Ask HN: How to learn mathematical proofs from scratch?

Many people recommend learning to write proofs in the context of a class or text that's focussed on another topic, e.g. geometry or linear algebra or real analysis or abstract algebra. But I preferred learning from a book that was more proof-focussed and added context along the way. Here's the text I learned proof-writing from, and which I highly recommend: https://www.whitman.edu/mathematics/higher_math_online/

I did math in undergrad and struggled with proofs at first.

I learned a lot from working through An Infinite Descent into Pure Mathematics by Clive Newstead. It's designed to get someone with minimal math background started with the basics of pure math. Becoming comfortable with proofs happens along the way.

https://infinitedescent.xyz/

Lately, I have also really enjoyed 99 Variations on a Proof by Philip Ording, which is not a textbook. Rather it is an Exercises in Style type book that explores many different ways to express a mathematical proof of the same simple fact. Some of the proofs are whimsical, and others offer genuine insight. If you're looking for something lighter than a textbook that is still interesting and somewhat useful, this book is more approachable.

https://www.amazon.com/99-Variations-Proof-Philip-Ording/dp/...

2 books I recommend: how to prove it, and the book of proof, get a real analysis book, and if possible take a course. it took me around 5 years to be able to prove stuff... you need relaxation, fluid thinking, and a breadth of knowledge of facts you can use to prove stuff

we can try your example

we want to show that (a^b)^c = a^bc

let's work on the LHS

by the definition of an exponent, we know that a^b is just a * a * ... * a b times, so we can rewrite it as:

(a_0 * a_1 * ... * a_b)^c

by the same definition, we can multiply the quantity inside the parenthesis by itself c times:

(a_0 * a_1...a_b)_0 * (a_0 * a_1 * ... * a_b)_1 * ... * (a_0 * a_1 * ... * a_b)_c

now, use the fact that a^m * a^n = a^(m+n) to consolidate the parenthesis, since each factor has an exponent of 1 we can use simple counting:

(a^b)_0 * (a^b)_1 * ... * (a^b)_c

Repeat the previous step c times, we end up with

a^(b_0 + b_1 + ... + b_c)

which of course is just

a^(bc)

therefore (a^b)^c = a^(bc)

I disagree regarding Euclidean geometry. Euclid never does any proofs by induction, which is enough on its own to disqualify Euclid as a good introduction to proof.

What you want is a book that combines an introduction to logic with a bunch of different proofs from different areas of math, such as set axioms, relations, functions, sequences, construction of real numbers, etc. There are many books like this, here is one that includes all of that plus a little number theory and algebra towards the end: http://libgen.rs/book/index.php?md5=7E4D97D2F58B91D052595E68...

The equation x/a = b/c has equivalent fractions on either side of the "equals" sign. You can make equivalent fractions by multiplying or dividing both numerator and denominator by the same non-zero number.

Take a square that has a equal parts, with x parts shaded. It represents x/a.

You can split up the whole square in a different way but keep the same area. This square has c equal parts, with b parts shaded. It represents b/c.

You can represent the work by multiplication and division, too.

Take, for example, 6/12. Divide both numerator and denominator by 3. You get the equivalent fraction 2/4.

The area does not change. Hope it makes sense.