2. Number System
Used by Used in
System Base Symbols humans? computers?
Decimal 10 0, 1, … 9 Yes No
Binary 2 0, 1 No Yes
Octal 8 0, 1, … 7 No No
Hexa- 16 0, 1, … 9, No No
decimal A, B, … F
3. Binary?
– Uses only two digits, 0 and 1
– It is base or radix of 2
In State 0 In state 1
4. Binary?
• Each digit has a value depending on its
position:
102 = (1x21)+(0x20) = 210
112 = (1x21)+(1x20) = 310
1002 = (1x22)+ (0x21)+(0x20) = 410
5. Why Binary ?
• digital
on" and "off“ digits – 0 and 1
• binary use more storage than decimal
• Easier to handle 2-digits for circuits,
transistors i.e (1,0) rather then more
6. Why Binary?
• Recall: we can use numbers to represent
marital status information:
• 0 = single
• 1 = married
• 2 = divorced
• 3 = widowed
7. Binary Addition Rules
Rules:
0+0 =0
0+1 =1
1+0 =1 (just like in decimal)
1+1 = 210
= 102 = 0 with 1 to carry
1+1+1 = 310
= 112 = 1 with 1 to carry
9. Decimal Addition Explanation
What just happened?
111 1 1 1 (carry)
3758 3 7 5 8
+4 6 5 7
+ 4657 -
8 14 11 15 (sum)
10 10 10 (subtract the base)
8 4 1 5
8415
So when the sum of a column is equal to or greater than the base, we
subtract the base from the sum, record the difference, and carry one to the
next column to the left.
10. Binary Addition Example 1
Col 1) Add 1 + 0 = 1 Write 1
Example 1: Add
binary 110111 to 11100 Col 2) Add 1 + 0 = Write
1
Col 3) Add 1 + 1 = 2 (10 in binary)
Write 0, carry 1
Col 4) Add 1+ 0 + 1 = 2
Write 0, carry 1
1 1 1 1
1 1 0 1 1 1 Col 5) Add 1 + 1 + 1 = 3 (11 in binary)
Write 1, carry 1
+ 0 1 1 1 0 0 Col 6) Add 1 + 1 + 0 = 2
10 1 00 1 1 Write 0, carry 1
Col 7) Bring down the carried 1
Write 1
11. Binary Addition Explanation
What is actually In the first two columns,
happened when we there were no carries.
carried in binary? In column 3, we add 1 + 1 = 2
Since 2 is equal to the base, subtract
the base from the sum and carry 1.
In column 4, we also subtract
1 1 1 1 the base from the sum and carry 1.
1 1 01 1 1 In column 5, we also subtract
the base from the sum and carry 1.
+ 0 1 11 0 0
In column 6, we also subtract
2 3 22 the base from the sum and carry 1.
- 2 2 22 . In column 7, we just bring down the
carried 1
1 0 1 0 0 1 1
12. Binary Addition Verification
You can always check your Verification
answer by converting the 1101112 5510
figures to decimal, doing the +0111002 + 2810
addition, and comparing the 8310
answers.
64 32 16 8 4 2 1
1 0 1 0 0 1 1
1 1 0 1 1 1 = 64 + 16 + 2 +1
+ 0 1 1 1 0 0 = 8310
1 0 1 0 0 1 1
19. Positive / Negative Combination
9 00001001
Positive / Negative
Positive Answer
+ (-5) + 11111011
4 1]00000100
1-Positive / 1-Negative
8th Bit = 0 : Answer is Positive
Take 2’s Complement Disregard 9th Bit
Of Negative Number (-5)
00000101
2’s
11111010 Complement
Process
+1
11111011 19
20. Negative / Positive Combination
(-9) 11110111
Positive / Negative
Negative Answer
+ 5 + 00000101
- 4 11111100
1-Positive / 1-Negative
8th Bit = 1 : Answer is Negative
Take 2’s Complement Take 2’s Complement to Check Answer
Of Negative Number (-9)
11111100 00001001
2’s 2’s
Complement 00000011 11110110 Complement
Process Process
+1 +1
00000100 11110111 20
21. Negative / Negative Combination
2’s Complement
(-9) 11110111 Numbers, See
Conversion Process
Negative / Negative
Negative Answer
+ (-5) + 11111011 In Previous Slides
- 14 1]11110010
2-Negative
Take 2’s Complement Of 8th Bit = 1 : Answer is Negative
Disregard 9th Bit
Both Negative Numbers Take 2’s Complement to Check Answer
11110010
2’s
Complement 00001101
Process +1
00001110 21
22. 2’S Complement Quick Method
Example: 11101100
1) Start at the LSB and write down all zeros moving
to the left.
2) Write down the first “1” you come to.
3) Invert the rest of the bits moving to the left.
0 001 0 1 0 0
22
24. Binary Subtraction
Explanation
In binary, the base unit is 2
So when you cannot subtract, you borrow from the
column to the left.
The amount borrowed is 2.
The 2 is added to the original column value, so
you will be able to subtract.
25. Binary Subtraction
Example 1
Col 1) Subtract 1 – 0 = 1
Example 1: Subtract Col 2) Subtract 1 – 0 = 1
binary 11100 from 110011 Col 3) Try to subtract 0 – 1 can’t.
Must borrow 2 from next column.
But next column is 0, so must go to
column after next to borrow.
2 1 Add the borrowed 2 to the 0 on the right.
0 0 2 2 Now you can borrow from this column
(leaving 1 remaining).
1 1 0 0 1 1 Add the borrowed 2 to the original 0.
Then subtract 2 – 1 = 1
- 1 1 1 0 0 Col 4) Subtract 1 – 1 = 0
1 0 1 1 1 Col 5) Try to subtract 0 – 1 can’t.
Must borrow from next column.
Add the borrowed 2 to the remaining 0.
Then subtract 2 – 1 = 1
Col 6) Remaining leading 0 can be ignored.
