George Boole published a paper in 1854 detailing Boolean algebra, which proved instrumental in the development of the binary system used in electronics. Binary uses only two digits, 0 and 1, rather than the base-10 system of decimals. Computers store information using binary, with each bit (0 or 1) representing a single character and eight bits equaling one byte.
2. Binary
In 1854, British mathematicianIn 1854, British mathematician
George BooleGeorge Boole published a paperpublished a paper
detailing a system of logic that woulddetailing a system of logic that would
become known asbecome known as Boolean algebraBoolean algebra..
His logical system proved instrumentalHis logical system proved instrumental
in the development of the binaryin the development of the binary
system, particularly in itssystem, particularly in its
implementation in electronic circuitry.implementation in electronic circuitry.
3. Binary
A numbering systems that only usesA numbering systems that only uses
two digits.two digits. 00 andand 11..
Rather than a base ten that we are allRather than a base ten that we are all
familiar with.familiar with.
Computers use binary to storeComputers use binary to store
information in a digital format.information in a digital format.
Each digit (Each digit ( 00 oror 11) represents one bit) represents one bit
Eight bits are equal to one byte.Eight bits are equal to one byte.
4. Bit
One Binary Digit
abbreviation is “b”
Can be thought of as one character
Either a 1 or a 0
5. Byte
Eight bits make up one byte
Abbreviation “B”
Combination of 1’s and 0’s
Can be thought of as one character
11101010
10. terabyte
Represented by TB
Slang “tera”
Equal to 1,000,000,000,000 Bytes
One Trillion bytes
240
11. petabyte
Represented by PB
Slang “peta”
Equal to 1,000,000,000,000,000 Bytes
One Thousand Trillion bytes
250
12. exabyte
Represented by EB
Slang “exa”
Equal to 1,000,000,000,000,000,000 Bytes
One Million Trillion bytes
260
Allprintedmaterialintheworld
would use about 5 Exabytes
14. All eight of these Light bulbs would represent one byte
One Light bulb represents oneOne Light bulb represents one
BitBit
15. Think of Binary as light bulbs
that are either ON
or Off
11 00 00 00 00 00 00 1111 00
16. Binary ExerciseBinary Exercise
Bit Postion Bit 8 Bit 7 Bit 6 Bit 5 Bit 4 Bit 3 Bit 2 Bit 1 Total Binary Value
Position Value
if ON
128 64 32 16 8 4 2 1 255
Position Value
if OFF
0 0 0 0 0 0 0 0 0
Turn a Postion
ON
1 0 0 0 0 0 0 1
Here we would
ADD
The Postion Value
Here we bring
The Postion
Value
DOWN
128 1 129
In this incidence our binary Number 10000001 would have a value of 129
Because Postion 8 is ON Postion 7 is OFF Postion 6 is OFF Position 5 is OFF Position 4 is Off
Postion 3 is OFF Postion 2 is OFF and Position 1 is ON.
Binary Exercise
17. Binary
Figuring Binary.
Starting on the right going to the left
The first digit will be 1
The second digit will be 2
The third digit will be 4
The fourth digit will be 8
The fifth digit will be 16
The sixth digit will be 32
The seventh digit will be 64
The eighth digit will be 128
20. Example
10000000
The 1st
– 7th
digit would be Off
The Eighth digit would be On
The first digit will be 1 0
The second digit will be 2 0
The third digit will be 4 0
The fourth digit will be 8 0
The fifth digit will be 16 0
The sixth digit will be 32 0
The seventh digit will be 64 0
The eighth digit will be 128 +128
Add the bits
The value of the number would be Total 128
21. Example
10000001
The 1st
digit would be On
The 2nd
– 7th
digit would be Off
The Eighth digit would be On
The first digit will be 1 1
The second digit will be 2 0
The third digit will be 4 0
The fourth digit will be 8 0
The fifth digit will be 16 0
The sixth digit will be 32 0
The seventh digit will be 64 0
The eighth digit will be 128 +128
Add the bits
The value of the number would be Total 129
22. Example
10000011
The 1st
digit would be On
The 2nd
digit would be On
The 3rd
– 7th
digit would be Off
The Eighth digit would be On
The first digit will be 1 1
The second digit will be 2 2
The third digit will be 4 0
The fourth digit will be 8 0
The fifth digit will be 16 0
The sixth digit will be 32 0
The seventh digit will be 64 0
The eighth digit will be 128 +128
Add the bits
The value of the number would be Total 131
23. Example
10000111
The 1st-
3rd
digit would be On
The 4th
– 7th
digit would be Off
The Eighth digit would be On
The first digit will be 1 1
The second digit will be 2 2
The third digit will be 4 4
The fourth digit will be 8 0
The fifth digit will be 16 0
The sixth digit will be 32 0
The seventh digit will be 64 0
The eighth digit will be 128 +128
Add the bits
The value of the number would be Total 135
24. Example
11000000
The 1st-
6th
digit would be Off
The 7th
digit would be On
The 8th
digit would be On
The first digit will be 1 0
The second digit will be 2 0
The third digit will be 4 0
The fourth digit will be 8 0
The fifth digit will be 16 0
The sixth digit will be 32 0
The seventh digit will be 64 64
The eighth digit will be 128 +128
Add the bits
The value of the number would be Total 192
25. Think of Binary as light bulbs
that are either ON
or Off
11 11 00 00 00 00 00 00
What is theWhat is the
value?value?
192192
26. Example
11111111
The 1st-
8th
digit would be On
The first digit will be 1 1
The second digit will be 2 2
The third digit will be 4 4
The fourth digit will be 8 8
The fifth digit will be 16 16
The sixth digit will be 32 32
The seventh digit will be 64 64
The eighth digit will be 128 +128
Add the bits
The value of the number would be Total 255
27. Think of Binary as light bulbs
that are either ON
or Off
11 11 11 11 11 11 11 11
What is theWhat is the
value?value?
255255
128128 6464 3232 1616 88 44 22 11
28. Using Calculator
to figure
Binary Numbers
First we would open Calculator
Start/All Programs/Accessories/Calculator
From the Calculator go to View and down
To SCIENTIFIC
41. Decimal to Binary
It follows a starightforward method.
It involves dividing the number to be
converted, say N by 2 (since binary is in base
2) until we reach the division of (1/2), also
making note of all remainders.
42. Example 1: Convert 98 from
decimal to binary
Divide 98 by 2, make note of all the
remainder.
Continue dividingquotientsby 2, making
notes of the remainder.
Also, note the star beside the last remainder.
43. Division Remainder, R
98/2 = 49 R=0
49/2 = 24 R=1
24/2 = 12 R=0
12/2 = 6 R=0
6/2 = 3 R=0
3/2 = 1 R=1
1/2 = 0 R=1
The sequance of remainders going up gives the answer.
Starting from 1*, we have 1100010.
Therefore, 98 in decimals is 1100010 in binary
44. Example 2: Convert 21 into
binary
Division Remainder, R
21/2 = 10 R=1
10/2 = 5 R=0
5/2 = 2 R=1
2/2 = 1 R=0
1/2 = 0 R=1
Therefore, 21 in decimals is 10101 in binary
45. Binary to decimal
Conversion follows the same steps as decimal
to binary, except in reverse order.
We can begin by multiplying 0 x 2 and adding
1.
We continue to multiply the numbers in the
“results” column by 2, and adding the digits
from left to right in our binary numbers.
46. Example 1: Convert 11101
from binary to decimal
Operations Result
0 x 2 + 1 1
1 x 2 + 1 3
3 x 2 + 1 7
7 x 2 + 0 14
14 x 2 + 1 29
Therefore, 11101 in binary is 29 in decimal.