hamilton’s principle still holds. What it says is that, if we regard $\mathcal S$ as a functional of the function $q$, then the problem of finding the actual motion $q$ of the system is equivalent to solve the optimization problem: constraint the value of $q$ on $\partial M$, and minimize $\mathcal S$.

In this case, the Euler–Lagrange equation is (according to a previous post) $$\sum_j\partial_j\frac{\partial\mathcal L}{\partial\!\left(\partial_jq_k\right)}=\frac{\partial\mathcal L}{\partial q_k}.$$ Therefore, the momentum is defined to be the matrix $$p_{j,k}\coloneqq\frac{\partial\mathcal L}{\partial\!\left(\partial_jq_k\right)}.$$ Note that now the Euler–Lagrange equation is a set of second-order PDEs.

We try performing Legendre transformation on $\mathcal S$ and get Hamiltonian $\mathcal H$.

As a function of space coordinates and velocities, the total derivative of $\mathcal L$ is $$\mathrm d\mathcal L=\sum_{j,k}\frac{\partial\mathcal L}{\partial\!\left(\partial_jq_k\right)}\,\mathrm d\!\left(\partial_jq_k\right) +\sum_k\frac{\partial\mathcal L}{\partial q_k}\,\mathrm dq_k.$$ Substitute the Euler–Lagrange equation and the definition of momenta, and we have $$\mathrm d\mathcal L=\sum_{j,k}p_{j,k}\,\mathrm d\!\left(\partial_jq_k\right)+\sum_{j,k}\partial_jp_{j,k}\,\mathrm dq_k.$$ By the product rule, the first term in the formula above can be written as $$p_{j,k}\,\mathrm d\!\left(\partial_jq_k\right)=\mathrm d\!\left(p_{j,k}\partial_jq_k\right)-\partial_jq_k\,\mathrm dp_{j,k},$$ and then we have $$\mathrm d\!\left(\sum_{j,k}p_{j,k}\partial_jq_k-\mathcal L\right) =-\sum_{j,k}\partial_jp_{j,k}\,\mathrm dq_k+\sum_{j,k}\partial_jq_k\,\mathrm dp_{j,k}.$$ If we let $\mathcal H\coloneqq\sum_{j,k}p_{j,k}\partial_jq_k-\mathcal L$, then we have $$\partial_jq_k=\frac{\partial\mathcal H}{\partial p_{j,k}},\quad\sum_j\partial_jp_{j,k}=-\frac{\partial\mathcal H}{\partial q_k}.$$ This is the new Hamiltonian equations, or canonical equations. We may find that it lacks the beauty of the form in one-dimensional time.

Problem 1: Prove that if we add the Lagrangian by the “total divergence” w.r.t. time of some function defined on space and time, the new Lagrangian describes the same mechanical system as the original. In other words, $\mathcal L'\coloneqq\mathcal L+\sum_j\partial_jf$ has the same equation of motion as $\mathcal L$, where $f$ is an arbitrary function defined on the spacetime.

Problem 2: Prove by principle of Galilean relativity that the Lagrangian of a single free particle system is $\mathcal L=\sum_{j,k}\frac{m_j}2\left(\partial_jq_k\right)^2$, where $m_j$ are constants (their physical meaning is mass, which means that mass is not scalar in time of higher dimensions), and $q_k$ are Cartesian coordinates. Find and solve its equation of motion, and hence derive the law of inertia.

Problem 3: Does the conservation of energy still hold?

A problem that I am too lazy to consider: Consider the Minkowski universe. Poincaré transformations are defined as those affine transformations that preserve the spacetime distance between events, and the spacetime distance is defined as the difference of the square of Euclidean distance in time space and the square of Euclidean distance in space space. The principle of special relativity guarantees that the equation of motion of closed systems is invariant under Poincaré transformations. Find the Lagrangian of a single free particle system.

Have you ever thought of how can we describe the basic principles of our world (or universe), especially in a physical or mathematical way?

The thought itself seems like a philosophical problem (and is actually thought over by philosophers for thousands of years). However, maybe it can be interesting to think it over in another perspective.

Note that most of the definitions used below are different from the popular definitions!

Galilean world

Here is the basic principle of the Galilean world:

Principle 1. The world is a Galilean structure with $3$-dimensional space and $1$-dimensional time.

Here is the definition of a Galilean structure. A Galilean structure with $\chi$-dimensional space and $\iota$-dimensional time is a $3$-tuple $\left(\mathscr A,\tau,\rho\right)$ with the following principles:

1. $\mathscr A$ is a $\nu$-dimensional affine space associated with the vector space $\mathbb R^\nu$, where $\nu\coloneqq\chi+\iota$;
2. $\tau:\mathbb R^\nu\rightarrow\mathbb R^\iota$ is a linear mapping;
3. For $a,b\in\mathscr A$ such that $\tau\!\left(a-b\right)=0$, the mapping $\rho$ satisfies $\rho\!\left(a,b\right)=\sqrt{\left(a-b\right)^2}$.

To make the physical meanings of the above mathematical stuff clear, we

• call $\mathscr A$ the universe or a Galilean space,
• call the points in the universe the events,
• call $\tau$ the time,
• say two events $a,b$ are simultaneous iff $\tau\!\left(a-b\right)=0$,
• call $\rho\!\left(a,b\right)$ the distance between simultaneous events $a,b$.

Here is the second principle of the Galilean world:

Principle 2 (Galileo’s principle of relativity). Laws of nature remain the same under Galilean transformation.

Here is the definition of a Galilean transformation. An affine transformation $g$ over the Galilean space $\mathscr A$ is called a Galilean transformation iff both of the following are satisfied:

1. $\forall a,b\in\mathscr A: \tau\!\left(a-b\right)=\tau\!\left(ga-gb\right)$ (preservation of intervals of time),
2. $\forall a,b\in\mathscr A: \tau\!\left(a-b\right)=0\Rightarrow \rho\!\left(a,b\right)=\rho\!\left(ga,gb\right)$ (preservation of distance between simultaneous events).

Galilean transformations form a group (why?) called the Galilean group, which is an $\left(\iota+\frac{\chi\left(\chi+3\right)}2\right)$-dimensional Lie group (why?).

$$\begin{align*} \mathscr A&\coloneqq\mathbb R^\nu,\\ \tau&\coloneqq\left(t,x\right)\mapsto t,\\ \rho&\coloneqq\left(a,b\right)\mapsto\sqrt{\left(a-b\right)^2} \end{align*}$$ is a Galilean structure (why?). Here $\mathscr A$ is called the Galilean coordinate space.

The following transformations on the Galilean coordinate space are Galilean transformations (why?):

1. $\left(t,x\right)\mapsto\left(t,x+vt\right)$, where $v\in\mathbb R^\chi$ (uniform motion),
2. $\left(t,x\right)\mapsto\left(t+s,x+d\right)$, where $s\in\mathbb R^\iota$ and $d\in\mathbb R^\chi$ (translation),
3. $\left(t,x\right)\mapsto\left(t,Gx\right)$, where $G\in\mathrm O\left(\chi\right)$ (rotation).

Every Galilean transformation of the Galilean coordinate space can be represented uniquely as the composition of a rotation, a translation, and a uniform motion (why?).

All Galilean spaces with the same dimensions are isomorphic to each other (why?).

In fact, the two principles above are not enough to build up the whole classical mechanics. We need to define motion, velocity, and acceleration in our $3+1$ universe. Then, to describe how the motion is determined, we need the following principle:

Principle 3 (Newton’s principle of determinacy). The motion is uniquely determined by initial positions and initial velocities.

With this principle, we can conclude that the motion can be depicted by Hamilton’s principle $$\delta\int\mathcal L\!\left(q,\dot q,t\right)=0$$ (why?), which leads to Euler–Lagrange equation $$\frac{\mathrm d}{\mathrm dt} \frac{\partial\mathcal L}{\partial\dot q}= \frac{\partial\mathcal L}{\partial q}$$ (why?).

According to Principle 2, for a closed system, its Euler–Lagrange equation should remain unchanged after a Galilean transformation (in a specific coordinate system, which in most cases is the Cartesian coordinate system utilized by Galilean coordinate space) acts on it, from which we can see that the universe is time-homogeneous (invariance under time translation), space-homogeneous (invariance under space translation), and space-isotropic (invariance under space rotation).

The rest (deriving the Lagrangian for some typical mechanical systems, and solving them) is just the normal classical mechanics, and is not related to the topic today.

Einsteinian world

We can build up the Einsteinian world similarly as we have done for the Galilean world.

Principle 1. The world is an Einsteinian structure with $3$-dimensional space and $1$-dimensional time.

Here is the definition of an Einsteinian structure. An Einsteinian structure with $\chi$-dimensional space and $\iota$-dimensional time is a $4$-tuple $\left(\mathscr A,\tau,\sigma,\rho\right)$ with the following principles:

1. $\mathscr A$ is a $\nu$-dimensional affine space associated with the vector space $\mathbb R^\nu$, where $\nu\coloneqq\chi+\iota$;
2. $\tau:\mathbb R^\nu\rightarrow\mathbb R^\iota$ and $\sigma:\mathbb R^\nu\rightarrow\mathbb R^\chi$ are linear mappings;
3. The linear mapping $a\mapsto\left(\tau\!\left(a\right), \sigma\!\left(a\right)\right): \mathbb R^\nu\rightarrow\mathbb R^\nu$ has full rank;
4. $\forall a,b\in\mathscr A:\rho\!\left(a,b\right)= \sqrt{\tau\left(a-b\right)^2-\sigma\left(a-b\right)^2}$.

To make the physical meanings of the above mathematical stuff clear, we

• call $\mathscr A$ the universe or an Einsteinian space,
• call the points in the universe the events,
• call $\tau$ the time,
• call $\sigma$ the space,
• call $\rho\!\left(a,b\right)$ the spacetime interval between events $a,b$.

Here is the second principle of the Einsteinian world:

Principle 2 (Einstein’s principle of relativity). Laws of nature remain the same under extended Poincaré transformation.

Here is the definition of a Poincaré transformation. An affine transformation $g$ over the Einsteinian space $\mathscr A$ is called a Poincaré transformation iff $\forall a,b\in\mathscr A: \rho\!\left(a,b\right)=\rho\!\left(ga,gb\right)$. Well, the definition is much simpler than that of Galilean transformation.

Poincaré transformations form a group (why?) called the Poincaré group, which is a $\frac{\nu\left(\nu+1\right)}2$-dimensional Lie group (why?).

$$\begin{align*} \mathscr A&\coloneqq\mathbb R^\nu,\\ \tau&\coloneqq\left(ct,x\right)\mapsto t,\\ \sigma&\coloneqq\left(ct,x\right)\mapsto x,\\ \rho&\coloneqq\left(a,b\right)\mapsto \sqrt{\tau\!\left(a-b\right)^2-\sigma\!\left(a-b\right)^2} \end{align*}$$ is an Einsteinian structure (why?), where the constant $c\in\mathbb R$ is called the speed of light. Here $\mathscr A$ is called the Minkowski space.

The following transformations on the Minkowski space are Poincaré transformations (why?):

1. $a\mapsto a+d$, where $d\in\mathbb R^\nu$ (translation),
2. $a\mapsto Ga$, where $G\in\mathrm O\!\left(\iota,\chi\right)$ is an indefinite orthogonal matrix (rotation).

Every Poincaré transformation of the Minkowski space can be represented uniquely as the composition of a translation and a rotation (why?).

All Einsteinian spaces with the same dimensions are isomorphic to each other (why?).

The rest is just the same as what we have done with Galilean world. You can find that the Einsteinian world is also space-homogeneous, time-homogeneous, and space-isotropic. Further more, it is time-isotropic.

Aristotelian world (imagination)

Although the Aristotelian world is not real, we can think of what it may look like.

Principle 1. The world is an Aristotelian structure with $3$-dimensional space and $1$-dimensional time.

According to Aristotle’s theory about the natural place, the world has something like a “center”, so the world cannot be space-homogeneous. However, he admits the invariance of natural laws over time, so the world is still time-homogeneous. It may be also reasonable to assume that the world is space-isotropic.

Thus, our definition of the Aristotelian structure should be non-affine, and the Aristotelian transformations should be composed of rotation and time translation.

However, although the space is non-affine, the time is affine. This makes it tricky to mix space and time together into an “$\mathscr A$”. However, there is a workaround. We can define the universe still an affine space, while give it an origin. Since this origin only add limitations to space transformation instead of time transformation, we can make it an $\iota$-dimensional affine subspace instead of a single point.

Here is the definition of an Aristotelian structure. An Aristotelian structure with $\chi$-dimensional space and $\iota$-dimensional time is a $4$-tuple $\left(\mathscr A,\tau,o,\rho\right)$ with the following principles:

1. $\mathscr A$ is a $\nu$-dimensional affine space associated with the vector space $\mathbb R^\nu$, where $\nu\coloneqq\chi+\iota$;
2. $\tau:\mathbb R^\nu\rightarrow\mathbb R^\iota$ is a linear mapping;
3. $o$ is an $\iota$-dimensional affine subspace of $\mathscr A$;
4. For $a,b\in\mathscr A$ such that $\tau\!\left(a-b\right)=0$, the mapping $\rho$ satisfies $\rho\!\left(a,b\right)= \sqrt{\left(a-b\right)^2}$.

To make the physical meanings of the above mathematical stuff clear, we

• call $\mathscr A$ the universe or an Aristotelian space,
• call the points in the universe the events,
• call $\tau$ the time,
• call $o$ the center of space,
• say two events $a,b$ are simultaneous iff $\tau\!\left(a-b\right)=0$,
• call $\rho\!\left(a,b\right)$ the distance between simultaneous events $a,b$.

Then the task is to define the Aristotelian transformations.

Principle 2. Laws of nature remain the same under Aristotelian transformation.

Here is the definition of an Aristotelian transformation. An affine transformation $g$ over the Aristotelian space $\mathscr A$ is called an Aristotelian transformation iff all of the following are satisfied:

1. $\forall a,b\in\mathscr A: \tau\!\left(a-b\right)=\tau\!\left(ga-gb\right)$,
2. $\forall a,b\in\mathscr A: \tau\!\left(a-b\right)=0\Rightarrow \rho\!\left(a,b\right)=\rho\!\left(ga,gb\right)$.
3. $\forall a\in o:ga\in o$.

Notice the third condition, which makes it different from a Galilean transformation.

Aristotelian transformations from a group (why?) called the Aristotelian group, which is an $\left(\iota+\frac{\chi\left(\chi-1\right)}2\right)$-dimensional Lie group (why?).

$$\begin{align*} \mathscr A&\coloneqq\mathbb R^\nu,\\ \tau&\coloneqq\left(t,x\right)\mapsto t,\\ o&\coloneqq\mathbb R^\iota\times\left\{0\right\},\\ \rho&\coloneqq\left(a,b\right)\mapsto\sqrt{\left(a-b\right)^2} \end{align*}$$ is an Aristotelian structure (why?). Here $\mathscr A$ is called the Aristotelian coordinate space.

The following transformations on the Aristotelian coordinate space are Aristotelian transformations (why?):

1. $\left(t,x\right)\mapsto\left(t+s,x\right)$, where $s\in\mathbb R^\iota$ (time translation),
2. $\left(t,x\right)\mapsto\left(t,Gx\right)$, where $G\in\mathrm O\!\left(\chi\right)$ (rotation).

Every Aristotelian transformation of the Aristotelian coordinate space can be represented uniquely as the composition of a time translation and a rotation (why?).

All Aristotelian spaces with the same dimensions are isomorphic to each other (why?).

After building up the Aristotle world, how can we develop the mechanics here? Maybe it can be interesting.

Other imaginations

Here are some other imaginations of a world:

• What about a space-anistropic universe?
• What about defining the spacetime interval by multiplying space interval and time interval?
• What about a time-heterogeneous universe?
• What about making laws of nature unchanged under uniform acceleration?
• What about…

These can be materials for science fiction (novels or video games).