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November 9th, 2015, 06:52 AM   #11
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From: Barto PA

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Quote:
Originally Posted by Chikis View Post
I was asked to write down the first 1111111111 in binary counting system.
I could not attempt the problem because I don't understand what they want me to do. Could anyone put me through please?
Just in case you have not yet seen the pattern
that others have tried to get you to see, here
is a simple way to write such a list: Write it
vertically down the page. Start at the extreme
upper right-hand side of your paper and write
a column of alternating 0's and 1's:
010101010101010101010...

Write another column down the page to the
left of the first column:
00110011001100110011...

Another column to the left of the last one:
000011110000111100001111...

One more:
00000000111111110000000011111111...

Now you should see the pattern (if you already
haven't). Just double the number of 0's and
1's in the sequence for each column you write.
Your list should look like this (for just 17 num-
bers including 0):

0 --- ................000000
1 --- ................000001
2 --- ................000010
3 --- ................000011
4 --- ................000100
5 --- ................000101
6 --- ................000110
7 --- ................000111
8 --- ................001000
9 --- ................001001
10 - ................001010
11 - ................001011
12 - ................001100
13 - ................001101
14 - ................001110
15 - ................001111
16 - ..............0010000
.
.
.
and so on.

It certainly isn't necessary to write such a list
if all one wants to do is convert some given
base-two number to decimal.
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November 9th, 2015, 02:07 PM   #12
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When you write the first positive integers and zero in base ten, you find that there are ten digits, 0,1,2,3,4,5,6,7,8,9. To express ten, you write 10 which means 1 ten and no ones. eleven is 1 ten and 1 one, etc. After 99, you will need an extra digit, so that you would write 100. The 1 represents 10×10 or a hundred

If you were writing the first positive integers and zero in base 4, there are four digits,
0,1,2,3. To express four, you would write 10 which means 1 four and no ones. To express five, you write 11, which is 1 four and 1 one, etc. To write nine, you would write 21, which means 2 fours and 1 one. you will need an extra digit after 33 (3 fours and 3 ones, namely 100. This time the 1 represents 4×4 or a sixteen.

101 seventeen
102 eighteen
103 nineteen
110 twenty (1 sixteen and 1 four)
111 twenty-one
etc. etc. etc.


In writing the first positive integers and zero in base 2 (the binary system) there are two digits, 0 and 1

The number two would be written as 10 which means 1 two and no ones.
Three would be written as 11, that is, 1 two and 1 one. After 11, you will need and extra digit, namely 100. This time the 1 represents 2×2 or a four.
Thus 101 is 1 four, no twos, and 1 one (five)
110 is 1 four, 1 two, and no ones (six).
111 is 1 four, 1 two, and 1 one (seven).

After that you need an extra digit that will represent 2×2×2×2 or eight. Thus
1000 eight
1001 nine
1010 ten
1011 eleven
1100 twelve
1101 thirteen
1110 fourteen
1111 fifteen

After that you need an extra digit that will represent 2×2×2×2×2 or sixteen
10000 sixteen

etc. etc. etc.

You can use the same addition, subtraction, multiplication, and division algorithms in other bases that you normally use in base 10.
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