Numerical Systems

We are used to using the decimal system in our daily lives. Digital technology requires an understanding of several other number systems the most important being the Binary system. The decimal system has its origin in the fact that man has ten fingers (or digits) hence the decimal system has ten digits (0 through 9). The binary system and Boolean Algebra have their origin in that simple switches or valves have two highly distinguishable states. They are OFF or ON. The binary system has two digits. They are 0 and 1.

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Octal and Hexadecimal are two additional number systems frequently used when dealing with digital systems. The octal system has eight digits (0-7) and the hexadecimal system has sixteen (0-9, A, B, C, D, E, F) These numbering system are particularly useful in dealing with large binary numbers, for example, A9B3 (HEX) is far easier to remember than 1010 1001 1011 0011(Binary) or 76751(Octal) is easier to remember than 111 110 111 101 001(Binary) .

Decimal     Binary     Octal      Hexadecimal

0                  0000          0              0

1                  0001          1              1

2                  0010          2              2

3                  0011          3              3

4                  0100          4              4

5                  0101          5              5

6                  0110          6              6

7                  0111          7              7

8                  1000          10             8

9                  1001          11             9

10                1010          12            A

11                1011          13             B

12                1100          14              C

13                1101          15              D

14                1110          16              E

15                1111          17              F

16            1 0000          20              10

Adding Binary Numbers

The rules of addition of binary numbers are essentially the same as the rules for decimal addition only simpler.

BINARY     DECIMAL     REMARKS

0 + 0 = 0     0 + 0 = 0          SAME

0 + 1 = 1     0 + 1 = 1          SAME

1 + 0 = 1     1 + 0 = 1          SAME

1 + 1 = 10   1 + 1 = 2         In binary we generated a zero and a carry.

We see that the rules are exactly the same for decimal and binary addition until we reach one and one equals two. In binary we cannot generate the digit "2" since there are only two digits in Binary "0" and "1". Therefore when you add one to highest binary digit "1" you generate a carry and a result of zero, just as when you add one to the highest decimal digit "9" you generate a carry and a result of zero. The four rules for Binary addition 0 + 0 = 0, 0 + 1 = 1, 1 + 0 = 1 and 1 + 1 = 0 and a carry are fundamental to implementing binary addition electronically.

0000 1100     0110 0011     0011 1111      0011 0011
0000 0011    0110 1111    0000 0010     1010 0011

Add the above binary numbers. and record your results on paper and then check your answer against answers at bottom of page.