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Basic Mathematics

a roadmap for a firm foundation in basic principles of mathematics

status: In Progress
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certainty: certain
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importance: 6/10
AnalysisAlgebraCalculus
Filed under: stable-syllabi
To learn and not to do is really not to learn. To know and not to do is really not to know.
– Stephen R. Covey

On the Mathematical Life

It is to the suprise of no one that has put serious effort into the meticulous study of mathematics that the subject requires the balancing of a great deal of temperance, and creative thought. I have found that even obsession to a reasonable degree is not enough. One must become immersed in the subject, wholly engaging both mind and faculties in tandem. Therefore, and for that cause I recommend my essay Toward a Mathematical Life. A guide into coaxing the unconcious mind into a state of mathematical thought through enviornmental factors, and intention.

Aspects of Modularity in Structure

As mentioned above in "Who is this for?". This roadmap was designed for the Undergrad Mathematics Major. So there is to a extend a assumption of natural talent, giftedness, perseverance, and/or passion. I suggest that most work through this list linearly as those that are naturally quick to mathematical understanding will move through this fairly quickly. I do understand that this is not possible for everyone as you may be in the middle of a semester struggling to keep up with your current work. In that case, the structure of this Syllabis was designed in such a modular way that allows for the simultaneous learning of several areas. If your goal is to fill in most of the gaps that a poor elementary, middle, and highschool education left you with. Then this is the recommended way.

Mathematical History

Mathematical Writing

I first and foremost reccomend that each one of you start a blog. Mathematical Exposition is not in of itself more complex than most other types of exposition. It is the dull expositional insticts of one that lend to the difficulty of more formal writing. For that reason I suggest first the reading of a series of essays by Terrance Tao with a list of additional essays he reccomends prior to our main curricula here. I will do my best to present them in the most logical fashion.
The contents of Mathematical Writing, Like Mathematical History or quite unique in the fashion that they have no prerequisite requirements for there completion. Do not worry about having something "meaningful", or "Impactful" to write. That will come later. Just write.
In mathematics, the art of proposing a question must be held of higher value than solving.
– David Hilbert

Foundations of Basic Mathematics

Basic MathematicsBasic MathematicsSerge LangTrigonometryTrigonometryI. M. Gelfand and M. SaulKiselev's Geometry I: PlainimetryKiselev's Geometry I: PlainimetryAlexander GiventalKiselev's Geometry II: StereometryKiselev's Geometry II: StereometryAlexander Givental

Discrete Mathematics, Proof Writing, Naive Set Theory, Formal Logic

A Mathematical Introduction to LogicA Mathematical Introduction to LogicHerbert B. EndertonNaive Set TheoryNaive Set TheoryPaul R. HalmosDiscrete MathematicsDiscrete MathematicsSusanna EppHow to Prove ItHow to Prove ItDaniel J. Velleman

Combinatorics, Graph Theory, and Number Theory

Introductory CombinatoricsIntroductory CombinatoricsRichard BrualdiA Path to Combinatorics for UndergraduatesA Path to Combinatorics for UndergraduatesTitu Andreescu and Zuming FengIntroduction to Graph TheoryIntroduction to Graph TheoryDouglas B. WestAn Introduction to the Theory of NumbersAn Introduction to the Theory of NumbersG. H. Hardy and E. M. Wright

Calculus, Linear Algebra, Abstract Algebra

CalculusCalculusMichael SpivakCalculus, Volume ICalculus, Volume ITom M. ApostolCalculus, Volume IICalculus, Volume IITom M. ApostolLinear Algebra Done RightLinear Algebra Done RightSheldon AxlerAlgebraAlgebraSerge Lang

Differential Equations and Dynamical Systems

Ordinary Differential EquationsOrdinary Differential EquationsMorris Tenenbaum and Harry PollardPartial Differential Equations: An IntroductionPartial Differential Equations: An IntroductionWalter A. StraussDifferential Equations, Dynamical Systems, and an Introduction to ChaosDifferential Equations, Dynamical Systems, and an Introduction to ChaosMorris Hirsch, Stephen Smale, and Robert DevaneyIntroductory Functional Analysis with ApplicationsIntroductory Functional Analysis with ApplicationsErwin Kreyszig

Real Analysis, and Metric Spaces

Principles of Mathematical AnalysisPrinciples of Mathematical AnalysisWalter RudinUnderstanding AnalysisUnderstanding AnalysisStephen AbbottElementary AnalysisElementary AnalysisKenneth A. RossMetric SpacesMetric SpacesVictor Bryant
a2+b2=c2a^2 + b^2 = c^2
∫0∞e−x2 dx=π2\int_0^\infty e^{-x^2}\,dx = \tfrac{\sqrt{\pi}}{2}

Classic Literature Collection

Pride and PrejudicePride and PrejudiceJane AustenSense and SensibilitySense and SensibilityJane AustenEmmaEmmaJane AustenPersuasionPersuasionJane AustenMansfield ParkMansfield ParkJane AustenNorthanger AbbeyNorthanger AbbeyJane AustenLady SusanLady SusanJane AustenLove and FriendshipLove and FriendshipJane AustenThe WatsonsThe WatsonsJane AustenSanditonSanditonJane AustenPride and PrejudicePride and PrejudiceJane AustenSense and SensibilitySense and SensibilityJane AustenEmmaEmmaJane AustenPersuasionPersuasionJane AustenMansfield ParkMansfield ParkJane AustenNorthanger AbbeyNorthanger AbbeyJane AustenLady SusanLady SusanJane AustenLove and FriendshipLove and FriendshipJane AustenThe WatsonsThe WatsonsJane AustenSanditonSanditonJane Austen

On the Beauty of Mathematical Notation

Mathematics is not merely a language—it is a symphony of symbols, a choreography of logic and form. To test the robustness of our Math component, we present a flurry of expressions both elementary and esoteric.

Consider the Pythagorean Theorem in its most basic form:
a2+b2=c2a^2 + b^2 = c^2

Triple Integrals and Volume

A fundamental result in multivariable calculus is the computation of volume via triple integrals. For a region EE in 3D space, the volume is given by:

∭Ef(x,y,z) dx dy dz\iiint_E f(x, y, z) \, dx\, dy\, dz

In spherical coordinates, where x=ρsin⁡ϕcos⁡θx = \rho\sin\phi\cos\theta, y=ρsin⁡ϕsin⁡θy = \rho\sin\phi\sin\theta, and z=ρcos⁡ϕz = \rho\cos\phi, the Jacobian determinant introduces an extra factor of ρ2sin⁡ϕ\rho^2\sin\phi. Thus:

∭Ef(ρ,ϕ,θ)ρ2sin⁡ϕ dρ dϕ dθ\iiint_E f(\rho, \phi, \theta) \rho^2 \sin\phi \, d\rho\, d\phi\, d\theta

Summation and Limits

The Basel problem, solved by Euler, explores the value of the infinite sum of reciprocals of squares:

∑n=1∞1n2=π26\sum_{n=1}^{\infty} \frac{1}{n^2} = \frac{\pi^2}{6}

Nested summations can test vertical alignment and large limits:

∑i=1n∑j=1maijbji=tr⁡(AB)\sum_{i=1}^n \sum_{j=1}^m a_{ij}b_{ji} = \operatorname{tr}(AB)

Or even:

∑k=1∞(−1)k+1k=ln⁡(2)\sum_{k=1}^{\infty} \frac{(-1)^{k+1}}{k} = \ln(2)

Stirling's Approximation and Limits

The behavior of factorials at infinity:

lim⁡n→∞n!2πn(ne)n=1\lim_{n \to \infty} \frac{n!}{\sqrt{2\pi n} \left(\frac{n}{e}\right)^n} = 1

Fourier Transform

A classic in signal analysis:

f^(ξ)=∫−∞∞f(x)e−2πixξ dx\hat{f}(\xi) = \int_{-\infty}^{\infty} f(x) e^{-2\pi i x \xi} \, dx

And the inverse:

f(x)=∫−∞∞f^(ξ)e2πixξ dξf(x) = \int_{-\infty}^{\infty} \hat{f}(\xi) e^{2\pi i x \xi} \, d\xi

Dirac Delta Function

In distribution theory:

∫−∞∞f(x)δ(x−a) dx=f(a)\int_{-\infty}^\infty f(x)\delta(x-a)\,dx = f(a)

Gaussian Integral

A gem of analysis:

∫0∞e−x2 dx=π2\int_0^\infty e^{-x^2}\,dx = \tfrac{\sqrt{\pi}}{2}

Tensor Notation and Einstein Summation

Stress test with indices:

Tμν=gμαgνβTαβT^{\mu\nu} = g^{\mu\alpha}g^{\nu\beta}T_{\alpha\beta}

Or an expression involving the Levi-Civita symbol:

εijkεlmn=det⁡[δilδimδinδjlδjmδjnδklδkmδkn]\varepsilon_{ijk}\varepsilon^{lmn} = \det \begin{bmatrix} \delta_i^l & \delta_i^m & \delta_i^n \\ \delta_j^l & \delta_j^m & \delta_j^n \\ \delta_k^l & \delta_k^m & \delta_k^n \\ \end{bmatrix}

Conclusion

This post serves both as an ode to the expressive power of mathematics and a rigorous stress test for symbol rendering.

■\blacksquare

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  3. Focus on the end-user

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Bibliography

John Doe (2023). Example Reference. Example Publisher. View resource
Jane Smith (2021). Artificial Intelligence: A Modern Approach. Academic Press. View resource
Alex Johnson (2022). The Future of Space Exploration. Science Today. View resource
Maria Garcia (2023). Quantum Computing: Principles and Practice. Tech Publications. View resource
David Lee (2022). Neural Networks and Deep Learning. AI Research Press. View resource
Sarah Wilson (2023). The Ethics of Artificial Intelligence. Philosophy Today. View resource
Michael Brown (2022). Understanding Machine Learning. Tech Education Press. View resource
Emily White (2023). The Future of Human-AI Interaction. Digital Horizons. View resource
Tao, Terence (2009). On Writing Mathematical Papers. What's New. View resource
Gowers, Timothy (2007). My Favourite Pedagogical Principle: Examples First. Gowers's Weblog. View resource
Hilbert, David (1900). Mathematical Problems. Bulletin of the American Mathematical Society. View resource

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Citation

Cited as:

Yotam, Kris. (Jun 2025). Basic Mathematics. krisyotam.com. https://krisyotam.com/notes/stable-syllabi/basic-mathematics

Or

@article{yotam2025basic-mathematics,
  title   = "Basic Mathematics",
  author  = "Yotam, Kris",
  journal = "krisyotam.com",
  year    = "2025",
  month   = "Jun",
  url     = "https://krisyotam.com/notes/stable-syllabi/basic-mathematics"
}