1. Number Systems
2. Orders of Magnitude
3. Integers
4. Floating Point
5. ASCII

There are several number systems that are important to computer science. Primary among them is the binary (base 2) number system. Other systems used are the octal (base 8) system and the hexadecimal or hex (base 16) system. Below is a table of values that you need to memorize.

Seeing a number written as 10 does not tell you what its value is. Its value may be ten or two or eight or sixteen depending on its base. You must be told what base you are working in!

Every number system as a set of allowable digits. The decimal system uses the digits 0-9; the binary system uses the digits 0 and 1; the octal system uses the digits 0-7; the hex system uses 0-9 and A-F.

Converting from One System to Another

Other Bases into Decimal

45297 =
4 * 104 = 4 * 10,000 = 40,000
5 * 103 = 5 *  1,000 =  5,000
2 * 102 = 2 *    100 =    200
9 * 101 = 9 *     10 =     90
7 * 100 = 7 *      1 =      7
                        45297

11011101 = 
1 * 27 = 1 * 128 = 128
1 * 26 = 1 *   64 =  64
0 * 25 = 0 *   32 =   0
1 * 24 = 1 *   16 =  16
1 * 23 = 1 *     8 =   8
1 * 22 = 1 *     4 =   4
0 * 21 = 0 *     2 =   0
1 * 20 = 1 *     1 =   1
                          221

45261 = 
4 * 84 = 4 * 4,096 = 16,384
5 * 83 = 5 *    512 =   2,560
2 * 82 = 2 *      64 =     128
6 * 81 = 6 *        8 =      48
1 * 80 = 1 *        1 =        1
                             19,121

285BCE = 
2 * 165 =   2 * 1,048,576 = 2,097,152
8 * 164 =   8 *     65,536 =    524,288
5 * 163 =   5 *       4,096 =      20,480
B * 162 = 11 *          256 =       2,816
C * 161 = 12 *           16 =          192
E * 160 = 14 *             1 =           14
                                      2,644,942

You need to be able to convert a number from binary, octal, or hex to decimal. Luckily we don't do this very often.

Binary to Octal

Octal uses three bits so we divide up the binary number into groups of three. Remember to start from the right!

11011101 = 11 011 101
            3   3   5

Binary to Hex

11011101 = 1101 1101
              D    D

Decimal to Binary Or Hex

Converting a decimal value into binary (or hex) is a repeated process of dividing, using the remainder, and dividing again. We are going to use the division you did in third grade for this algorithm. This is also called integer division.

  110 r 1
2)221

   55 r 0
2)110

I find it easier to start at the bottom and stack up my divisions, keeping track of my remainders. Read this example as a series of divisions from the bottom up, but read the conversion as the list of remainders from the top down.

    0 r 1
  2)1 r 1
  2)3 r 0
  2)6 r 1
 2)13 r 1
 2)27 r 1
 2)55 r 0
2)110 r 1
2)221 

The subtraction method for converting decimal to binary will be illustrated in class.

To convert from decimal to hex, divide by 16 and remember to convert the remainder into hex digits (0-F). For example, let's convert 2644942₁₀ into hex.

         0 r 2
      16)2 r 8
     16)40 r 5
    16)645 r 11 (B)
  16)10331 r 12 (C)
 16)165308 r 14 (E)
16)2644942

This division is more difficult to do by long hand, but it is possible. Or you can use your calculator.

If your calculator will do modulo division, then you can get through the calculations easier. Modulo (or mod) gives the remainder from integer division. Using the same problem we would get:
2644942 mod 16 = 14 or E.
You must compute the integer quotient portion so you can do the next part: 2644942 / 16 = 165308.875. Throw away the fraction part.
Now 165308 mod 16 = 12 or C.
165308 / 16 = 10331.75. Then 10331 mod 16 = 11 or B. And so on.
10331 / 16 = 645.6875; 645 mod 16 = 5.
645 / 16 = 40.3125; 40 mod 16 = 8.
40 / 16 = 2.5; 2 mod 16 = 2. 2 / 16 = 0. So we quit.