wpe41.gif (23084 bytes)CIS3355: Business Data Structures
Fall, 2008
 

How many messages we could transmit for a given number of bits?

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  • The number of digits available dictates how many messages can be transmitted. For example, suppose I had two (2) decimal (i.e., 0, 1, 2, 3, 4, 5, 6, 7, 8, 9) digits available, and I could use them in any combination (I could have 08, 99, 83, 71, or any other combination of digits).
     
  • In decimal, we could calculate the maximum number of combinations as:
Number of Digits Value (number Messages) Exponential Representation
1 10 (0 .. 910) 101
2 100 (0 .. 9910) 102
3 1,000 (0 .. 99910) 103
4 10,000 (0 .. 9.99910) 104
5 100,000 (0 .. 99.99910) 105
10 10,000,000,000 1010
  • Notice that the power to which we must raise the base (in this case, base 10) indicates how may digits are available:
Number of Digits Value (number Messages) Exponential Representation
1 2 (0 .. 12) 21
2 4 (0 .. 112) 22
3 8 (0 .. 1112) 23
4 16 (0 ..11112) 24
5 32 (0 .. 111112) 25
10 1,024 (0 .. 11111111112) 210
  • The General Formula is:   I = Bn

where

    I  = The amount of information (message) available.
     B = The base we are working in (Decimal or Binary)
   n  = The number of digits (Decimal or bits) we have
  • As Stated, this works for ANY Base. Suppose that we wished to calculate how many representations (combinations) we would have if we could only use the digits 0, 1, 2, 3, 4, 5, 6, 7 (i.e., we had only 8 digits to work with)
Number of Digits Value (number Messages) Exponential Representation
1 8 (0 .. 78) 81
2 64 (0 .. 778) 82
3 512 (0 .. 7778) 83
4 4,096 (0 ..77778) 84
5 32, 768 (0 .. 77778) 85
10 1,073,741,824 810

Why would I care about Base 8 ???

Stay tuned --- We will get to that

Some good references include:

  1.   How Bits and Bytes Work
  2.   Data Structures And Number Systems
  3.   Bits and Bytes : An Explanation

At this point in time, you should be able to Answer the following questions:

  1. How do the number of digits I have influence how many different combinations of digits I can have?
    Answer: The number of digits available dictates how many different combinations there are. For example if I could use up to three decimal (base 10) digits, the maximum number of combinations I could have would be 103 = 1,000. That is, I could represent all of the numbers from 000 to 999. If I had 4 binary digits (bits), The maximum number of combinations I could have would be 24 = 16.
     
  2. If I had 7 different light switches, how many different combinations of on/off switches would I have?

    a.  7
    b.  14
    c.  49
    d.  128
    e.  10000000

    (answer: d since 27 = 128)
     

  3. Suppose I were working in Base 5, and I had only the digits 0, 1, 2, 3, and 4 (a total of 5 digits) to work with. If I could combine any 3 of these digits, what is the maximum number of combinations I could have?

a.  5
b.  32
c.  125
d.  1000
e.  3125

(answer: c since 53 = 125)

This page was last updated on 08/16/06