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In my Math World story, all numbers are of a single race, no matter the base, irrational or rational, complex or geometry, same race.

Sexual Reproduction is different in 3 ways actually, meiosis, hermaphrodites and determination.

Meiosis

Meiosis is not all that different in female numbers or in the female parts of hermaphroditic numbers. Males or male parts of hermaphroditic numbers is where the real difference is.

The first division is the same, division into 2 identical cells, 1 of which will become the sperm and another a spermatogonium(basically a sperm producing stem cell).

Following the spermatocyte we have asymmetric division. This is true in every case.

The divisions can happen in any order but first here are their chromosomes:

Sex Chromosomes: XXXY or XYYY(both viable and both male)

Operational Chromosomes: + - * / ^ nth root

Size Chromosome: Determines how big or small the number gets and is unrelated to the digits

Color Chromosome: Determines what color the number is

Circular chromosome: Other functions

First the spermatocyte divides into 1 sperm and another spermatocyte. This continues until the last division where 2 sperm are produced.

Then you end up with something like this(isn't the same every time, just showing a possibility):

Sperm from primary spermatocyte: Addition chromosome and nothing else

Sperm from secondary spermatocyte: Subtraction chromosome and nothing else

Sperm from tertiary spermatocyte: Multiplication chromosome and X chromosome

Sperm from quarternary spermatocyte: Division chromosome and X chromosome

Sperm from quinary spermatocyte:

First sperm: Exponentiation chromosome, Size chromosome, Color chromosome, Circular chromosome, and X chromosome

Second sperm: Nth root chromosome and Y chromosome

Hermaphrodites

There are actually 2 kinds of hermaphrodites but they all have these sex chromosomes:

XXYY

And they are all bifertile(both male and female parts are fertile) which requires a special form of male and female hormones that only affect what they are supposed to affect(male parts and female parts respectively) and not their opposite.

The difference between the 2 types of hermaphrodites is whether the gonads are separated or merged.

In the separated case, effort is required every time to impregnate either another number or itself.

In the merged case it is much more interesting. Here are all the possibilities in the merged case:

Sperm through male parts, egg through female parts: Effort required to impregnate

Sperm and egg both through female parts: Pregnancy without any effort whatsoever

Sperm and egg both through male parts: Temporary ectopic pregnancy that over time, via pressure of fluid is gotten rid of like how a stone is gotten rid of. Painful but not life threatening

Egg through male parts and sperm through female parts: Sperm flow out fast and the egg, just as before, is like a stone

Fertilization

This time of fertilization is where math is involved.

So sperm arrive and 6 or more sperm fertilize the egg(remember, the spermatogenesis is asymettric thus requiring multiple sperm to fertilize the egg). Any extra chromosomes are broken down via a series of enzymes into nucleotides the cell can use for energy or more importantly DNA replication. But the operation done, the math done to it is completely determined by the first sperm to fertilize the egg. Lets take 2 and 3 as an example:

3 is female and 2 is male. There are 6 different possibilities here. Every operation has the male number as the first number.

Here are the possibilities:

2+3 = 5

2-3 = -1

2*3 = 6

2/3 = $\frac{2}{3}$

2^3 = 8

2(nth root) 3 = sqrt(3)

Sex determination is much easier though. The egg has just 1 sex chromosome and the sperm are what supply the other 3. Of course 4 get in there because of the need for operational chromosomes and so this extra sex chromosome is part of what gets broken down into nucleotides.

XXXX = female

XXXY = male

XXYY = hermaphroditic

XYYY = male

Is this mathematical operations in DNA reasonable for a mathematical race? I mean to me this is the only way to get all possible numbers from a small population(301 numbers from 0 to 300, including 0 and 300) with gender restrictions(male always comes first in the operations) and without it being too random or too predictable.

Caters
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    What I'm having trouble understanding is what these numbers represent? You are asking whether it is feasible that a 8 mating with a 4 can produce a 12 or a 4 or a 32 but how can this be answered unless we understand biologically what a 4 is? You use cell divisions and chromosomes which suggests a biological organism, are these beings made of cells or are they abstract mathematical constructs? What does it mean for an organism to be a 17? Is it a big floating number 17? – Mike Nichols Dec 02 '16 at 17:11
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    What defines "reasonable?" I mean, the reality of DNA is literally millions of times more nuanced than what you are trying to create, so it's very reasonable to assume this system will not produce the extraordinary variety of life. However, to ask whether it is "reasonable" requires us to understand what grounds you are coming from? Is this an effort to have a self-sustaining colony? Is this something that might accidentally come about in a computer (a-la Core Wars)? Is this trying to prove Peano arithmetic? What metric should we use for "reasonable?" – Cort Ammon Dec 02 '16 at 19:21
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    I’m also confused. Your first sentence mentions “base”. Are $16$, $\rm0x10$, $020$, $100_4$ and $31_5$ five different numbers? Are $\rm0xf$ and $\rm0xF$ different? Is $e^{i\pi}$ a number or just $-1$’s nickname? In your first sentence (and only there) you mention “geometry”. How does this relate to your world? About “color”: are $\color{red}{42}$ and $\color{blue}{42}$ different numbers? Can $\color{red}6$ and $\color{blue}7$ mate and produce $\color{purple}{42}$ (purple)? You list “meiosis, hermaphrodites and determination”, but your headings are “Meiosis,” “Hermaphrodites” & “Fertilization.” – Peregrine Rook Dec 02 '16 at 19:22
  • Is YYYY illegal operation? why? I think a 4 bits DNA can churn out 16 combination so I presume the sequence isn't important too? Also after 1 or 2 generations since you included 0 then what happens to their offspring if one of the parent possessed the "/" operational chromosomes? – user6760 Dec 03 '16 at 06:04
  • @PeregrineRook Yes, those numbers are in my mind different because while they all have the same value, they are in different bases. Also the only time I use a lowercase x in a non-base ten number is in a base like base 30(really high base). For things like hexadecimal I just write 0F and yes to me 0f and 0F mean totally different things. 0F, typically hexadecimal but can occur in higher bases. 0f, minimum of base 42. Geometry relates to the story as a whole because $π$ came from geometry because of $τ$ trying to kill him. – Caters Dec 03 '16 at 16:01
  • And yes different colors means different numbers with or without same value. 2 numbers of different colors can mate to produce offspring with a mixture of those colors. And yes YYYY is illegal. No fertilized YYYY egg will develop into a number. And no the sequence isn't important, just the X:Y ratio. Since 0 is male in my case, if he fertilized an egg with the division chromosome first, the offspring is going to be 0(because 0/any number except 0 = 0) – Caters Dec 03 '16 at 16:05
  • Why are males as common as females and hermaphrodites combined? Why not males and females the same and hermaphrodites different? Why are both XXXY and XYYY male? – Brythan Dec 04 '16 at 03:47
  • I figured that XXXX = female and XYYY = male because those are obvious. But I also figured that if there is just 1 Y chromosome, it should be male and if the number is XXYY, it should be hermaphroditic. And according to these rules for assigning gender to the first generation of numbers(all based on factors) hermaphroditic is rare. In the first generation, the only numbers that are hermaphroditic are powers of male numbers. Primes alternate in gender(so 2 is male, 3 is female, etc.) So females are very common, males are less common, and hermaphrodites are rare in the first generation. – Caters Dec 04 '16 at 04:06
  • So chromosome ratio in the first generation has nothing to do with how common it is. And it would make sense that as population increases over time it evens out to a 2:1:1 ratio of male:female:hermaphroditic due to chromosome ratio, but that is over time, not in the first few generations. – Caters Dec 04 '16 at 04:08

2 Answers2

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Addition and Subtraction

Adding two sets of DNA together is relatively easy for cells along with destroying DNA, so for your mathematical race these are reasonable.

All other operations

So what does 2*3 really mean? Our brains can quickly process that it should be 6, but how does it get to that? By either doing 2 + 2 + 2 or 3 + 3. Now for fertilization it only has one copy for the two and one copy for the three, so in order for the math to work the cell would need to read the second number translate it to mean the number of times it needs to add the first. In order to do this the cell would essentially need an organic computer in it to perform the operation.

Organic Computers

Even though no species on our planet have organic computers in their cells does not mean you can't in yours. We are even trying to develop our own organic molecular computers. In your case, how it could work is that at the point of fertilization the chromosomes would be fed in as input to the organic computer (quite literally in this case) which would perform the operation based off of the operation chromosome and spit out the copies of the chromosomes as a result. The result would then trigger the various next steps of the fertilization process. You could even establish in the evolutionary history of the mathematical species that it used to only be able to add, subtract, multiply, and divide but it eventually evolved to have all the various operations.

Irrational numbers

These cannot be truly stored, they would eventually get rounded off. So I recommend not including them in your mathematical species.

Note that I was confused by the question so I may have totally misunderstood it...

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Anketam
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  • Many irrational numbers can be "stored" in the sense that they can be perfectly represented. "sqrt(2)" is a perfect representation of the square root of two. Trying to figure out its digits will not be possible, sure, but you don't need to know its digits to represent it. Only a countable subset of the irrationals can be represented this way, but it seems the OP may only need roots and powers, so that should be fine. – cbh Dec 02 '16 at 19:43
  • @cbh: Worldbuilding Stack Exchange supports MathJax — if you type $\sqrt 2$, it displays as $\sqrt 2$. Also note that math has a concept of algebraic numbers, which are a subset of the complex numbers, but which include some irrational numbers. They are numbers that are a root of a non-zero polynomial in one variable with integer coefficients; e.g., $\sqrt2$ is a root of $x^2-2=0$. However, transcendental numbers, like $\pi$ (which the OP wants) and $e$, are not included. – Peregrine Rook Dec 03 '16 at 21:50
  • Well I solved that "I want $π$ but it isn't algebraic" by having $π$ come from the geometry and trigonometry planet after confronting $τ$ about the equations and how they look nicer with $π$ than with $τ$ and having to escape with his circular house materials, circular house plan(the only circular house plan), and seeds of pi based plants. – Caters Dec 06 '16 at 20:48
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No, it's not. At least when you name it X and Y, which implies human-like functions.

In humans, X contains everything needed for female. Y contains what's needed to be male and to suppres female traits. Of course this is a simplification,but workable one. That's why XYY boys are rather normal, and tends to be taller and stronger, and XXX girls have bigger boobs. XXY boys tends to look girly because there is more of woman recipe to be suppressed and single Y can't do it that well.

Why one and three "recipes for masculinity" would give a pure male, but two allowed female traits, I have no idea. No matter what I try to think of, it makes no sense.

What you describe would need totally different mechanism, so naming it X and Y will be confusing.

Could this evolve naturally? Not feasible. Having sex makes mixing genes faster, so it's evolutionary beneficial for a species. But more than two? The more complicated scheme of reproduction, the bigger chance your specie will fail to reproduce even if conditions are right. Fertile hermaphrodite would get better results if would act like a male, spreading his genes without the cost of pregnancy. So woman fertility would probably die out in them, leaving them as just a bit weird version of male. Unless you have weird pressure, like trees - inability to decide with whom you reproduce, being both increases your chance. But for those trees this pressure mostly eliminated both males and females, leaving hermaphrodites.

Mołot
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  • I am not asking about whether having 3 genders is reasonable. I am asking about whether having mathematical operations done during fertilization is reasonable, in other words if something like 2+3 being done if the addition chromosome is the first operational chromosome to get into the egg and 3 is female and 2 is male is reasonable. – Caters Dec 02 '16 at 16:45
  • @Caters but this is closely related. If you can't expect more than two genders you can't expect any complicated gender calculation either. But evolution pressure is in the direction of simplest, energy-cheapest, error-resistant solution that gets the job done. Math calations during conception aren't it. I'll expand my answer in the evening. – Mołot Dec 02 '16 at 17:11
  • as a fyi in humans having three x chromosomes (trisomy X) does not result in having larger breasts. Here is a full list of the symptoms of Trisomy X. – Anketam Dec 02 '16 at 18:44