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Binary and Hexadecimal

  1. Sep 1, 2004 #1
    Hello all

    I need help understanding binary and hexadecimal numbers. What exactly are they? Could you give me some examples?

    Thanks
     
  2. jcsd
  3. Sep 1, 2004 #2

    dduardo

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    Staff Emeritus

    I'm sure your familiar with the decimal(base 10) system: 0,1,2,3,4,5,6,7,8,9
    Binary(base 2) is: 0,1
    Hexadecimal (base 16) is : 0,1,2,3,4,5,6,7,8,9, A, B, C, D, E, F

    To convert a number in decimal to any other base, all you have to do is take the decimal number and continualy divide the number by the base keeping track of the remainder

    Example:

    8 (Base 10) to (Base 2)

    8/2 = 4 R 0
    4/2 = 2 R 0
    2/2 = 1 R 0
    1/2 = 0 R 1

    Starting from the last remainder work your way up:

    1000 (base 2) = 8 (base 10)

    200 (base 10) to (base 16)

    200/16 = 12 R 8
    12 / 16 = 0 R 12 = C

    C8 (base 16) = 200 (base 10)
     
  4. Sep 1, 2004 #3

    robphy

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    in base 10, 123 = 1x102+2x101+3x100=100+20+3=123 (base 10)

    ("hexadecimal")
    in base 16, 123 = 1x162+2x161+3x160= 256+32+3= 291 (base 10)

    ("binary")
    in base 2, 10010 = 1x24+0x23+0x22+1x21+0x20=16+0+0+2+0= 18 (base 10)
     
  5. Sep 1, 2004 #4
    Fairly detailed response

    we use a base 10 system like he said which is 0-9 or ten digit. you may not realize it but numbers are created by the rules of base ten.
    our numbers say 352 are made up of combinations of base ten


    3------5---------2
    10^ 2 --10^1-----10^0

    for biger numbers u would just continue the pattern
    number
    143,251


    1---------4---------3---------2---------5-------1
    10^5------10^4-------10^3-------10^2------10^1---10^0


    we know we need to add another digit when we get higher than 9 which is the same when its in other bases except the digit you can go up to is different
    in binary like he said you can only use 0 or 1

    so you would count 0,1,2,3,4,5,6,7,8,9,10 in base ten and
    Base 2 Base 10 equivilent Hex
    (binary)
    0 1 1
    01 2 2
    10 3 3
    11 4 4
    100 5 5
    101 6 6
    110 7 7
    111 8 8
    1000 9 9
    1001 10 A
    1010 11 B
    1011 12 C
    1100 13 D
    1101 14 E
    1110 15 F
    1111 16 10
    1001 17 11





    It goes the same way
    lets say we have 352 in base 10
    what is that in binary

    well lets do what we did for base 10 except with base 2


    2^9--2^8----2^7----2^6----2^5---2^4---2^3----2^2-----2^1----2^0
    512--256----128-----64------32----16-----8-------4-------2------1


    now we take 352 and substract the first number under it that it can go into
    so 512 is to big-- so we move to 256 and we can take 1 256 out of 352 (in binary it will always be one or 0 but it changes for differnent bases-- you will always at max be able to take the base number of times out of the orgiinal number) so

    lets do it
    352-256=96 (and you put a one on top of the 2^8 to represent you can take the 256 out)

    0 ---1
    2^9--2^8---- 2^7---- 2^6--- 2^5-- 2^4-- 2^3-- 2^2---2^1-----2^0
    512----256--- 128---- 64----32-----16----8-----4-------2-------1

    So now your left with 96
    you can't take 128 out of 96 so you put a 0 there
    continue in this fashion until you get the following

    352-256=96
    128 to big so 0 in 2^7
    96-64=32 so one on top of 2^6
    32-32=0 so one on top of 2^5
    therefore since your are out the rest are zeros


    0----1-------0------1 -------1-----0 ----- 0-------- 0--------0------0
    2^9-- 2^8--2^7----2^6-----2^5--- 2^4--- 2^3----2^2-----2^1----- 2^0
    512---256---128 -----64------32 ----- 16--- 8-------4--------2------1

    so the binary representation of 352 is
    0101100000
    u don't need the leading 0 so
    101100000

    you can use this method to test the following numbers

    13 = 1101
    234 = 11101010
    456 = 111001000
    2547 = 100111110011



    Now lets do the same thing for hex
    take 352 again
    we set it up the same way
    we say 4096 is too big so we move down to 256 and see 256 can be taken out once so we put a one there
    352-256=96
    Now how many times can 16 go into 96:
    the answer is 6 so we put a 6 there
    96- 6(16)=0

    0-------1-------6-------0
    16^3----16^2---16^1----16^0
    4096-----256-----16-------1

    so the hex representation of 352 base 10 is 160

    now lets try a harder one
    take
    9453

    65536 is obviously too big so we move on to 4096
    how many times can 4096 go into 9453
    the answer is 2 so u put a 2 in on top of it

    9453-2(4096)=1261

    so we continue
    how many times can 256 go into 1261
    the answer is 4
    so 1261-4(256) = 237

    --0--------2--------4
    16^4-----16^3----16^2---16^1----16^0
    65536-----4096----256------16-----1


    Now is where it gets tricky
    how many times does 16 go into 237
    the answer is 14--- but you can't put 14 in there because it needs to be a single digit-- thats when you remember that unlike base 2 which only has 2 digits or base 10 which has 10 digits (0-9) base 16 has 16 digits
    0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F
    just remember you treat the letter as you would the base 10 number -- just continue the sequence
    so A is 10
    B is 11
    C is 12
    D is 13
    E is 14
    F is 15

    so instead of us saying 16 goes into 237 14 times we would say it goes into it E times


    ---0-------2--------4------E
    16^4-----16^3----16^2---16^1---16^0
    65536----4096------256----16 ------ 1

    then we continue in the same fashion

    so 237-E(16) or 237-14(16)= 13
    so how many times does 1 go into 13
    13 times
    but remember what we just learned
    we can't put 13 so we put D instead
    10-A
    11-B
    12-C
    13-D


    ----0------2--------4------E-------D
    16^4----16^3----16^2---16^1---16^0
    65536----4096-----256-----16------1


    so the hex representation of 9453 is 24ED

    here are some more for practice

    173 - AD
    8923 - 22DB
    21723 - 54DB
    66123 - 024B

    ok that should do it for now i need to finish my homework
    anyway here are some final tips
    when you go from higher base to lower base
    aka say from base 10 to 2
    the number will always increase
    when you go from lower base to higher the number will always decrease

    note hopefully it will all line up
    it did when i made it
     
    Last edited: Sep 1, 2004
  6. Sep 1, 2004 #5
    Is there anyway I can easily convert Hex into base 10, base 10 to Hex, or using any base converting it to another base, similar to dduardo's method?
     
    Last edited: Sep 1, 2004
  7. Sep 1, 2004 #6
    just follow what i did
    it is the easyist way possible

    unless you want a little trick

    first convert into binary
    say

    you take 57 into binary which is 0111001
    then seperate into groups of 4 starting from the back
    0111 1001
    then you convert these small numbers into hex
    so
    0111=3
    1001=9
    and u get 39
     
  8. Sep 1, 2004 #7
    Hey thanks for all your help guys
     
  9. Sep 1, 2004 #8
    your welcome
     
  10. Sep 3, 2004 #9
    any other questions on base conversions?
     
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