Samuel Morse developed a character coding system that was used to send messages along telegraph lines and also for light signalling between ships. Morse code is based on a series of dots and dashes (short of long bursts). The most frequently used letter, E, is coded as a single dot. The least frequently used letter, Q, is coded as dash, dash, dot, dash.
a) How many letters can be coded by a single burst? two bursts? three bursts? four?
b) How many characters can be coded, in total, by up to four bursts?
single burst = 2
two bursts = 4 , (2x2)
three bursts = 8 , (2x2x2)
four bursts = 16
total = 16+8+4+2 = 30
To answer these questions, we need to understand how many different combinations of dots and dashes can be formed with a certain number of bursts.
a) To determine the number of letters that can be coded by a single burst, we need to consider all possible combinations of dots and dashes. Since each burst can be encoded as either a dot or a dash, there are 2 possibilities. Therefore, a single burst can code for 2 letters: dot or dash.
To find how many letters can be coded by two bursts, we need to consider all possible combinations of dots and dashes for two bursts. With two bursts, each burst can be one of two possibilities (dot or dash). So, we have 2 * 2 = 4 possible combinations: dot-dot, dot-dash, dash-dot, dash-dash. Therefore, two bursts can code for 4 letters.
Similarly, for three bursts, each burst can be one of two possibilities, giving us 2 * 2 * 2 = 8 possible combinations: dot-dot-dot, dot-dot-dash, dot-dash-dot, dot-dash-dash, dash-dot-dot, dash-dot-dash, dash-dash-dot, dash-dash-dash. Thus, three bursts can code for 8 letters.
Finally, for four bursts, each burst can be one of two possibilities. So, we have 2 * 2 * 2 * 2 = 16 possible combinations. Therefore, four bursts can code for 16 letters.
To summarize:
- Single burst can code for 2 letters
- Two bursts can code for 4 letters
- Three bursts can code for 8 letters
- Four bursts can code for 16 letters
b) To find the total number of characters that can be coded by up to four bursts, we sum up the number of characters that can be coded by each burst.
Total = 2 + 4 + 8 + 16 = 30
Therefore, up to four bursts can code for a total of 30 characters.