Editing Maths

=Misc=
[https://www.quantamagazine.org/the-map-of-mathematics-20200213/ A Map of Maths (2020)]
[https://www.desmos.com desmos geometry]

[https://jeremykun.com/primers/ Read this] - About maths, vector spaces, fields, etc... and programming.

[https://github.com/JulianGaal/python-cheat-sheet/blob/master/NumPy.md NumPy cheatsheet]

[https://projecteuler.net/ Project Euler] - Math challenges.

[https://www.youtube.com/watch?v=Ezn3xCyyyIs A variation on an Ulam Spiral: a Sacks Spiral]

[http://www.naturalnumbers.org/sparticle.html More on Stacks Spiral]

[https://en.wikipedia.org/wiki/Fundamental_theorem Wikipedia: Fundamental theorem]

[https://en.wikipedia.org/wiki/Ford_circle Wikipedia: Ford circle]

    x_prime = var('x_prime', latex_name=r'x^\prime') # This is apparently in sage, does it work in Jupyter?

[https://www.physicsforums.com/insights/self-study-algebra-linear-algebra/ How to Self-study Algebra: Linear Algebra Reference]

[https://www.youtube.com/watch?v=S4Qg2CsiIj8 Solve non-linear equations with python]

=Parabola point=
* [https://www.youtube.com/watch?v=nEGl2JoJ38I Optimize a point to a parabola -- Juda math]
* [https://www.youtube.com/watch?v=QEF6RteZ4FU Optimization The Closest Point on the Graph]
* [https://www.youtube.com/watch?v=mamH094uw_U Optimization - Calculus (KristaKingMath)]

=Notation and Symbols=

==Alphanumeric==
* '''Superscript:''' 𝓍⁰¹²³⁴⁵⁶⁷⁸⁹ ⁺⁻⁼⁽⁾ ᵃᵇᶜᵈᵉᶠᵍʰⁱʲᵏˡᵐⁿᵒᵖʳˢᵗᵘᵛʷˣʸᶻ ᴬᴮᴰᴱᴳᴴᴵᴶᴷᴸᴹᴺᴼᴾᴿᵀᵁⱽᵂ 
* '''Subscript:''' 𝓍₀₁₂₃₄₆₇₈₉ ₊₋₌₍₎ ₐₑₒₓₔₕₖₗₘₙₚₛₜ ᵢᵣᵤᵥₓ ᵦᵧᵨᵩᵪ
* '''Blackboard Bold:''' 𝔸𝔹ℂ𝔻𝔼𝔽𝔾ℍ𝕀𝕁𝕂𝕃𝕄ℕ𝕆ℙℚℝ𝕊𝕋𝕌𝕍𝕎𝕏𝕐ℤ 𝕒𝕓𝕔𝕕𝕖𝕗𝕘𝕙𝕚𝕛𝕜𝕝𝕞𝕟𝕠𝕡𝕢𝕣𝕤𝕥𝕦𝕧𝕨𝕩𝕪𝕫 𝟘𝟙𝟚𝟛𝟜𝟝𝟞𝟟𝟠𝟡 ℾℽℿℼ⅀ ⅅⅆⅇⅈⅉ
* '''Script:''' 𝒜ℬ𝒞𝒟ℰℱ𝒢ℋℐ𝒥𝒦ℒℳ𝒩𝒪𝒫𝒬ℛ𝒮𝒯𝒰𝒱𝒲𝒳𝒴𝒵 𝒶 𝒷 𝒸 𝒹 ℯ 𝒻 ℊ 𝒽 𝒾 𝒿 𝓀 𝓁 𝓂 𝓃 𝓅 𝓆 𝓇 𝓈 𝓉 𝓊 𝓋 𝓌 𝓍 𝓎 𝓏
* '''Script (Bold):''' 𝓐𝓑𝓒𝓓𝓔𝓕𝓖𝓗𝓘𝓙𝓚𝓛𝓜𝓝𝓞𝓟𝓠𝓡𝓢𝓣𝓤𝓥𝓦𝓧𝓨𝓩𝓪𝓫𝓬𝓭𝓮𝓯𝓰𝓱𝓲𝓳𝓴𝓵𝓶𝓷𝓸𝓹𝓺𝓻𝓼𝓽𝓾𝓿𝔀𝔁𝔂𝔃 
* '''Fraktur:''' 𝔄𝔅ℭ𝔇𝔈𝔉𝔊ℌℑ𝔍𝔎𝔏𝔐𝔑𝔒𝔓𝔔ℜ𝔖𝔗𝔘𝔙𝔚𝔛𝔜ℨ𝔞𝔟𝔠𝔡𝔢𝔣𝔤𝔥𝔦𝔧𝔨𝔩𝔪𝔫𝔬𝔭𝔮𝔯𝔰𝔱𝔲𝔳𝔴𝔵𝔶𝔷
* '''Fraktur (Bold):''' 𝕬𝕭𝕮𝕯𝕰𝕱𝕲𝕳𝕴𝕵𝕶𝕷𝕸𝕹𝕺𝕻𝕼𝕽𝕾𝕿𝖀𝖁𝖂𝖃𝖄𝖅𝖆𝖇𝖈𝖉𝖊𝖋𝖌𝖍𝖎𝖏𝖐𝖑𝖒𝖓𝖔𝖕𝖖𝖗𝖘𝖙𝖚𝖛𝖜𝖝𝖞𝖟
* '''Serif (Bold):''' 𝐀𝐁𝐂𝐃𝐄𝐅𝐆𝐇𝐈𝐉𝐊𝐋𝐌𝐍𝐎𝐏𝐐𝐑𝐒𝐓𝐔𝐕𝐖𝐗𝐘𝐙𝐚𝐛𝐜𝐝𝐞𝐟𝐠𝐡𝐢𝐣𝐤𝐥𝐦𝐧𝐨𝐩𝐪𝐫𝐬𝐭𝐮𝐯𝐰𝐱𝐲𝐳
* '''Sans-serif:''' 𝖠𝖡𝖢𝖣𝖤𝖥𝖦𝖧𝖨𝖩𝖪𝖫𝖬𝖭𝖮𝖯𝖰𝖱𝖲𝖳𝖴𝖵𝖶𝖷𝖸𝖹𝖺𝖻𝖼𝖽𝖾𝖿𝗀𝗁𝗂𝗃𝗄𝗅𝗆𝗇𝗈𝗉𝗊𝗋𝗌𝗍𝗎𝗏𝗐𝗑𝗒𝗓
* '''Sans-Serif (Bold):''' 𝗔𝗕𝗖𝗗𝗘𝗙𝗚𝗛𝗜𝗝𝗞𝗟𝗠𝗡𝗢𝗣𝗤𝗥𝗦𝗧𝗨𝗩𝗪𝗫𝗬𝗭𝗮𝗯𝗰𝗱𝗲𝗳𝗴𝗵𝗶𝗷𝗸𝗹𝗺𝗻𝗼𝗽𝗾𝗿𝘀𝘁𝘂𝘃𝘄𝘅𝘆𝘇
* '''Mathematical Italic:''' 𝐴𝐵𝐶𝐷𝐸𝐹𝐺𝐻𝐼𝐽𝐾𝐿𝑀𝑁𝑂𝑃𝑄𝑅𝑆𝑇𝑈𝑉𝑊𝑋𝑌𝑍𝑎𝑏𝑐𝑑𝑒𝑓𝑔ℎ𝑖𝑗𝑘𝑙𝑚𝑛𝑜𝑝𝑞𝑟𝑠𝑡𝑢𝑣𝑤𝑥𝑦𝑧𝛩
* '''Mathematical Italic (bold):''' 𝑨𝑩𝑪𝑫𝑬𝑭𝑮𝑯𝑰𝑱𝑲𝑳𝑴𝑵𝑶𝑷𝑸𝑹𝑺𝑻𝑼𝑽𝑾𝑿𝒀𝒁𝒂𝒃𝒄𝒅𝒆𝒇𝒈𝒉𝒊𝒋𝒌𝒍𝒎𝒏𝒐𝒑𝒒𝒓𝒔𝒕𝒖𝒗𝒘𝒙𝒚𝒛
* ıȷ

* [https://en.wikipedia.org/wiki/Mathematical_Alphanumeric_Symbols Wikipedia: Mathematical Alphanumeric Symbols].
* [https://www.compart.com/en/unicode/block/U+1D400?page=3 more greek italics here]
* [https://en.wikipedia.org/wiki/Unicode_subscripts_and_superscripts wikipedia]

==Links==
* [http://homepages.math.uic.edu/~jyang06/stat401/handouts/handout2.pdf helpful pdf]
* [https://meta.wikimedia.org/wiki/Help:Displaying_a_formula#Derivatives MediaWiki: Displaying a formula]
* [https://www.mediawiki.org/wiki/Extension:Math MediaWiki: Extension:Math]
* [https://en.wikipedia.org/wiki/Mathematical_operators_and_symbols_in_Unicode Wikipedia: Mathematical operators and symbols in Unicode].
* [https://en.wikipedia.org/wiki/List_of_mathematical_symbols Wikipedia: List of mathematical symbols].
* [https://en.wikipedia.org/wiki/List_of_mathematical_symbols_by_subject Wikipedia: List of mathematical symbols by subject]
* [http://www.rapidtables.com/math/symbols/Basic_Math_Symbols.htm RapidTables - Mathematical Symbols].
* [http://xahlee.info/comp/unicode_math_font.html Unicode Math Font].
* [https://en.wikipedia.org/wiki/List_of_logic_symbols Wikipedia: List of logic symbols]
* [http://unicode.org/notes/tn28/UTN28-PlainTextMath.pdf unicode.org] - This is weird
* [http://www.unicode.org/reports/tr25/ unicode.org - UNICODE SUPPORT FOR MATHEMATICS]
* [https://www.tug.org/TUGboat/Articles/tb18-1/tb54becc.pdf Some ISO standard]
* [http://sites.psu.edu/symbolcodes/accents/math/mathchart/ mathchart]

===Latex===
* [http://www.sas.rochester.edu/psc/thestarlab/help/latextut2.pdf latex Tutorial PDF]
* [https://www.cl.cam.ac.uk/~mgk25/ucs/examples/TeX.txt TeX codes for various Unicode 3.2 characters]
* [http://matplotlib.org/users/mathtext.html Matplotlib: mathtext]

==Operations==
[https://en.wikipedia.org/wiki/Mathematical_operators_and_symbols_in_Unicode Wikipedia: Mathematical operators and symbols in Unicode]

[https://en.wikipedia.org/wiki/Supplemental_Mathematical_Operators Wikipedia: Supplemental Mathematical Operators]

* ⊕ - XOR. Direct Sum.
* ⊖ - Symmetric difference.
* ⊗ - [https://en.wikipedia.org/wiki/Tensor_product Tensor product]. 
* ⊘ - 
* ⊙ - Circled dot operator. ⨀ n-ary circled dot operator.
* ∗ - Astrisk Operator. Also ✱.
* ∘ - Function composition. Ring operator. (Also ∙ bullet operator)
* ⋈ - Natural join.
* ⋆ - Star Operator.
* ζ(s) - The Riemann zeta function.
* Arithmetic: − + ± × ⋅ · ÷ ⁄ √ ∛ ∜

<math>f(x) = x^2\,</math>

==Calculus==
* ∆ - ∆x change in x. Sometimes symmetric difference.
* ∇ - Gradient.
* ∂ - Partial Derivative.
* ∫ ∬ ∭ ∮ ∯ ∰ - Integrals (double+triple). ∮ ∯ ∰ - Contour, surface and volume integrals. ∱ ∲ ∳ - Clockwise integral, clowise contour integral. [http://www.fileformat.info/info/unicode/block/mathematical_operators/images.htm]

==Other==
* ≠ ≈ ≡ ∀ ∃ ∄ ∧ ∨ ¬ ≤ ≥ ≮ ≯ ⇔ ⇒ ∈ ∉ ∌ ∩ ∪ ⊃ ⊂ ⊇ ⊆ ⊢ ⋂ ⋃
* '''∈''' 'is an element of', '''∉''' 'is not an element of', '''∌''' does not contain as a member
* ◅ ▻ - Normal sub group of. ⟨ ⟩ generator.
* x′ - x prime.

* Misc: ∑ π φ ℵ ℶ ω ∞ ∴ ∵ ! ∎ → ↦ ㏒ * ∥ ∦ θ

* 𝓋⃗, v⃗ - Vector 𝓋 + (U+20D7).
* ∅ - Empty set
* δ - [https://en.wikipedia.org/wiki/Kronecker_delta Kronecker Delta]
* ∏ - Product over (Like ∑). ∐ for [https://en.wikipedia.org/wiki/Coproduct coproduct].
* '''⊤ / ⊥''' - True / False
* '''∖''' - Set minus.
* '''σ''' - Selection
* '''↔⇔≡''' - If and only if.
* '''⇒''' - If A is true then B must be true.
* '''ℜ(z)''', '''ℑ(z)''' - Return the real & imaginary parts of a complex number.
* {♠, ♦, ♥, ♣} 
* '''↪''' - [https://en.wiktionary.org/wiki/%E2%86%AA Embedding]... 𝒻: 𝐗↪𝐘. 𝒻 is the identity map, taking 𝓍∈𝐗 to 𝓍∈𝐘 and 𝐗⊂𝐘. Maybe 'lift'.
* '''↣''' - 𝒻: 𝐗↣𝐘 is an injective function from 𝐗 to 𝐘. '''Injection'''.
* '''↠''' - 𝒻: 𝐗↠𝐘, [https://en.wiktionary.org/wiki/↠ ↠ is a surjection] ([https://math.stackexchange.com/questions/46678/what-are-usual-notations-for-surjective-injective-and-bijective-functions possibly] only an 'onto' '''surjection'''. See also [https://en.wikipedia.org/wiki/Surjective_function#Definition this.] and [http://homepages.math.uic.edu/~jyang06/stat401/handouts/handout2.pdf pdf]
* '''≡''' - '''Congruence'''. 23 ≡ 45 (mod 11)

* |𝐴|×|𝐵|

* [https://en.wikipedia.org/wiki/Geometric_Shapes Wikipedia: Unicode Geometric Shapes]

==Elementary Algebra==
[http://tutorial.math.lamar.edu/ Goodish Overview]

=Number Systems & Groups=
* ℕ - The set of all '''natural numbers'''. No zero. No negatives. Subtraction and division aren't always possible. ℕ = {1, 2, 3, ...}.
* ℤ - The set of all '''integers'''. Subtraction is always possible. ℤ = {..., −3, −2, −1, 0, 1, 2, 3, ...}. ℤ is an infinite '''ring'''.
* ℤₘ - The set of integers modulo m. A '''finite ring'''.
* ℚ - The set of all '''rational numbers''' (Quotient). Fractions. Division is always possible. ℚ is a '''field''' and an infinite ring.
* ℝ - The set of all '''real numbers'''. Includes irrational and transcendental numbers. √(2), π, e, φ. ℝ is an extension of ℚ to a larger '''field'''. Also an infinite '''ring'''.
* ℝ⁺ - The set of all '''positive reals'''.
* ℂ - The set of all '''complex numbers'''. 𝒾²=−1. Is a '''field''' and an infinite '''ring'''. Is not ordered because there is no way to tell which complex number is > another.
* ℍ - The set of all '''Quaternions'''. Are '''noncommutative'''. 𝒾𝒿=𝓀, 𝒿𝒾=−𝓀. Not a field.
* 𝕆 - '''Octonions'''. An 8 dimension number system. '''Noncommutative''' and '''nonassociative'''. Not a field.
* 𝔽ₚ - Set of '''integers modulo a prime p'''. Addition and multiplication also defined modulo p. A '''field'''.

* 𝕂 - Field of real or complex numbers.
* 𝔽 - Finite field.
* ℚ(𝛾) - 𝛾 is the root of the polynomial 𝓍³-𝓍-1. When 𝛾³ appears it is replaced with 𝛾+1. See: 2.2 in Princeton companion of mathematics. AUTOMORPHISM
* ℙ - Primes.
* ℙⁿ - n-dimensional '''[https://en.wikipedia.org/wiki/Projective_space projective space]'''. ℝⁿ/scaling. If you can rescale one vector to another, they are the same.
* 𝔸ⁿ - '''[https://en.wikipedia.org/wiki/Affine_space Affine Space]'''. A n-dimensional real vector space without origin. ℝⁿ/translations. 2 Objects points and vectors. [https://www.youtube.com/watch?v=uAFzRRiLSWA ~3:00min in]
* ℤ/pℤ - The set of integers modulo some prime p. Or just ℤ/p for short.
* p-adic - Changes the measurement metric to one modulus a prime.
* ℓ-adic - 'el-adic'. - [https://www.youtube.com/watch?v=aj4FozCSg8g New Theories Reveal the Nature of Numbers] - [https://en.wikipedia.org/wiki/%C3%89tale_cohomology Étale cohomology]
* [https://en.wikipedia.org/wiki/Non-standard_positional_numeral_systems Wikipedia: Non-standard positional numeral systems]
* 𝒢ₙ, 𝒢(p,q) - The geometric algebra generated by the vector space of signature (p,q) is 𝒢(p,q). 𝒢ₙ refers to all of them.  𝒢(2,0) / 𝒢(3,0) is a 2D/3D Euclidean algebra. "Geometric Algebra for Physicists".
* [https://en.wikipedia.org/wiki/Golden_ratio_base phinary] in base-φ
* [https://en.wikipedia.org/wiki/Non-integer_representation Non-integer representation]

=Axioms=
[https://en.wikipedia.org/wiki/List_of_axioms List of axioms]

==Peano Arithmetic==
The axioms that define the natural numbers. - Good Math pg5.

* Initial Value Rule - There is one object called 0 and 0 is a natural number.
* Successor rule: For every natural number n there is exactly one other natural number called its sucessor, s(n).
* Uniqueness Rule: No two natural numbers have the same sucessor.
*Equality Rules: Numbers can be compared for equality.
** Equality is reflexive: - Every number is equal to itself
** Equality is symmetric: a=b then b=a
** Equality is transitive: if a=b and b=c then a=c
* Induction rule: For P, P is true for all natural numbers if.
** 1. P is true about 0. P(0)=true.
** 2. If you assume P is true for a natural number n(P(n) is true). Then you can prove that P is true for the sucessor s(n) of n, P(s(n) == true.

===Addition===
* Commutative: n+m = m+n
* Identity: n+0 = 0+n=n
* Recursion: m+s(n) = s(m + n)

==Naive Set Theory==
∀𝓍(𝓍 ∈ 𝓎 → φ(𝓍))

[https://stanford.library.sydney.edu.au/archives/win2011/entries/settheory-alternative/#NaiSetThe Alternative Axiomatic Set Theories]
[http://scienceblogs.com/goodmath/2007/05/20/the-axioms-of-set-theory-1/ The Axioms of Set Theory]

==ZFC Axioms==
[https://arxiv.org/pdf/1603.08916.pdf The new axiom of set theory and Bell inequality.]

There are a few differnt symbolisms of it. Jech and Kunen seem to be popular.

[https://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_set_theory Wikipedia: Zermelo–Fraenkel set theory]
[http://mathworld.wolfram.com/Zermelo-FraenkelAxioms.html Wolfram MathWorld: Zermelo-Fraenkel Axioms

===Axiom of empty set===
Seems to be missing from many versions of ZFC since it can apparently be derived from the axiom of infinity.
[https://en.wikipedia.org/wiki/Axiom_of_empty_set Wikipedia: https://en.wikipedia.org/wiki/Axiom_of_empty_set].

* ∃𝓍∀𝓎¬(𝓎 ∈ 𝓍) - There is a set that no set is a member of it.

* ∃∅∀𝓍(𝓍∉∅)
* ∃∅ : ∀𝓍¬∈ ∅

===1. Axiom of extensionality===
∀𝓍∀𝓎[∀𝓏(𝓏 ∈ 𝓍 ⇔ 𝓏 ∈ 𝓎) ⇒ 𝓍 = 𝓎 - Sets are equal if they share the same elements.

=Algebraic Structures=
==Properties==

* '''associative''' - 𝒶(𝒷𝒸) = (𝒶𝒷)𝒸. '''∀𝓍,𝓎,𝓏. 𝓍∘(𝓎∘𝓏)=(𝓍∘𝓎)∘𝓏'''
* '''commutative''' - 𝒶𝒷 = 𝒶𝒷.  '''∀𝓍,𝓎. 𝓍∘𝓎=𝓎∘𝓍'''
* '''Identity element''': There exists an element ℯ such that for each element 𝓍, ℯ∘𝓍 = 𝓍 = 𝓍∘ℯ ;   formally: '''∃ℯ ∀𝓍. ℯ∘𝓍=𝓍=𝓍∘ℯ'''. The identity element of multiplication is 1 as 1×𝓍=𝓍=𝓍×1. The identity of addition is 0 as 0+𝓍=𝓍=𝓍+0.
* '''Inverse element''': It can easily be seen that the identity element is unique. If this unique identity element is denoted by ℯ then for each 𝓍, there exists an element 𝒾 such that 𝓍∘𝒾=ℯ=𝒾∘𝓍;   formally: '''∀𝓍 ∃𝒾. 𝓍∘𝒾=ℯ=𝒾∘𝓍'''. The multiplicative inverse: '''𝒶𝒶⁻¹=1=𝒶⁻¹𝒶'''


* '''domain''' - The set of elements on which a function has a valid definition is it's domain. For '''𝒻: X→Y''', X is the domain.
* '''codomain / target''' - The possible set that a function outputs. For '''𝒻: X→Y''', X is the target. There may be things in the target that aren't actually reachable by the function.
* '''image / range''' - The set of elements that a function maps to and no more. This is like the '''target''' but only the possible elements are in the set. For example the '''cos''' function could be defined as having a target of the real numbers, but the range could only be between -1 and 1. It is not always possible to define the range. Concepts of Modern Mathematics (pg. ~67). Some ambiguity as sometimes 'range' refers to the codomain.


* '''homomorphism''' - A category of function. Preserves the structure. '''𝒻: X→Y'''. '''𝒻(a)𝒻(b)=𝒻(c)''' and '''𝒶𝒷=𝒸'''. A homomorphism of vector spaces is referred to as a '''linear map'''.
* '''injection''' - Each target element must be reachable from only one element in the domain. It does not need to be '''surjective''', meaning that it's possible to have elements in the target that aren't reachable at all. But you can not have an element in the domain that is reachable from more than one target element. cos is not an injection because an infinite number of inputs get mapped to values between -1 and 1.
* '''surjection (onto)''' - If every element of a target set T is reachable by a function, that function is a surjection '''onto''' T. This means the target is also the range. Does not need to be an injection. - Concepts of Modern Mathematics (pg. 70).
* '''bijection''' - A bijection is invertible. It is an '''injection''' and a '''surjection'''. A function which relates each member of a set S (the domain) to a separate and distinct member of another set T (the range), where each member in T also has a corresponding member in S. A mapping that is both one-to-one (an injection) and onto (a surjection).
* '''isomorphism'''  - A '''homomorphism''' that is also a '''bijection'''. An operation-preserving bijection. A homomorphism with an inverse that is also a homomorphism. '''𝒻: X→Y, ℊ: Y→X'''
* '''automorphism''' - Galios Groups related. An isomorphism from a structure to itself.
* '''Abelion''' - An abelian group/operation is '''commutative'''.

==Structures==
==='''Set'''===
* A collection of elements.

==='''Group'''===
An algebraic structure that is a set with a single binary operation.

An '''infinite group''' has an unlimited number of elements. (ℤ, +)

A '''finite groups''' has a finite number of elements. (ℤₘ, +) - The integers mod m.

The order of an element is the number of elements in the subgroup it generates. |2| = 3 

* '''(𝔾, ∗)''' - Group '''𝔾''' with operation '''∗'''.

* Axioms
** The operation must be '''closed'''. - '''a,b ∈ 𝔾 → a∗b ∈ 𝔾'''
** The operation must be '''associative'''. - '''(a∗b)∗c = a∗(b∗c)'''
** Operation must have an '''identity''' element that has no effect on any other element under operation. '''∃e(a∗e = e∗a = a)'''.
** Every element must have an '''inverse'''. An element when combined with the original will return the identity. '''∀a∃a⁻¹(a∗a⁻¹=a⁻¹∗a=e)'''

* '''(ℤ, +)''' - The integers under addition.
** Closure: '''a,b ∈ ℤ → a+b ∈ ℤ'''
** Associativity: '''(a+b)+c = a+(b+c)'''
** Identity: '''a+0 = 0+a = a'''
** Inverse: '''a⁻¹ = -a'''

* '''(ℤ, ×)''' - The integers under multiplication it '''not a group'''
** Closure: '''a,b ∈ ℤ → a×b ∈ ℤ'''
** Associativity: '''(a×b)×c = a×(b×c)'''
** Identity: '''a×1 = 1×a = a'''
** '''NO''' Inverse: '''2x=1 has no solution in ℤ'''

* If the operation is also '''commutative''' then it is an '''Abelian''' group.
* [https://www.youtube.com/watch?v=WwndchnEDS4 Bill Shillito].

==='''Ring'''===
Similar to a field but multiplication doesn't require an inverse.

'''ℤ, ℚ, ℝ, ℂ, ℤₘ''' are all '''unital commutative rings.'''

'''𝕄₂(ℝ)''' the set of all '''2×2 real matrices''' is a non-commutative '''ring''' as matrix multiplication is not commutative.

* '''+''' must be '''abelian'''.
* '''×''' must be '''closed''' and '''associative'''.
* [https://www.youtube.com/watch?v=syHBApgJhsA Bill Shillito - Rings and Fields]

* '''(R, +, ∗)'''
** '''(R, +)''' is an '''abelian group''' - '''addition'''
*** '''Closed'''
*** '''Associative'''
*** Identity - '''Additive identity'''
*** Inverse - '''Additive inverse −a'''
*** '''Commutative'''
** '''(R, ∗)'''
*** Closed under '''×'''
*** Operation '''×''' must be '''associative'''.
** '''+''' and '''×''' must be linked by the '''distributive property'''. Multiplication distributes over addition.
*** a×(b+c) = a×b+a×c
*** (a+b)×c = a×c+b×c
**** 0×a = a×0 = 0 - Zero times anything is zero
**** a×−b = −a×b = −(a×b) - A positive times a negative is a negative.
**** −a×−b = a×b - A negative times a negative is a positive.

* '''×''' '''might''' not be '''commutative'''.  If it is then it is a '''commutative ring''' (not called abelian which is only for groups).
* If the ring has a '''multiplicative identity''' then it is a '''unital ring'''. For '''ℤ, ℚ, ℝ, ℂ, ℤₘ''' the '''multiplicative identity''' is 1. For '''𝕄ₘ(ℝ)''' it is the '''identity matrix'''.
* An element of a unital ring that has a multiplicative inverse is called a '''unit'''. Not every element is necessarily a unit. In ℤ only 1 and −1.
* If '''every element other than 0 is a unit'''. It is a '''division ring'''. ℚ is a division ring.
* A '''zero divisor''' of a ring is a nonzero element that can be multiplied by some other nonzero element to produce 0. In ℤ₆, 2, 3 and 4 are zero divisors. 2×3=0
* An element '''can not''' be a '''unit''' and a '''zero divisor'''.
* An '''integral domain''' is a '''commutative, unital ring''' with '''no zero divisors'''. '''ℤ, ℚ, ℝ, ℂ''' are '''integral domains'''.
** ℤₘ is not necessarily an integral domain. '''ℤₚ''' is an integral domain '''if p is a prime number'''.
** In an integral domain you can '''cancel factors''' from both sides of an equation. '''<strike>2</strike>×x=<strike>2</strike>×3'''

====Polynomial Ring====
A '''monomial''' is the product of a number and a non-negative integer power of a variable. eg: 2x³

A '''polynomial''' is a finite sum of monomials. a₀+a₁x+a₂x²+a₃x³+…+aₙxⁿ

n is the degree of the polynomial. It's highest power.

Some polynomial rings: ℤ[𝓍], ℚ[𝓍], ℝ[𝓍], ℂ[𝓍], ℤₘ[𝓍]

In '''ℝ[x]''' we can take congruents to the modulus '''𝓍²+1'''. Two polynomials are congruent if their difference is divisible by '''𝓍²+1'''. This makes the polynomial ring behave like complex numbers. When using a ''''prime' polynomial''' as modulus, the '''polynomial ring''' is a '''field'''.

'''𝓍³+𝓍²-2𝓍+3 ≡ -3𝓍+2   (mod 𝓍²+1)'''

Mentioned in a concrete guide to modern mathematics, page 90.

==='''Field'''===
Like a ring with stricter multiplication axioms.

A '''field''' is a division ring where '''×''' is '''commutative, unital''' and has '''no zero divisors''' and '''every nonzero element is a unit'''.

A field forms an abelian group under both addition and multiplication.

* A set with 2 binary operations (+, ×)
* Both must be '''commutative''' and '''associative'''
* Operations must have '''identity elements''' ('''+''' is '''0''' and '''×''' is '''1''')
* Every element must have an '''inverse''' (x = -x, x = 1/x)
* Must follow the '''distributive law'''.
* Must be '''closed under addition, multiplication, taking of inverses''' (Results must in in the same set).

'''ℚ, ℝ, ℂ''' are '''fields'''.

'''ℤₚ''' is a field '''if p is prime'''.

'''ℤ''' is '''not a field'''.

'''ℝ''' is the '''completion''' of ℚ. It allows for '''calculus'''.

'''ℂ''' is '''algebraically closed'''. Every polynomial equation in ℂ[x] has solutions in ℂ.

But '''ℂ''' is '''not ordered'''.

==='''Vector Space'''===
* Like a 2D or 3D plane. Can be built from 'unit vectors' or '[https://www.youtube.com/watch?v=P2LTAUO1TdA basis]'. for example (1,0), (0,1).
* There was a Youtube on the topic (Blue2Brown1).
* '''Scalar''' - A number used to multiply a vector. Infinite dimensional vector spaces exist such as when vectors are functions.
* ℝⁿ - A vector space of n-dimensions over the field ℝ. ℝ is the scalar type. ℝ², ℝ³, etc...

==='''Common Groups'''===
* '''Aₙ - Alternating Group''' - The group of even permutations of a finite set. A₄ is symmetries of a Tetrahedron. - "Fearless Symmetry"
* '''Cₙ - Cyclic Group''' - 
* '''Dₙ - Dihedral Groups''' - A '''dihedral group''' is the group of '''symmetries''' of a regular polygon, which includes '''rotations and reflections'''. '''D₃''' is the '''dihedral group''' of the '''triangle'''. '''D₁≅Z₂''', '''D₂≅K₄'''.
* '''E⁺(n)''' -  The '''direct isometries''', i.e., isometries '''preserving orientation''', also called '''rigid motions'''; they are the moves of a rigid body in n-dimensional space. These include the translations, and the rotations, which together generate '''E⁺(n)'''. Also called a '''special Euclidean group''', and denoted '''SE(n)'''.
* '''T - Translation Group''' - T is a normal subgroup of E(n). T ◅ E(n).
* '''Sₙ - Symmetry Groups''' - '''S₃''' the '''symmetric group''' of 3 elements. '''S₃''' is '''isomorphic''' to '''D₃'''. ('''S₃ ≅ D₃''')
* '''O(n) - Orthogonal group''' - Reflections & rotations. The group of distance-preserving transformations of a Euclidean space of dimension n that preserve a fixed point, where the group operation is given by composing transformations ('''∘'''). It is equivalent to the group of n×n orthogonal matrices, where the group operation is given by matrix multiplication; an orthogonal matrix is a real matrix whose inverse equals its transpose. O(n, ℝ) is a subgroup of E(n), contains those that leave the origin fixed.
* '''SO(n) - Special Orthogonal''' - Rotation group. '''SO(2)''' is the [https://en.wikipedia.org/wiki/Circle_group circle group]. '''SO(3)''' is the group of all rotations about the origin of three-dimensional Euclidean space '''ℝ³''' under the operation of composition ('''∘'''). eg '''Roll, Pitch, Yaw'''.   '''Non-commutative'''. '''SO(n) ≤ O(n)'''.
* '''POₙ(ℝ)''' - '''Projective orthogonal group'''. '''PSO(V)''' for '''projective special orthogonal group'''.
* '''GLₙ(𝔽) - General Linear''' - '''Linear''' means '''matrix'''. The group of all n×n matrices with non-zero determinants under matrix multiplication. '''nonabelian''' group because matrix operations are non-commutative. The set of n×n invertible matrices, together with the operation of ordinary matrix multiplication. The group '''GLₙ(ℝ)''' over the field of real numbers is a real '''Lie group''' of dimension '''n²''' as the set of all n×n real matrices, '''Mₙ(ℝ)''', forms a real vector space of dimension '''n²'''.
* '''Special Linear: SL(n)''' - Is the subgroup of '''GL(n, 𝔽)''' consisting of '''matrices with a determinant of 1'''. These form a group because the product of two matrices with determinant 1 again has determinant 1.
* '''ΓLₙ(ℝ)''' - General semilinear group. Contains '''GL'''.
* '''Sp(n)''' - The '''compact symplectic''' group '''Sp(n)''' is often written as '''USp(2n)''', indicating the fact that it is '''isomorphic''' to the '''group of unitary symplectic matrices'''.
* '''Sp(2n, 𝔽)''' - The '''symplectic group''' of degree '''2n''' over a field '''𝔽''' is the group of '''2n × 2n symplectic matrices''' with entries in '''𝔽''', and with the group operation that of matrix multiplication. Since all symplectic matrices have determinant 1, the symplectic group is a subgroup of the special linear group '''SL(2n, 𝔽)'''.
* '''Uₙ(ℝ) - The unitary group''' - A group preserving a sesquilinear form on a module. There is a subgroup, the special unitary group '''SUₙ(ℝ)'''.
* <math>\Sigma_A</math> - The permutation group of set A.
* ℚᵃˡᵍ - The solutions of polynomials. Is a '''field'''. Its Elements are called "'''algebraic numbers'''". It is a subset of ℂ. - Fearless Symmetry p48


* [https://en.wikipedia.org/wiki/Classical_group Wikipedia: Classical group]
* [https://en.wikipedia.org/wiki/List_of_finite_simple_groups Wikipedia: List of finite simple groups]
* [https://en.wikipedia.org/wiki/List_of_simple_Lie_groups Wikipedia: List of simple Lie groups]

==='''Common Lie Algebras'''===
* '''𝔰𝔬(3)''' - The '''Lie algebra''' of '''SO(n)''' and would be the '''rate of change''' for Roll, Pitch and Yaw. '''𝔰𝔬(n)''' is equal to '''𝔬(n)''' - Princeton Mathematics (p234)
* '''𝔤𝔩ₙ(ℂ)''' is the '''Lie algebra''' for '''GLₙ(ℂ)''', the space of all '''n×n complex matrices'''.
* '''𝔰𝔩ₙ(ℂ)''' - The '''Lie algebra''' of the special linear group '''SLₙ(n)'''. "Is the subspace of all matrices with '''trace zero'''" - Princeton Mathematics (p234)

==='''Cayley table'''===
A table showing all the results of all possible operations on a finite group.

==='''Eignvectors'''===
* '''Eigenvector''' - A vector in a vector space doesn't get rotated by a linear transformation. It stays on it's 'span'. It's the axis of rotation.
* '''Eigenvalue''' - The value an Eigenvector is scaled by. 𝛢𝓋⃗=λ𝓋⃗. 𝛢 is transformation matrix. 𝓋⃗ is an Eigenvector. λ is the eigenvalue.


* [https://www.youtube.com/watch?v=8F0gdO643Tc Khutoryansky]
* [https://www.youtube.com/watch?v=ue3yoeZvt8E LeiosOS]
* [https://www.youtube.com/watch?v=PFDu9oVAE-g 3Blue1Brown]
* [https://www.youtube.com/watch?v=RW-Seu-yenQ MathTheBeautiful: Linear Algebra 15n: Why Eigenvalues and Eigenvectors Are So Important!]
* [https://www.youtube.com/watch?v=bOreOaAjDno Eigenvalues in under 6 minutes]- meh

=Lie Stuff=
* Like calculus for groups?

Lie Group - A finite, continuous group. - Symmetry and the monster, pg62.

=Analysis=
==Calculus==
* Derivatives - Amount of change.
* Integral - Area under a function.
* Partial derivative - For example, a 2D slice of a 3D surface.
* [https://en.wikipedia.org/wiki/Notation_for_differentiation Wikipedia: Notation for differentiation]

===Gradients===
* [https://www.youtube.com/watch?v=GkB4vW16QHI Gradients and Partial Derivatives]
* [https://www.youtube.com/watch?v=U7HQ_G_N6vo Gradient 1 | Partial derivatives, gradient, divergence, curl | Multivariable Calculus | Khan Academy]

=Manifolds=
* ℝⁿ - Real coordinate space (Normally the same as Eⁿ).
* Eⁿ - Euclidean Space.
* E¹ - The line.
* E² - The Euclidean plane.
* E³ - Three-dimensional Euclidean space.
* S¹ - The circle.
* S² - The Sphere.
* T² - The torus. Either a 'flat torus' or a 'donut surface' (wrap around, objects going off one side reappear on the other). T²#T² is a two holed torus. - The Shape of Space pg.96
* K² - The Klein bottle. P²#P²=K². Like a flat torus except one edge mirrors anything crossing it.
* P² - The projective plane (like a disc with opposite ends glued).
* D² - The disk.
* T³ - The three-torus.
* D³ - A solid ball.
* P³ - Projective three-space.
* I - The interval (A line segment with both end-points included.
* S¹ ⨯ I is a cylinder
* S¹ ⨯ S¹ is a torus

=Misc=
* Axiom of choice - Given a non-empty collection of non-empty sets you can form a new set by picking one element from each set. - Elementary Topology and it's applications pg20.

=Math Computer Programming=
* [http://primesieve.org/ primesieve] - Generates primes...

 >>> from primesieve.numpy import *
 
 # Generate a numpy array with the primes below 100
 >>> generate_primes_array(100)

=Videos=
==MOOCs==
* [http://maths.org.au/index.php/2013/105-events/education-events-2006/316-ice-emamsi-summer-school-2006 Algebraic Curves, Computability Theory, Measure Theory] - via Mathologger
* [https://www.youtube.com/playlist?list=PLZzHxk_TPOStgPtqRZ6KzmkUQBQ8TSWVX Introduction to Higher Mathematics] - Bill Shillito
* [https://www.coursera.org/learn/mathematical-thinking/home/welcome Coursera: Introduction to Mathematical Thinking] - Stanford
* [https://ocw.mit.edu/courses/find-by-topic/#cat=mathematics MIT OpenCourseware Maths]
* [https://ocw.mit.edu/courses/mathematics/18-06sc-linear-algebra-fall-2011/least-squares-determinants-and-eigenvalues/eigenvalues-and-eigenvectors/ MIT Linear Algebra]
* [https://ocw.mit.edu/courses/mathematics/18-03sc-differential-equations-fall-2011/ MIT Differential Equations]
* [http://www.extension.harvard.edu/open-learning-initiative/abstract-algebra Abstract Video Stuff]
* [https://courses.edx.org/courses/course-v1:UTAustinX+UT.5.05x+1T2017/info Linear Algebra - Foundations to Frontiers]
* [https://courses.edx.org/courses/DavidsonX/D003x.1/1T2015/courseware/658085dbe8d24fd3a7c334d77a76dfc0/b6f960db8b5746b3b330cbcf584b1f4e/ D003x.1 Applications of Linear Algebra (Part 1)]

* [https://www.youtube.com/user/itssoblatant/videos It's so blatant] - Chaos, Dimension, etc...]

===Geometry===
* [https://courses.edx.org/courses/course-v1:SchoolYourself+GeometryX+2T2016/info Introduction to Geometry] - SchoolYourself
* [https://www.youtube.com/watch?v=kClSutQlOFs Projective Geometry]
* [https://www.youtube.com/user/richardsouthwell/videos And the rest of these...]
* [https://www.youtube.com/watch?v=aSz5BjExs9o TEDxBoulder - Thad Roberts - Visualizing Eleven Dimensions]
* [https://www.youtube.com/channel/UC4zzTEL5tuIgGMvzjk1Ozbg Visualizing Mathematics with 3D Printing]

==YouTube==
* [https://www.khanacademy.org/math/math-for-fun-and-glory/vi-hart More vi-hart on khan academy]
* [https://www.youtube.com/playlist?list=PLi01XoE8jYoi3SgnnGorR_XOW3IcK-TP6 Socratica] - Abstract Algebra
* [https://www.youtube.com/channel/UCYO_jab_esuFRV4b17AJtAw/videos These Videos] - Good explanations of advanced concepts.
* [https://www.youtube.com/channel/UCd0dc7kQA1FUpJ76o1EjLqQ LeiosOS]
* [https://www.youtube.com/user/Vihart/videos Vi Hart]
* [https://www.youtube.com/channel/UC1_uAIS3r8Vu6JjXWvastJg Mathologer]
* [https://www.youtube.com/channel/UCYO_jab_esuFRV4b17AJtAw 3Blue1Brown]
* [https://www.youtube.com/channel/UCr22xikWUK2yUW4YxOKXclQ MathTheBeautiful]
* [https://www.youtube.com/channel/UCFsZ2CadKpAt_yInoTcVRnQ Higher Mathematics]
* [https://www.youtube.com/channel/UCkTDtyvWkyH0yfc4e1cgM6g mathisasport]
* [https://www.youtube.com/user/LadislauFernandes/videos Group Theory] [https://www.youtube.com/watch?v=c2DKoAsBAAw GT3]
* [https://www.youtube.com/watch?v=Qw5jonrLbPU Is this any good?]
* [https://www.youtube.com/watch?v=kpeP3ioiHcw Particle Physics stuff] [http://inside.mines.edu/~aflourno/Particle/423.shtml Notes] [https://www.youtube.com/channel/UCHAwDVSS8oDLLln07cNdU6A/videos?sort=dd&view=0&shelf_id=0 List] [https://www.youtube.com/watch?v=f_0JDilvvME ep1]
* [https://www.youtube.com/channel/UConVfxXodg78Tzh5nNu85Ew Complex Numbers]
* [https://www.youtube.com/watch?v=A2EdR_aA3mw&index=6&list=PLCTMeyjMKRkqPXevy84y6J7Zac8S-n5i6 Perspective Geometry]


* [https://www.youtube.com/watch?v=s65DSz78jW4 Fractals - Hunting The Hidden Dimension]
* [https://www.youtube.com/watch?v=h9MS9i-CdfY Hidden Dimensions: Exploring Hyperspace]
* [https://www.youtube.com/watch?v=JLQtSyPq3Q4 The Infinity - Science Documentary]
* [https://www.youtube.com/watch?v=KDCJZ81PwVM Infinity: The Science of Endless]
* [https://www.youtube.com/watch?v=BCUmeE8sIVo NOTHING: The Science of Emptiness]
* [https://www.youtube.com/watch?v=KI5wr6KkG7c MICROSCOPIC UNIVERSE | Quantum mechanics behind Simulation Hypothesis]

* [https://www.youtube.com/user/MathsSmart/videos MathsSmart] - Did a thing on polygonal numbers.
* [https://www.youtube.com/user/njwildberger/videos Bunch of videos] - Universal Hyper Geometry

=Unsorted=
[https://www.quantstart.com/articles/How-to-Learn-Advanced-Mathematics-Without-Heading-to-University-Part-1 HOW TO LEARN ADVANCED MATHEMATICS WITHOUT HEADING TO UNIVERSITY - PART 1] [https://www.quantstart.com/articles/How-to-Learn-Advanced-Mathematics-Without-Heading-to-University-Part-3 PART 3]

[http://4chan-science.wikia.com/wiki/Math_Textbook_Recommendations 4chan - Math Textbook Recommendations]

[http://www.extension.harvard.edu/open-learning-initiative/abstract-algebra Harvard Course of Abstract Algebra] (apparently goes well with the Artin book)

[http://math.stackexchange.com/questions/tagged/book-recommendation?sort=votes&pageSize=15 List of books]


[https://www.csee.umbc.edu/portal/help/theory/group_def.shtml Order of abstract algebra]


[http://betterexplained.com/articles/linear-algebra-guide/ This guide on algebra]

[http://betterexplained.com/articles/a-visual-intuitive-guide-to-imaginary-numbers/ This guide to imaginary numbers]

[http://betterexplained.com/articles/developing-your-intuition-for-math/ Math intuition]

[http://betterexplained.com/articles/category/popular/ These guides]


[http://www.math.uconn.edu/~kconrad/blurbs/ These notes are recommended] 

<strike>[http://online.stanford.edu/course/how-to-learn-math-for-students-s14 HOW TO LEARN MATH: FOR STUDENTS]</strike>

=Books=
==Reading==

* Fearless Symmetry - Good book. Got hard around ch9 ish.

* The Shape of Space - Fairly Easy read and actual content.

* [[What is Mathematics?]] - Seems like a good overall math introduction. Seems to have non-stupid exercises.

* The Princeton Companion to Mathematics - Covers basically everything.

* Good Math - Seems like a basic approachable introduction.

* [[The Foundations of Mathematics]] by Ian Stuart. - It's mostly just about logic, set theory, proofs, etc... Not badly written but the concepts aren't that interesting. Probably give up?

* <strike>Symmetry and the Monster - Finished. Didn't have too much learning content but wasn't a long read. Forgotten a lot of it already though...</strike>

==To Read==
* Mathematician's Delight - Several recommendations. Apparently a good introduction to maths.

* A Mathematician's Apology

* Measurement - Paul Lockhart (Also Wrote Mathematicians Lament about math education).

* Summing It Up: From One Plus One to Modern Number Theory - Same person as Fearless Symmetry. Also "Elliptic Tales".

* How to Solve It - Bunch of recommendations. (also some books on [[TODO]]).

* Galois' dream: group theory and differential equations - Pop culture thing in Japan?

* Parallel Coordinates: Visual Multidimensional Geometry and Its Applications - Recommended by Steven Hawking

* Indras Pearls - Hyperbolic Geometry, companion site http://klein.math.okstate.edu/IndrasPearls/

* Pi the next generation - Seems cool. Can calculate pi to any arbitrary digit.

* The Elements of Mathematics - Seems to be a Princeton book.

* Handbook of Practical Logic and Automated Reasoning - Recommended for programmers.

* Concrete mathematics - Knuth

* The Princeton Companion to '''Applied''' Mathematics

* [https://www.amazon.co.uk/Bridging-University-Mathematics-Edward-Hurst/dp/1848002890/ref=pd_bxgy_14_img_3?_encoding=UTF8&psc=1&refRID=XER5WCKNGYWJ7J2K1P01 Bridging the Gap to University Mathematics] - Seems to have good reviews and says it has tests to see what level of knowledge student has.

* Grad, Div, Curl and all that by H. M. Schey - Recommend on [https://mathoverflow.net/questions/31879/are-there-other-nice-math-books-close-to-the-style-of-tristan-needham Stack Overflow].

* Nonlinear dynamics and chaos by Steven Strogatz

* The Essence of Chaos - Edward Lorenz

* The Shape of Space - Has some simple manifold examples...

===Topology 🍩===
* A Panoramic View of Riemannian Geometry

* Topology - Jänich - Seems approachable while in depth.

* Elementary Applied Topology - Robert Ghrist - Seems not too bad. Lots of pictures and starts with manifolds.

* Experiments in Topology - Seems not too bad again.

* Topology - James R. Munkres - [http://www.maa.org/press/maa-reviews/browse?field_tags_tid=37395&field_bll_rating_tid=36864&field_maa_review_value=All MMA Recommendation]

* A Combinatorial Introduction to Topology - Michael Henle - [http://www.maa.org/press/maa-reviews/browse?field_tags_tid=37395&field_bll_rating_tid=36864&field_maa_review_value=All MMA Recommendation]

===Groups===
* Groups and Symmetry: A Guide to Discovering Mathematics - The tile maps puzzle.

* Visual Group Theory - Was approachable from memory. Recommend on [https://mathoverflow.net/questions/31879/are-there-other-nice-math-books-close-to-the-style-of-tristan-needham Stack Overflow].

* Shattered Symmetry: Group Theory from the Eightfold Way to the Periodic Table - Looks like a mid level introduction to stuff.

* Group Theory for Physicists

* Naive Lie Theory - John Stillwell - Recommended on [https://mathoverflow.net/questions/31879/are-there-other-nice-math-books-close-to-the-style-of-tristan-needham Stack Overflow]. But seems kind of formal to me.

===Misc===
* [http://www.worldscientific.com/action/showPublications?pubType=bookSeries&startPage=1 World Scientific Series] - [http://www.worldscientific.com/page/mathematics2017 2017]
* [https://hbpms.blogspot.com/ How to Become a Pure Mathematician (or Statistician)] - a List of Undergraduate and Basic Graduate Textbooks and Lecture Notes

* Springer Undergraduate Texts (Jänich Topology has some recomendations)
* [http://www.maa.org/press/ebooks/dolciani-mathematical-expositions Dolciani Mathematical Expositions] - Same series as the "New Horizons in Geometry" book
* [http://www.maa.org/publications/books/book-series MAA series] - [http://www.maa.org/taxonomy/term/37012 MMA: Elementary Number Theory] - [http://www.maa.org/taxonomy/term/37183 MMA: Mathematics for the General Reader] - [http://www.maa.org/press/maa-reviews/browse?field_tags_tid=All&field_bll_rating_tid=36864&field_maa_review_value=All MMA: Most Recommended Books]
* [https://en.wikipedia.org/wiki/Euler_Book_Prize Euler Book Prize]

* [http://bookstore.ams.org/MAWRLD Mathematical World] - This accessible series brings the beauty and wonder of mathematics to the advanced high school student (Gotten everything of use?)

* Steven Strogatz
** The Joy of X
** Sync
** Nonlinear Dynamics
** The Calculus of Friendship

* [https://www.amazon.com/W.-W.-Sawyer/e/B001ITYQIY/ref=dp_byline_cont_book_1 W. W.-Sawyer] - Old but apparently wrote decent introduction books.

* The Mathematical Mechanic - Mark Levi - Using Physics to solve math problems. Mentioned on a 3Blue2Borown YouTube video.

* [http://libgen.io/search.php?req=Universitext&column=series universitext]
* Dover books. [http://store.doverpublications.com/by-subject-mathematics-aurora.html Dover Aurora Originals]
* [https://www.springer.com/gp/mathematics Spinger books]
* Wiley Books
* Princeton University Books - [http://libgen.io/search.php?&req=Princeton&phrase=1&view=simple&column=publisher&sort=year&sortmode=DESC&page=1 libgen] - [http://press.princeton.edu/math/subjects/mgen.html official site]
* [http://libgen.io/search.php?&req=World+Scientific&phrase=1&view=simple&column=def&sort=year&sortmode=DESC World Scientific] books
* [http://dme.ufro.cl/docmate/deliciouslibrary/ Some Library] - Has a few decent books ive seen, others worth a look.