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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 1However, 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
-12831
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
-12831 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
-12831 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
-64420 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
-64420 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
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