Should I start with Euclid geometry?? Iam doing electrical engineering course(undergrad first year). I like to see how mathematics works in it's core. From grade 6(when I was 11), when I was introduced algebra, I did my maximum to know how things worked..I would want to know how a particular formula was derived. I made a special attempt to know how pythagoras theorem is proved and so on....I was introduced geometry in grade 7. it appeared like learning straight lines, about different angles(like alternate angle) and so on. A brief overview was given. We had a lot of theorems and proofs on congruency of triangles in grade 8. I just took the theorems on similarity and congruency just for granted(ie..like axioms) and proved many other theorems using it. For grade 9, I had to shift my school where there was a different education board. The students in this board already learnt the axioms on Euclid geometry in grade 8 itself. But they did not have rigorous method of proving. In my new school, we had to prove things like how the the triangle whose one edge is on the circumference of the semicircle and whose another side is the semicircle's diameter is right angled... We had used the following axioms or theorems like.. 1.Area of rectangle is product of length and breadth.. 2. Congruency and similarity or triangles. 3.Theorems on angles(like how the measure of vertically opposite angles when two lines are equal). Though we were presented only this much, I looked these proving exercises as logically conistent and self-contained without going into rigour of euclidean geometry. After few chapters on geometry, we had concept of cartesian co-ordiante system. By using these theorems, I found that we can prove all the formulas like distance-formula, section formula and so on... My book did not mention the proof of Hero's formula but I worked on it for days and proved it myself without using trignometry. My mind was convinced that I learnt mathematics(except calculus where Iam learning real analysis from Apostol) from scratch but now, I came to know that basic things like congruency and similarity of triangles can be proved by the axioms of Euclid.(from Wikipedia) I saw those axioms and they were simple. I regret of not learning those axioms. This has put my spirits down. My already self-convinced mind is not peaceful now. Now, should I learn euclid geometry again or am I missing something in not learning it? Is my approach logically incorrect? Please help me...Also help me what to learn(in mathmatics) for logical continuity to convince my mind....
hi sharan swarup! a) do you need euclidean geometry for any engineering or physics course … will it help you? no b) should you learn euclidean geometry? yes yes, because it obviously interests you cartesian geometry (x y and z coordinates, and equations using them) made euclidean geometry very nearly redundant, because most practical problems are much easier to solve using cartesian geometry (eg proving that cutting a cone gives you an ellipse) so euclidean geometry isn't much taught nowadays, even in maths courses (and i can't think of any applications in electrical engineering) but euclidean geometry is fairly easy, and a lot of the proofs are neat, and if it interests you, you should definitely get a book and go through it you should for example be able to prove that opposite angles of a quadrilateral inscribed in a circle add up to 180° my favourite euclidean geometry proof is that cutting a cone gives you an ellipse (or parabola or hyperbola) … google "Dandelin spheres" (and to answer your next question, no sorry i don't have any books to recommend)
What about http://en.wikipedia.org/wiki/Parabolic_antenna? The ancient greeks worked those properties of conic sections by geometry. Or, plotting complex numbers as points on a plane - there are plenty of complex numbers in EE! The whole of trigonometry is just "geometry for people who can't draw" but sometimes, drawing the pictures gives insights that you can't see from the equations.
Perhaps I can suggest a book. http://www.amazon.com/Euclids-Elements-Euclid/dp/1888009195/ref=sr_1_1?ie=UTF8&qid=1387333593&sr=8-1 Pretty cheap ($19 for a 500 pg. book if you have Prime), decent margin sizes to write notes, and plenty of pictorial representations of the propositions. On the flip side, you can find websites with the full translation of all the books including interactive applets which may be better for the reader than just pictures.
If something interests you, you should pursue it. If you want a resource for Euclid's The Elements, there is an interactive text for free: http://aleph0.clarku.edu/~djoyce/java/elements/elements.html Regarding your pursuit of rigor, I think you mean deductive or axiomatic reasoning. There are a lot of holes in Euclid's treatment. For some rigor means understanding that there are alternate axiomatic systems. For example, many triangle congruence properties depend on the definition of the distance formula. There are several other concepts of rigor, and some might claim an individual is never the creator of rigor, but it is the community of math and science that builds rigor. For sciences you may want to consider geometry in terms of rigid motions, or Helmholtz axioms. This is geometry on the basis of translation, symmetry, reflection, etc. These are useful for manipulating ODEs, PDEs, and calculus of complex variables.
There are different levels of mathematical rigor. As thelema418 points out, Euclid's Elements doesn't meet the highest modern standards of mathematical rigor Nevertheless, many mathematicians learned plane geometry in secondary school from textbooks loosely based on Euclid's elements. Euclid's Elements may satistfy your personal standard of rigor. The book Elementary Geometry From An Advanced Standpoint by Edwin Moise attempts to do plane geometry from a modern standard of rigor. Unless you have the instincts of a really pure mathematician, you won't like that book. The idea that mathematics is developed in a logical manner, beginning with "elementary" topics and proceeding to advanced topics may be true if we are talking about a cultural activity carried out by thousands of people. It is not true of how most individuals (including mathematicians) do mathematics. For example, a rigorous treatment of the the theory of the real numbers is an advanced topic - usually presented in a few chapters of a graduate course on "analysis". By the time students reach that level, they have already seen enough non-rigorous presentations of the properties of the real numbers to be familiar with the bottom line results.