binary numbers ... binary notation → 11000000 . 10101000 . 10000011 . 01101001 to convert...

Click here to load reader

Post on 24-Apr-2020

3 views

Category:

Documents

0 download

Embed Size (px)

TRANSCRIPT

  • BINARY NUMBERS

    Converting decimal numbers to binary numbers

    What are binary

    numbers and why do

    we use them?

  • Thousands Hundreds Tens ones

    4 3 5 1

    The number system we commonly use is decimal numbers, also known as

    Base 10. Ones, tens, hundreds, and thousands.

    For example, 4351 represents 4 thousands, 3 hundreds, 5 tens, and 1 ones.

  • Thousands Hundreds Tens ones

    4 3 5 1 However, a computer does

    not understand decimal

    numbers. It only understands

    “on and off,” “yes and no.”

  • Thousands Hundreds Tens ones

    4 3 5 1

    In order to convey “yes and

    no” to a computer, we use the

    numbers one (“yes” or “on”)

    and zero (“no” or “off”).

  • DECIMAL NUMBERS (BASE 10)

    4351

    4x1000 3x100 5x10 1x1

    To break it down further, the

    number 4351 represents

    1 times 1, 5 times 10,

    3 times 100, and 4 times 1000.

    Each step to the left is another

    multiplication of 10. This is why

    it is called Base 10, or decimal

    numbers. The prefix dec-

    means ten.

  • DECIMAL NUMBERS (BASE 10)

    4351

    4x1000 3x100 5x10 1x1

    103=1000 102=100 101=10 100=1

    One is 10 to the zero power.

    Anything raised to the zero power

    is one.

    Ten is 10 to the first power (or 10).

    One hundred is 10 to the second

    power (or 10 times 10).

    One thousand is 10 to the third

    power (or 10 times 10 times 10).

  • Base 10

    103 102 101 100

    1000 100 10 1

    Base 2

    23 22 21 20

    8 4 2 1

    Binary numbers, or Base 2,

    use the number 2 instead

    of the number 10.

    The prefix bi- means two.

  • Base 10

    103 102 101 100

    1000 100 10 1

    Base 2

    23 22 21 20

    8 4 2 1

    Two raised to the zero

    power is one.

    Two raised to the first

    power is two.

    Two raised to the second

    power is four (or 2 times 2).

    Two raised to the third

    power is eight

    (or 2 times 2 times 2).

  • Base 2

    23 22 21 20

    8 4 2 1

    27 26 25 24

    128 64 32 16

    BINARY NUMBERS (BASE 2)

    And so on…

    Eight times two is

    sixteen, or two to the

    fourth power.

    Sixteen times two is

    thirty-two, or two to

    the fifth power.

  • Base 2

    23 22 21 20

    8 4 2 1

    27 26 25 24

    128 64 32 16

    BINARY NUMBERS (BASE 2) Thirty-two times two

    is sixty-four, or two to

    the sixth power.

    And sixty-four times

    two is one hundred

    twenty eight, or two

    to the seventh power.

  • DECIMAL

    15

    1

    101=10

    5

    100=1

    BINARY

    15

    1

    23=8

    1

    22=4

    1

    21=2

    1

    20=1

    15 1111

    The number fifteen is written

    in decimal as one ten and five

    ones. In binary, the number fifteen

    is written as one eight, one four,

    one two, and one one.

    These are called bits, and they are

    either one (on) or zero (off).

  • 8 BITS = 1 BYTE = 1 OCTET

    27 26 25 24 23 22 21 20

    128 64 32 16 8 4 2 1

    Eight bits make a byte.

    This is also known as an octet.

    When you see an IP address, it is

    made up of four octets (or 32 bits).

  • 8 BITS = 1 BYTE = 1 OCTET

    27 26 25 24 23 22 21 20

    128 64 32 16 8 4 2 1

    x x x x x x x x

    0 0 0 0 0 0 0 0

    = = = = = = = =

    0 + 0 + 0 + 0 + 0 + 0 + 0 + 0

    = 0

    If every bit is a zero…that’s eight zeros…

    and we multiply each power of two by zero,

    and add them up…

    the decimal equivalent of that octet is zero.

  • 8 BITS = 1 BYTE = 1 OCTET

    27 26 25 24 23 22 21 20

    128 64 32 16 8 4 2 1

    x x x x x x x x

    0 0 0 0 0 0 0 0

    = = = = = = = =

    0 + 0 + 0 + 0 + 0 + 0 + 0 + 0

    = 0

    x x x x x x x x

    1 1 1 1 1 1 1 1

    = = = = = = = =

    128 + 64 + 32 + 16 + 8 + 4 + 2 + 1

    = 255

    If every bit is a one…that’s eight ones…

    and we multiply each power of two by one,

    and add them up…

    the decimal equivalent is two hundred and

    fifty-five.

    Therefore, each octet can have a value

    between 0 and 255.

  • Decimal notation → 192 . 168 . 131 . 106

    Binary notation → 11000000 . 10101000 . 10000011 . 01101001

    Let’s look at an IP address.

    It is easier for us to recognize decimal numbers, so

    we write the IP address as 192.168.131.106.

    However, a computer sees the IP address in binary

    notation as four octets of ones and zeros.

  • 27 26 25 24 23 22 21 20

    128 64 32 16 8 4 2 1

    Binary notation → 11000000 . 10101000 . 10000011 . 01101001

    To convert binary numbers to decimal numbers, we use

    the powers of two again.

    Write the octet below…one in the 128 column, one in

    the sixty-four column, and zeros for the rest.

    1 1 0 0 0 0 0 0

  • 128 64 32 16 8 4 2 1

    1 1 0 0 0 0 0 0

    128 + 64 + 0 + 0 + 0 + 0 + 0 + 0 =

    Binary notation → 11000000 . 10101000 . 10000011 . 01101001

    196

    Then multiply each column

    and add across…128 plus 64

    plus zero equals 196.

  • 128 64 32 16 8 4 2 1

    1 0 1 0 1 0 0 0

    128 + 0 + 32 + 0 + 8 + 0 + 0 + 0 =

    Binary notation → 11000000 . 10101000 . 10000011 . 01101001

    Decimal notation → 196

    168

    Write the second octet, multiply down and add across.

    128 plus 0 plus 32 plus 8 plus 0 equals 168.

  • Decimal notation → 192 . 168 . 131

    128 64 32 16 8 4 2 1

    Binary notation → 11000000 . 10101000 .

    131

    -128 31

    Now we’ll convert the other way…from decimal to

    binary…for the third and fourth octets.

    To convert 131 to binary…we start from the left.

    Can we subtract 128 from 131?

    Yes. So we put a one in the 128 column, and we are

    left with three.

  • Decimal notation → 192 . 168 . 131

    128 64 32 16 8 4 2 1

    Binary notation → 11000000 . 10101000 .

    131

    -128 31 0 0 0 0 0

    Can we subtract 64 from 3? No. So we put a

    zero in the 64 column.

    Can we subtract 32 from 3? No. Another

    zero for the 32 column.

    Zero in the 16 column, the 8 column, and the

    four column.

  • Decimal notation → 192 . 168 . 131

    128 64 32 16 8 4 2 1

    Binary notation → 11000000 . 10101000 .

    131

    -128 31 0 0 0 0 0 1 1 -2

    1 -1

    0

    10000011 .

    Can we subtract a 2 from 3? Yes, and

    we put a one in the two column. We

    are left with one in the one column.

    So 131 in binary is 10000011.

  • Decimal notation → 192 . 168 . 131 . 106

    128 64 32 16 8 4 2 1

    Binary notation → 11000000 . 10101000 .

    106

    -64 420 1 1

    -32

    10

    10000011 .

    Now we’ll convert the fourth octets. Starting from the left.

    Can we subtract 128 from 106? No.

    Can we subtract 64 from 106? Yes, and we are left with 42.

    Can we subtract 32 from 42? Yes, leaving 10.

  • Decimal notation → 192 . 168 . 131 . 106

    128 64 32 16 8 4 2 1

    Binary notation → 11000000 . 10101000 .

    106

    -64 420 1 1 0 1 0 1 0

    -32

    10 -8

    2

    10000011 . 01101010

    Can we subtract 16 from 10? No.

    Can we subtract 8 from 10? Yes, leaving 2.

    Can we subtract 4 from 2? No.

    Can we subtract a 2 from 2? Yes, leaving 0.

    So, 106 written in binary is 01101010.

  • I hope this has helped you understand a little bit

    about converting binary numbers.

    Thanks for watching!

    North Campus Learning Lab

    Room NA-113i

View more