my first questions on pre calculus (summer stuff)

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• 06-22-2003
Silvercord
my first questions on pre calculus (summer stuff)
We had some written problems. Here is the first, tell me what you think. The forumla for the 'gaps' is just taken from the book, so don't ask me about it :O)

I thought I answered the question well, but I want to impress my math teacher to let him know I'm still interested and to help keep him motivated. So if there's anything cool I should add please tell me.

Quote:

1. Discuss the relationships/differences between screen coordinates and xy – plane coordinates. Are they the same? Different? Why? Include a definition of a pixel and why we should care.

Your typical graphing calculator displays many of the points on the xy plane and is useful for many purposes. However, not all points on the xy plane can be displayed. For example, when you enlarge your viewing window ‘gaps’ between pixels appear, and are represented by the formula:
(Xmax – Xmin) / (Columns – 1)
and
(Ymax – Ymin) / (Rows – 1)

Computers are limited by the data types that they use. For example, it is impossible to accurately represent a point on the xy plane where one of the numbers is irrational. This is because on your typical graphing calculator you have a mantissa with about 9 decimal places accuracy (on a personal computer it is typically between 5 and 7). Another problem is the fact that pixels are of a finite size. A pixel (which stands for picture element) is the smallest component of a graphics display, and thus is the smallest section that can be lit up. In reality, numbers go on forever and can be infinitely small, large, or precise. If you had an infinite level of accuracy (which you don’t) you could take a single pixel and graph ten trillion numbers within it.
Computers also have built in data types that can hold a finite number of values (in powers of two). For example, on your typical 32 bit system (a PC) the highest value of an (unsigned) integer (4 bytes ~ 32 bits) is 2^31 or 2147483648 (the 0 power is the first bit). If you try to add 1 to that number, it will overflow and actually start back at 0. This can pose a problem on your graphing computer (whether it be a TI 83 or a personal computer) because you can never represent a point that goes beyond these limitations unless new hardware or special software is introduced.
• 06-22-2003
Eibro
Actually, the highest value an unsigned long can hold on a 32 bit system is (2^32)-1 or 0xFFFFFFFF.
• 06-22-2003
Silvercord
uhh

no

cout << "largest int val " << numeric_limits<int>::max() << endl;
cout << "largest int val " << numeric_limits<long>::max() << endl;

4 bytes = 32 bits
the 32nd bit represents the 31st power

2^31 = the really big number in my paragraph

unless I'm missing something

a long and an int are the same thing on a 32 bit system

EDIT: ok, i guess that's for a signed long, my bad

EDIT1: That doesn't make sense to me though. I changed the int to unsigned int in that code above and received what you told me, but it doesn't seem like a possibility when the highest available power is 31. Maybe someone can explain it to me.
• 06-22-2003
Perspective
2^31 == 1000 0000 .......

(2^32)-1 == 1111 1111 111....

get the idea? 2^31 is the largest high order bit by itself, (2^32)-1 is all bits set.
• 06-22-2003
Silvercord
Oh ok, now I got it. I feel smart right now. No, really.
• 06-22-2003
ygfperson
IMHO, you could lose the last paragraph without affecting your answer.
• 06-22-2003
Zach L.
You could say that while mathematics (this type at least) is continuous in nature, it cannot be represented on a screen because length is not particularly meaningful below 1.6x10^-35 m (Planck length), and then go off on a tangent about not being able to see where the line defining the function is because you know fairly well how fast the pixels are moving.

... You might get a calc book thrown at you, but you could mention it. :D