Topic, Offtopic post

  Pumpkins, Scarecrows, Ravens, Skeletons, Witches and Goblins

  We're going to the pumpkin patch tonight for a Halloween party. Enjoy this holiday guys. You can get candy from trick-or-treaters.

  Pi, 3.14, #, circumference, radius and circles

  Happy Halloween everyone. Enjoy all the decorations and candy that's passed out by trick-or-treaters. However I'm too old to do that sort of thing now.

  Here’s a potentially trickier to solve equation if you want something even more difficult. I know of two methods that will work here, but it may not be easy to ‘guess’ one method, the other is similar to Nelson’s other solution - which is the typical way to solve this type of equation:

x^4-2x^3-6x^2+10x-3=0

(Again there are four separate solutions!)

  s^2, pi*r^2, pi*r^2*h, s^3, b*h/2, (3*sqrt3/2)*s^2 (regular).

  Squares, Circles, Cylinders, Cubes, Triangles and Hexagons

  Red Riding Hood, Three Little Pigs and the Big Bad Wolf.

  , ,

  4, 8, 12, 16 and 20

  0, 1, e, pi, i

  Grass, dirt, sand, water, ice.

  Rain, thunder, wind, hail, tornados and hurricanes.

  @nelson Very good! Polynomial division is definitely a possible alternative to my substitution, and honestly I didn't think the equation was too hard when I first set it, I guess I should have assumed most people didn't see it.

@suhangha I derived the formulae (there are technically two if you count the + and - versions as separate though for legitimate triangular numbers you should always use the + formula), they are very interesting and can provide a generalisation to the question - 'If 2, -1/4, or whatever were a triangular number, what place would it be?'

I'd be interested to see if I got the correct answer though, I checked it and it seems to work.

  @CG52: no, your equation wasn't too difficult, it's just that I didn't see it. I remember learning how to solve this kind of equation over 50 years ago :)

The general way to solve it is as you mentioned:


second solution:

  Anyone interested in the inverse formula of the Triangular Number?

I figured out the formula a while ago on my own.

  So, what are you all going to do this weekend? I might go to the mall tomorrow sometime.

  c=10

Most standard method:


Maybe my equation was too difficult…

Hint:

  Now solve for c.

10c+8=108

This will be an easy one for you guys.

  The equation x^2-(a+b)x+ab=0 will have the solutions x=a and x=b, that's how you can make a polynomial equation with 2 solutions. More than 2 requires higher order polynomials. There are other ways, like trigonometric equations but that might not be familiar to you anymore if you stopped studying mathematics many years ago.

  Well done with those methods CG52, now I proved to make an equation with more than 1 solution.

  a=4

Method 1 - Normal:


Method 2 - Factoring:


A bit harder, try to solve for x:

x^4-5x^2+4=0 (there are 4 solutions!)

  Solve for a.

2a-8=16-4a

  I don’t actually know where the word ‘transcendental’ comes from. It’s probably to do with the numbers ‘transcending’ the algebraic methods of constructing numbers. Similarly, irrational numbers are those that aren’t ratios of whole numbers (so can’t be represented as fractions).

Inequalities can be confusing, particularly if it is a rational function instead of a polynomial, but even a simple quadratic can be trickier than you might think.

Mathematics takes a lot of practice to learn and while some people are naturally gifted at it, I do think that 90-95% of people can understand maths to a level that would surprise themselves. Of course I understand that this stuff is very abstract for a lot of people without the background knowledge or the desire to learn it xD. Even simpler stuff though, practice is the only way you can get better, but when you do a lot of those problems, improvements will naturally occur.

  This is an interesting conversation going on about mathematics stuff. I got a B in my 6th grade math class. The hardest part was the advanced equations, inequalities and fractions, etc. Transcendential sounds like a crazy term that the Greeks or Trojans would use back in the medieval times.

  The true creepiness of transcendental numbers is that as I said before, since a lot of early mathematics was based on algebra, it doesn't feel very comforting that almost any random real number cannot be constructed that way. To hammer home the point, if you take the section of real numbers between 0 and 1, and then pick out a random one based purely on a uniform distribution, the chance that the number can be constructed algebraically is zero, which just feels completely strange.

I wonder if the people who discovered that transcendental numbers exist had similar reactions as the Ancient Greeks did who discovered they could make numbers like the square root of 2 that can't be represented cleanly as a ratio of two integers...

  That was weird and creepy in a way, but it's always a brilliant thing that suhangha has made for us to see!

  That is quite weird to be honest. I hadn’t made the link that they are the same number… it’s probably just an error in WolframAlpha, because it claims that it doesn’t know if 1 is transcendental, which it obviously isn’t.

 

  Transcendence is a very interesting property to be honest. In a way it makes me feel slightly weird, you can show that transcendental numbers are far more common than algebraic numbers. The problem with that is our immediate intuition for how we understand numbers would start from whole numbers of things, develop to involve ratios between those whole numbers, and eventually solutions to algebraic equations in the whole numbers, but almost every number can't be produced that way. Simple concepts like the ratio between a circle's circumference and its diameter surely feel like the should be easy to construct in these ways, but try as you might it is impossible, as pi is transcendental.