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)
(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.