Definitions of 10 Number Sets (Number Rules)

nbrlz.txt

## 1: Complex Numbers - Definition
The complex numbers are the field C of numbers of the form x + i y, where x and y are real numbers and i is the imaginary unit equal to the square root of -1, sqrt(-1). When a single letter z = x + i y is used to denote a complex number, it is sometimes called an "affix." In component notation, z = x + i y can be written (x, y). The field of complex numbers includes the field of real numbers as a subfield.
The set of complex numbers is implemented in the Wolfram Language as Complexes. A number x can then be tested to see if it is complex using the command Element[x, Complexes], and expressions that are complex numbers have the Head of Complex.
Complex numbers are useful abstract quantities that can be used in calculations and result in physically meaningful solutions. However, recognition of this fact is one that took a long time for mathematicians to accept. For example, John Wallis wrote, "These Imaginary Quantities (as they are commonly called) arising from the Supposed Root of a Negative Square (when they happen) are reputed to imply that the Case proposed is Impossible".
Through the Euler formula, a complex number
z = x + i y
may be written in "phasor" form
z = left bracketing bar z right bracketing bar (cos θ + i sin θ) = left bracketing bar z right bracketing bar e^(i θ).


## 2: Upgraded Numbers - Definition
Upgraded number's are a new set of "relevant" numbers invented by @mcnole25, a.k.a. horizontal_shading in September 2026. His form of numbers consists of 12 integers. Information from another text file is below.
How to Use my Upgraded Numbers, {UAₙ} - A Useful Guide
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[#1: The Basic Logic Behind this Set]
With the 12-integer list machine form, this is a number.
[a, b, n, r, c, d, p, q, m, s, x, y] in Standard Form, =
(a/b * eʳ * √n + c/d) + (p/q * eˢ * √m + x/y)i

where a, r, c, p, s, x ∈ ℤ, where b, d, q, y ∈ ℕ⁺
where n, m ∈ ℕ⁺, such that (ℕ⁺ > 1)² ∤ any n, m
where i² = -1, or i = √-1, perpendicularly on the plane
Fractions & square root surds must be in simplest form.

You cannot do any advanced stuff, like cube roots, Pi,
nested square roots, natural logs, trig, or what else.
Every Upgraded number must exist in this strict form.
You shouldn't add rationals to rationals times 1,
this is just another rational number. (Needs simple form)


[#2: Dealing with Addition & Subtraction]
Any rational number can smoothly be added or subtracted.
2 + 0.25 = 2.25, 1/13 - 1/16 = 3/208, -1 + 5/7 = -2/7

With square roots & exponentials, it can only be added
if n, r are the same values in each exp. (ℚ = rationals)
Process of (ℚ eʳ √n + ℚ) + (ℚ eʳ √n + ℚ)i =
((ℚ±ℚ) eʳ √n + (ℚ±ℚ)) + ((ℚ±ℚ) eʳ √n + (ℚ±ℚ))i

Adding complex numbers with this too may make these
certain rules stricter. Add the real & imag parts each.

You're NOT allowed to add surds to other surds in reals.
You're NOT allowed to combine eʳ by adding square roots.

e² + √2 = BROKEN, WRONG. e² √2 = Multiplying, correct.
√2 + √3 = NESTED SQUARE ROOT. BAD. √2 * √3 = √6
√2 + (√3)i is fine, because axes are perpendicular.


[#3: Dealing with Multiplying & Dividing]
Using complex numbers with this too may make these
certain rules stricter. Recall (a+bi)² = a² - b² + 2abi.

Any rational number can smoothly be multiplied.
7 * 4 = 28, -2/3 * -4/5 = 8/15, 1/6 * 0 = 0, 1 * -3 = -3

Squaring square roots or E's by themselves? Yep. Good.
(e√2)² = 2e², (0.5√5 + 0.5)² = 0.5√5 + 1.5, (e⁵)² = e¹⁰
Be careful with adding rationals as multiplying will
almost surely break something. Try using just products.

Rotating 1 = 0° to i = 90° can only have certain angles.
(some rational)² + (sqrt reversible)² = 1 should work.

0.5√2 + (0.5√2)i = 45°, 0.5√3 + 0.5i = 30°
0.5 + (0.5√3)i = 60°, (√11 + (√5)i)/4 ≈ 33.9878°

The multiplying pattern is quite complicated to show,
using what you know, be delicate with multiplying stuff.


[#4: More Restrictions to Maintain a Closed Set]
There are other restrictions needed to keep things clear.

Do NOT treat things like a polynomial! It's LINEAR!!
e² + e is banned, e + 1 is not. e⁻¹ is OK, though. OK?
Careful when using 1/(...). Result must be upgraded too.

Obvious restrictions like dividing by 0, no sqrt of E...
Also following BODMAS or PEMDAS & regular rules...
No introducing potential breaks, like factorials & sine!
NO using fractional exponents, only make surds fractions.

[...] The size of the list is 12 INTEGERS, no more/less.
The only exception is "i" which is the 2nd base vector.
Real numbers are from the 1st base vector, easy enough.

The absolute value represents the DISTANCE from 0 in a
complex number, the formula can have corruption! Caution!
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## 3: Numbers - Definition
Whenever I talk about a number, I will be referring to an upgraded number & all its subsets aswell by default. Downgraded, complex-er numbers are only reserved for like, <1% of all reality cases to exist, really.


## 4: Algebraic Numbers - Definition
If r is a root of a nonzero polynomial equation
a_n x^n + a_(n - 1) x^(n - 1) + ... + a_1 x + a_0 = 0, where the a_i s are integers (or equivalently, rational numbers) and r satisfies no similar equation of degree <n, then r is said to be an algebraic number of degree n.
A number that is not algebraic is said to be transcendental. If r is an algebraic number and a_n = 1, then it is called an algebraic integer.
Any algebraic number is an algebraic period, and if a number is not an algebraic period, then it is a transcendental number. Note there is a "gap" between those two statements in the sense that algebraic periods may be algebraic or transcendental.
In general, algebraic numbers are complex, but they may also be real. An example of a complex algebraic number is i, and an example of a real algebraic number is sqrt(2), both of which are of degree 2.
The set of algebraic numbers is denoted A (Wolfram Language), or sometimes Q^_, and is implemented in the Wolfram Language as Algebraics.


## 5: Rational Numbers - Definition
A rational number is a number that can be expressed as a fraction p/q where p and q are integers and q!=0. A rational number p/q is said to have numerator p and denominator q. Numbers that are not rational are called irrational numbers. The real line consists of the union of the rational and irrational numbers. The set of rational numbers is of measure zero on the real line, so it is "small" compared to the irrationals and the continuum.
The set of all rational numbers is referred to as the "rationals, " and forms a field that is denoted Q. Here, the symbol Q derives from the German word Quotient, which can be translated as "ratio, " and first appeared in Bourbaki's Algèbre (reprinted as Bourbaki 1998, p. 671).
Any rational number is trivially also an algebraic number.
Examples of rational numbers include -7, 0, 1, 1/2, 22/7, 12345/67, and so on. Farey sequences provide a way of systematically enumerating all rational numbers.
The set of rational numbers is denoted Rationals in the Wolfram Language, and a number x can be tested to see if it is rational using the command Element[x, Rationals].
The elementary algebraic operations for combining rational numbers are exactly the same as for combining fractions.


## 6: Terminating Decimals - Definition
noun | a proper or improper fraction whose denominator is a power of 10


## 7: Integers - Definition
noun | any of the natural numbers (positive or negative) or zero


## 8: Natural Numbers - Definition
The term "natural number" refers either to a member of the set of positive integers 1, 2, 3, ... (OEIS A000027) or to the set of nonnegative integers 0, 1, 2, 3, ... (OEIS A001477; e.g., Bourbaki 1968, Halmos 1974). Regrettably, there seems to be no general agreement about whether to include 0 in the set of natural numbers. In fact, Ribenboim states "Let P be a set of natural numbers; whenever convenient, it may be assumed that 0 element P."
The set of natural numbers (whichever definition is adopted) is denoted N.
Due to lack of standard terminology, the following terms and notations are recommended in preference to "counting number, " "natural number, " and "whole number."
set | name | symbol
..., -2, -1, 0, 1, 2, ... | integers | Z
1, 2, 3, 4, ... | positive integers | Z^+
0, 1, 2, 3, 4, ... | nonnegative integers | Z^*
0, -1, -2, -3, -4, ... | nonpositive integers | 
-1, -2, -3, -4, ... | negative integers | Z^-


## 9: Prime Numbers - Definition
A prime number (or prime integer, often simply called a "prime" for short) is a positive integer p>1 that has no positive integer divisors other than 1 and p itself. More concisely, a prime number p is a positive integer having exactly one positive divisor other than 1, meaning it is a number that cannot be factored. For example, the only divisors of 13 are 1 and 13, making 13 a prime number, while the number 24 has divisors 1, 2, 3, 4, 6, 8, 12, and 24 (corresponding to the factorization 24 = 2^3·3), making 24 not a prime number. Positive integers other than 1 which are not prime are called composite numbers.
While the term "prime number" commonly refers to prime positive integers, other types of primes are also defined, such as the Gaussian primes.
{2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 | 53 | 59 | 61 | 67 | 71 | 73 | 79 | 83 | 89 | 97 | 101 | 103 | 107 | 109 | 113 | 127 | 131 | 137 | 139 | 149 | 151 | 157 | 163 | 167 | 173 | 179 | 181 | 191 | 193 | 197 | 199 | 211 | 223 | 227 | 229 | 233 | 239 | 241 | 251 ...... (∞ primes) ......}


## 10: Powers of 2 - Definition
Powers of 2 usually involve using an integer on some power of 2: example = 2^5 = 2*2*2*2*2 = 32

However, @mcnole25 suggests having "biggest" & "smallest" possible numbers, a sub-infinite & a sub-infinitesimal:

ό ~ INT LIMIT, 2 ↑↑ 5, or 2^2^2^2^2, or 2^65536 (2E19728)
Ƀ ~ Binary Fraction Piece, precision of exactly 2^-256

For any simplicity needed at the moment, any subset of numbers can be formed by this limitation, whether it'd be the complex numbers, upgraded numbers, real numbers, rational numbers, or even just integers.

2^-(2^8) | Ƀ
...
2^-6 | 0.015625
2^-5 | 0.03125
2^-4 | 0.0625
2^-3 | 0.125
2^-2 | 0.25
2^-1 | 0.5
2^0 | 1
2^1 | 2
2^2 | 4
2^3 | 8
2^4 | 16
2^5 | 32
2^6 | 64
2^7 | 128
2^8 | 256
2^9 | 512
2^10 | 1024
2^11 | 2048
2^12 | 4096
2^13 | 8192
2^14 | 16384
2^15 | 32768
2^16 | 65536
2^17 | 131072
2^18 | 262144
2^19 | 524288
2^20 | 1048576
2^21 | 2097152
2^22 | 4194304
2^23 | 8388608
2^24 | 16777216
2^25 | 33554432
2^26 | 67108864
2^27 | 134217728
2^28 | 268435456
2^29 | 536870912
2^30 | 1073741824
2^31 | 2147483648
2^32 | 4294967296
2^33 | 8589934592
2^34 | 17179869184
2^35 | 34359738368
2^36 | 68719476736
2^37 | 137438953472
2^38 | 274877906944
2^39 | 549755813888
2^40 | 1099511627776
2^41 | 2199023255552
2^42 | 4398046511104
2^43 | 8796093022208
2^44 | 17592186044416
2^45 | 35184372088832
2^46 | 70368744177664
2^47 | 140737488355328
2^48 | 281474976710656
2^49 | 562949953421312
2^50 | 1125899906842624
2^51 | 2251799813685248
2^52 | 4503599627370496
2^53 | 9007199254740992
2^54 | 18014398509481984
2^55 | 36028797018963968
2^56 | 72057594037927936
2^57 | 144115188075855872
2^58 | 288230376151711744
2^59 | 576460752303423488
2^60 | 1152921504606846976
2^61 | 2305843009213693952
2^62 | 4611686018427387904
2^63 | 9223372036854775808
2^64 | 18446744073709551616
...
2^2^16 | ό

Binary Fraction pieces can also help round any nonintegers to a simple 256 terminating decimal places, with roughly 75 digits of precision. 1 is just 2^256 * Ƀ.

But for fractions, & most upgraded numbers, rounding a VERY tiny amount is necessary. The closest piece to 1/3 is exactly 38597363079105398474523661669562635951089994888546854679819194669304376546645 * Ƀ, or in decimal value:

0.3333333333333333333333333333333333333333333333333333333333333333333333333333304546104816351851248712160457332001429613332118545728716587655099438027322791909806761641655092372578131509020179501121343349670177226844354489987409806417417712509632110595703125