Table of Contents
Appendix
Mathematics
- I am of the opinion that many of the questions on the sample test are really a test of mathematical skills. For example, practice questions 17, 31, and 32 do not really require computer knowledge to solve – they mostly involve mathematical skills. I recommend reviewing your discrete mathematics text as preparation for this test.
- There are three types of equivalence relations.
- A reflexive relation (R) on x means that xRx.
- A symmetric relation R for x and y means that xRy Û yRx.
- A transitive relation R for x, y and, z means that (xRy and yRz) Þ xRz.
- Binary is the language and number system of the computer. Binary uses base 2 instead of Base 10. The easiest decimal numbers to represent using binary are some power of 2. Practice test question #2 requires you to realize that 0.5 = 2^{-1}. I highly recommend looking at this website that describes how to do basic binary counting on your fingers. http://www.johnselvia.com/binary/binary.html Furthermore, in the real world, there are certain numbers that appear regularly. It is useful to memorize this table:
Power | Value | Description |
2^{8} | 256 | Byte |
2^{10} | 1024 | Kilo- |
2^{16} | 65,536 | 2 Bytes |
2^{20} | ~1,000,000 | Mega- |
2^{24} | ~16.7 million | 3 Bytes |
2^{30} | ~1,000,000,000 | Giga- |
2^{32} | ~4,000,000,000 | 4 Bytes |
- ! is the factorial function. The notation x! (read as x factorial) represents a number whose factors are all the integers from 1 to x. For example, 6! = 6 * 5 * 4 * 3 * 2 * 1 = 720
- ëxû is a unary function called the floor function. We apply this function to real numbers and always create an integer. It is almost like forcing the real number to round down. We will see this most often when trying to find a minimum integer value for a function. For example, ëlog_{2} 10û = 3 because it takes at least 2^{3} to get to 10.
- éxù is a unary function called the ceiling function. We apply this function to real numbers and always create an integer. It is almost like forcing the real number to round up. We will see this most often when trying to find a maximum integer value for a function. For example, élog_{2} 10ù = 4 because it takes at least 2^{4} to get to 10.
- We often use hexadecimal (base 16) to make it easier to write long binary numbers. Base 16 converts easily to base 2 (binary).
Hex | Binary | Decimal |
0 | 0000 | 0 |
1 | 0001 | 1 |
2 | 0010 | 2 |
3 | 0011 | 3 |
4 | 0100 | 4 |
5 | 0101 | 5 |
6 | 0110 | 6 |
7 | 0111 | 7 |
8 | 1000 | 8 |
9 | 1001 | 9 |
A | 1010 | 10 |
B | 1011 | 11 |
C | 1100 | 12 |
D | 1101 | 13 |
E | 1110 | 14 |
F | 1111 | 15 |