Proofs of basic, common, or concrete theorems in many branches of mathematics

Question

I have several books of proofs such as:

$\color{blue}{\text{Book of Proof}}\text{ by }\color{green}{\text{Richard Hammack}}$

$\color{blue}{\text{Proofs: A Long-Form Mathematics Textbook}}\text{ by }\color{green}{\text{Jay Cummings}}$

and some other books in proofs. These are really excellent books, and there are many proof books that are good.

However I am looking for a book that contains many (basic) or (common) or (concrete, possibly hard)

$$\color{red}{\text{the following are just examples}}$$

• There are infinitely many primes.

• The irrationality of $\sqrt{2}, e, \pi, \dots$.

• Derivation of quadratic formula, cubic formula, quartic formula.

• There is no solution in radicals to general polynomial equations of degree five or higher with arbitrary coefficients (also known as Abel–Ruffini theorem).

• The area of a circle $A=\pi r^2$, the volume of a sphere $V=\frac{4}{3}\pi r^3, \dots$.

• The Pythagoras' theorem, $a^2+b^2=c^2$ where $a$ and $b$ are the legs, and $c$ is the hypotenuse of a triangle.

• $\sin(\alpha \pm \beta)=\sin(\alpha)\cos(\beta) \pm \cos(\alpha)\sin(\beta)$, $\cos(\alpha \pm \beta) = \dots, \tan(\alpha \pm \beta) = \dots$.

• Wilson theorem, any prime $p$ divides $(p-1)!+1$.

• Fermat's little theorem.

• Sine and cosine rules.

• Intermediate value theorem, mean value theorem, Rolle's theorem, ...

• The sum of the first $n$ natural numbers, the sum of the squares of the first $n$ natural numbers, the sum of the cubes of the first $n$ natural numbers.

• The sum of $\text{A.P, G.P}$.

• The derivation of Taylor series formula.

• The integral of (even function or odd function) from $-a$ to $a$.

• L'Hospital's rule.

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The examples mentioned above are not all easy, for example Abel–Ruffini theorem, or the quartic formula, they are difficult. But I believe the are basic (basic in the sense of "knowing" the proofs/derivations).

So what I am looking for is a book that contain the proofs of formulae in different branches of mathematics, such as arithmetic, geometry, trigonometry, algebra, calculus, number theory, ...

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I am afraid such book does not exist. But if exist, I highly recommend myself and everyone who is interested in maths to get this book.

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Your help would be appreciated. THANKS!

Answer 1

I agree with CyclotomicField in the comments that a book that contains topics on Euclidean Geometry, Abstract Algebra, and Analysis would be the closest thing to encompass the challenge/diversity of the examples you want. However, I do not believe there to be one single book that can encompass it all. However, a collection of books might be able to, where you can skim them to find the most challenging parts that fit your, and others, needs the best. Below, split by subject, is a list of Open-Access, immediate and free, books that may be of assistance that you can immediately skim to attack the most challenging parts.

Euclidean Geometry

• Project Euclid Documents
• MathCentre's Book on Euclidian Geometry

Abstract Algebra

• Abstract Algebra: Theory and Applications by Thomas W. Judson, et al.
• Algebra: Abstract and Concrete by Frederick M. Goodman
• Elements of Abstract and Linear Algebra by Edwin H. Connell

Real Analysis

• A Primer of Real Analysis by Dan Sloughter

• Analysis by W. Ted Mahavier

• Measure, Integration & Real Analysis by Sheldon Axler

• A small list of Classical Real Analysis Open-Access books

Open-Access Site That Cover Mentioned Topics Plus More

• Mathematics: Open Textbooks

In addition, if a book does exist that covers everything that you mentioned, it may be good enough to justify it to be ridiculously expensive, so that may be an additional factor that may impact a decision for that book. Nevertheless, here is a hyper-dense list of serious math books that this community had recommended long before.