I have split the articles about squares and quadratic equations into two separate articles:
One of them is now solely on quadratic equations, and I have extended it with more info and more geometrical explanations :-) Now it should be easier for you to understand how do they work and how to solve them.
The second one is what I moved from the article about quadratic equations to have it separate and to be able to refer to it from other articles. It's about squares and square roots: what does it mean geometrically, from the Pythagorean point of view. There's also a bit about short multiplication formulas, but I plan to make a yet separate article from it and extend it there soon.
Yesterday my friend Ćukasz came to me and asked for help with solving some physics problem in optics (you know, lenses and stuff… ;-J ). Here's that problem in a form he gave it to me:
We have an object, a lens, and the real image of that object. The distance between the object and its image is 1 meter, and the magnification is 5 times. What is the focal distance of this lens?
My friend didn't know how to solve this problem; he didn't know either any formulas from optics which could help him to solve it. However, when I draw a picture illustrating this problem, i concluded that no fancy formulas are needed! :-) We also won't have to know any physical properties of the lens, such as refractive index between glass and air etc. We don't even need to know the law of light refraction nor any formulas related to it! :-D The only thing we need is a little bit of geometry :-)
I'm sure you've heard at least once in your life that division by zero is impossible. At schools they've probably teached you some nursery rhymes to remind you that you can't do that, and to convince you once for all that you shouldn't even try. Like if dividing by zero were to cause some enormous cosmic catastrophy; eg. create a big black singularity which will suck in the whole Universe in an instant. Like in these funny pictures.
Side note:
Well… It's true that sometimes division by zero can cause problems: when a computer program tells the CPU in your computer to divide by zero, the CPU of course doesn't know how to do it (because none of his creators knew either), so it'll just give up and break the execution with an error. If the programmer hadn't foreseen that, the program will usually crash. If this program controls some serious process, eg. a nuclear chain reaction or a space shuttle, it might truly cause problems :-P But these problems lie not in the sole division by zero, but in the architecture of contemporary computers, which are not designed to do it.
Every self-respecting mathematician (the more self-respecting the worse ;-P) will tell you that it's impossible and absurd to divide by zero. He will show you thousands of mathematical proofs which show that such an operation is nonsense, illogical and it even couldn't be sensibly defined.
But is it really that way? Could the number system really exist with such a serious breach related to only one single number? (the zero) Is it right that such a "nothing", as zero, could really lead to so much problems?
Have you ever wondered why, when we raise something to the second power, we say that we "square" it?
Pythagoreans believed that everything is a number. In a sense that everything in our Universe could be represented and described by numbers. They also related numbers to shapes and geometric figures. So when they spoke about square of some number, they spoke of literally the square as a geometric shape – they were constructing a square with the lenght of its side equal to that number, ant then the area of this square were the result of raising the number to the second power.
Hovewer, besides squares and cubes of numbers, they also knew triangular numbers, pentagonal numbers, hexagonal numbers etc. It is an interesting topic and we'll get back to it time to time. In this lecture nevertheless, we'll deal only with squares for now.
Here's how the firs few perfect squares (those which sides are whole numbers) look like:
Up to this moment you have learned linear equations and linear functions.
In this article we will go a step further: we'll introduce the second-degree term, that is, the quadratic term:
When they teach you how to solve such equations at schools, they usually give you only a template formula, where you insert the coefficients , , from the equation 1 to compute the solution:
But I guess that when you look at this complicated formula, it probably doesn't tell you much :-P That's because at schools they don't explain where does this formula come from and what does it mean. So your only option is to memoize this formula and apply it mindlessly ;-/
Today it's gonna change! :-> Not only I'll show you how to get to this formula step-by-step, by investigating the geometry hidden in the quadratic equation, but you'll also see that in most cases this formula is totally superfluous and it unnecessarily complicates calculations, which you can easily perform in your head, with much less of work! It's enough if you understand how it all works.
I would like to recommend you a book which I've stumbled upon some time ago, and which impressed me very much. In a sense it has changed my life and views on Physics, the structure of matter and the inner workings of the Universe. It's written by Dr Milo Wolff, an ingenious scientist from California, who worked for NASA, Aerospace and some famous universities (MIT & Caltech etc.). Last year he has been also honored with a Sagnac Award.
Dr Wolff's book has a similar subject and style as Leon Lederman's world's bestseller "The God Particle: If the Universe Is the Answer, What Is the Question?". If you ever had any problems with understanding Physics to this day, then you'll like Milo Wolff's book for sure, because Dr Wolff wrote it for this reason: to allow every human, who hadn't dealt with Physics before, to easily and in a short time catch up with pioneers from the frontier of Science (maybe even outpace them?) and join the club of explorers of the mysteries of the Universe. Milo Wolff prooves that you don't need a PhD in Physics to understand its arcanes and be up-to-date with scientific news. One might say: "You too can become another Einstein!" :-)
If you're a computer artist or web designer, or if you know how video displays works, you're probably familiar with the RGB model of color vision, in which the three primary colors of light: red, green and blue, need to be mixed together to get with different amounts to get any other colors of light. And you probably know also that in order to get a white light, you need to mix all these three primary colors in equal amounts. Right?
Wrong! :-P
In 1852, a scientist named Helmholtz has made many experiments with light and colour and these experiments have bring him to very interesting, and perhaps surprising, conclusions.
On the English section of the "Quantasia" discussion board Milo Wolff, father of the WSM theory, started an interesting thread about microchips. He asked about our ideas and thoughts on what the WSM theory could introduce into the subject of the production and working of michrochips.
Larry Pruitt showed recent works on making three-dimensional integrated circuits. It would be based on stacking many layers of semiconductors in a form of a sandwich ;-)
Of course stacking microchips can save space and faciliate cooling. It's an interesting idea too to build a microchip as a real 3D structure. I hope we will see this technology soon.
But there are limits of that technology, which might be approached quicker than we suspect.
I recommend only the books I have read, and which I've found good and life-changing.