Binary Arithmetic Binary operations Binary
Description: Binary Arithmetic Binary operations Binary addition Binary subtraction Binary multiplication Binary division Binary addition Rules for binary addition: Binary addition Addition of large binary numbers Solve (12)10 (8)10 (15)10 (10)10
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slide1. Binary Arithmetic<br>
slide2. Binary operations Binary addition
Binary subtraction
Binary multiplication
Binary division<br>
slide3. Binary addition Rules for binary addition:<br>
slide4. Binary addition<br>
slide5. Addition of large binary numbers<br>
slide7. Solve (12)10 + (8)10
(15)10 + (10)10
(35)10 + (48)10
(10101)2 + (10110)2
(10111)2 + (11000)2<br>
slide8. Binary Subtraction<br>
slide9. Rules for binary subtraction<br>
slide10. Binary subtraction<br>
slide11. Subtraction of large binary numbers 11001 - 10111 = 00010<br>
slide12. Examples<br>
slide13. Binary subtraction Example 1: 0011010 – 001100
Solution:
1 1 Borrow
0 0 1 1 0 1 0
(-) 0 0 1 1 0 0
——————
0 0 0 1 1 1 0
Decimal Equivalent :
0 0 1 1 0 1 0 = 26
0 0 1 1 0 0 = 12
Therefore, 26 – 12 = 14
The binary resultant 0 0 0 1 1 1 0 is equivalent to the 14<br>
slide14. Logic for binary to decimal 0 0 1 1 0 1 0 =26
26 25 24 23 22 21 20
64 32 16 8 4 2 1
16 8 2 =26<br>
slide15. Binary Subtraction Example 2: 0100010 – 0001010
Solution:
1 1 Borrow
0 1 0 0 0 1 0 = 3410
(-) 0 0 0 1 0 1 0 = 1010
——————
0 0 1 1 0 0 0 = 2410<br>
slide16. Binary subtraction subtract 1010101.10 from 1111011.11
1 borrow
1111011.11 – 1010101.10 ——————– 0100110.01 ——————–<br>
slide17. Binary subtraction using 1’s and 2’s complement method<br>
slide18. How to find 1’s Complement of given number 1’s complement of a number is found by changing all 1’s to 0’s and all 0’s to 1’s.
Ex: 1’s complement of a number 10111 is = 01000
Solve---- Find 1’s complement of
11010 = 00101
101101= 010010
1010= 0101
1111=0000
1011001=0100110<br>
slide19. How to find 2’s Complement of given number The 2’s complement of a number is obtained by adding 1 to the LSB of 1’s complement of that number
2’s complement = 1’s complement + 1
Ex: obtain 2’s complement of a number (10110010)2
Solution:<br>
slide20. Solve Find 2’s complement of following numbers.
(1101)2 0010=1=0011
(10111)2 01000+1=01001
(101101)2 010010+1=010011
(1011111)2 010000+1=010001
(101111101)2 010000010+1=010000011<br>
slide21. Subtraction using 1s complement A) For subtracting a smaller number from a larger number, the 1s complement method is as follows:
1. Determine the 1s complement of the smaller number.
2. Add the 1s complement to the larger number.
3. Remove the final carry and add it to the result. This is called the end-around carry.<br>
slide22. Binary subtraction using 1’s complement method To perform subtraction (A)2 - (B)2
Step 1: convert number to be subtracted (B)2 to its 1’s complement.
Step 2: Add first number (A)2 and 1’s complement of (B)2 using rules of binary addition.
Step 3: if final carry is 1 then add it to the result of addition obtained in step 2 to get final result.
**If final carry in step 2 is 1 then result obtained in step 2 is Positive and in its true form no conversion required.
Step 4: if final carry in step 2 is 0 then result obtained in step 2 is negative and in 1’s complement form. So convert it to its true form.<br>
slide23. Binary subtraction using 1’s complement 10---------------------------------- 10
-3 - 11-----1’complement of 11 +00
--- -----
-1 10 result
1’s complement of result 01 final result
carry 0 so result sign is negative 0<br>
slide24. 3 11 11
-2 10---------01
----------
1 0 0
1
------------
01<br>
slide25. Binary subtraction using 1’s complement 9 1001 1001
15 1111 1’s complement +0000
------- -------
-6 1001
1’s complement of result 0110 ---6
carry is 0 so result sign is negative 0<br>
slide26. Binary Subtraction Questions Using 1’s Complement Question 1:
(110101)2 – (100101)2
Solution:
(1 1 0 1 0 1)2 = 5310------- minuend.
(1 0 0 1 0 1)2 = 3710 – subtrahend
Now take the 1’s complement of the subtrahend and add with minuend.
1 carry
1 1 0 1 0 1
(+) 0 1 1 0 1 0
——————
0 0 1 1 1 1
+ 1 carry
——————
0 1 0 0 0 0
Therefore, the solution is 010000
(010000)2 = 1610 1<br>
slide27. Binary Subtraction Questions Using 1’s Complement Question 2:
(101011)2 – (111001)2 43-57= -14
Solution:
Take 1’s complement of the subtrahend
1 1 1
1 0 1 0 1 1
(+) 0 0 0 1 1 0 (1’s complement)
——————
1 1 0 0 0 1
Now take the 1’s complement of the resultant since it does not carry 1
The resultant becomes 0 0 1 1 1 0
Now, add the negative sign to the resultant value
Therefore the solution is – (001110)2.<br>
slide28. Binary subtraction using 2’s complement method To perform subtraction (A)2 - (B)2
Step 1: convert number to be subtracted (B)2 to its 2’s complement.
Step 2: Add first number (A)2 and 2’s complement of (B)2 using rules of binary addition.
Step 3: if final carry is 1 then the result Positive and in its true form no conversion required.
Step 4: if final carry in step 2 is 0 then result obtained in step 2 is negative and in 2’s complement form. So convert it to its true form.
** Carry always be discarded.<br>
slide29. Binary subtraction using 2’s complement 5 101
-7 111 1’s comple.---000
---- + 1
-2 ------
2’s complement 001
+101
-------
result 110
1’s complement 001
+ 1
--------
2’s complement 010 final result
Carry is 0 so result is negative 0<br>
slide30. 13 1101
-10 1010 0101
------- 1
03 ---------
0110
1101
--------------
1 0011<br>
slide31. Subtraction using 2’s complement 33 100001
-45 101101 1’s 010010
----- + 1
-12 ------------
2’s 010011
+100001
------------
110100 result
1’s 001011
+ 1
-------------
2’s 001100 final result 0<br>
slide32. Solve the following binary subtraction using 1’s and 2’s complement method 15-23
25-18
23-18
18-9<br>
slide34. 10111 10111
-10010 01101 1’s
1
---------
01110
10111
-----------
100101<br>
slide35. Binary Multiplication<br>
slide36. Multiplication (1 of 3) Decimal (just for fun) 35 x 105 175 000 35 3675<br>
slide37. Multiplication Binary, two 1-bit values<br>
slide38. Example Binary Multiplication A = 910 = 10012
A3A2A1A0
B = 810 = 10002
B3B2B1B0
1001
× 1000
------------------
0000 multiply by B0
+ 0000 multiply by B1
+ 0000 multiply by B2
+ 1001 multiply by B3
--------------
1001000
Cross check
1 0 0 1 0 0 0
64 32 16 8 4 2 1 place value of result bits
64+8=72<br>
slide39. Perform the following multiplication in binary number system: 10112 × 1012 1 0 1 1
× 1 0 1
---------
1 0 1 1
+ 0 0 0 0
+ 1 0 1 1
carry 1
-------------------
1 1 0 1 1 1<br>
slide40. Perform the following multiplication in binary number system: 101.12 × 11.12 1 0 1 .1
× 1 1 .1
----------------
1 0 1 1
+ 1 0 1 1
+ 1 0 1 1
Carry 1 1 1 1
---------------------
1 0 0 1 1. 0 1<br>
slide41. Multiplication Binary, two n-bit values
As with decimal values
E.g., 1110 x 1011 1110 1110 0000 1110 10011010<br>
slide42. Binary Multiplication Perform the following multiplication in binary number system: 1510 × 810
Perform the following multiplication in binary number system: 10012 × 11012
Perform the following multiplication in binary number system: 111.112 × 101.12<br>
slide43. Solve (205)10 x (3)10
(1110101)2 x (1001)2
(110)2 x (10)2
(1111101)2 x (101)2
(15)10 x (8)10<br>
slide44. Binary Division<br>
slide45. Binary Division:110 ÷10 10) 110 ( 11
- 10
----------
010
- 10
------------
000<br>
slide46. Binary division Ex: (25)10 ÷ (5)10<br>
slide47. Perform the division 111110.1 ÷101 1100.1 quotient
111110.1 5)62.5(12.5
-101 -5
-------------- ---------
0101 12.5
- 101 - 10
------------- --------
0000101 02.5
- 101 - 2.5
-------------- --------
0000000 remainder 000<br>
slide48. Solve (205)10 ÷ (3)10
(1110101)2 ÷ (1001)2
(110)2 ÷ (10)2
(1111101)2 ÷ (101)2<br>
slide50. 100)1100(11
100
---------------
0100
100
----------------
0000<br>
slide51. END<br>
slide2. Binary operations Binary addition
Binary subtraction
Binary multiplication
Binary division<br>
slide3. Binary addition Rules for binary addition:<br>
slide4. Binary addition<br>
slide5. Addition of large binary numbers<br>
slide7. Solve (12)10 + (8)10
(15)10 + (10)10
(35)10 + (48)10
(10101)2 + (10110)2
(10111)2 + (11000)2<br>
slide8. Binary Subtraction<br>
slide9. Rules for binary subtraction<br>
slide10. Binary subtraction<br>
slide11. Subtraction of large binary numbers 11001 - 10111 = 00010<br>
slide12. Examples<br>
slide13. Binary subtraction Example 1: 0011010 – 001100
Solution:
1 1 Borrow
0 0 1 1 0 1 0
(-) 0 0 1 1 0 0
——————
0 0 0 1 1 1 0
Decimal Equivalent :
0 0 1 1 0 1 0 = 26
0 0 1 1 0 0 = 12
Therefore, 26 – 12 = 14
The binary resultant 0 0 0 1 1 1 0 is equivalent to the 14<br>
slide14. Logic for binary to decimal 0 0 1 1 0 1 0 =26
26 25 24 23 22 21 20
64 32 16 8 4 2 1
16 8 2 =26<br>
slide15. Binary Subtraction Example 2: 0100010 – 0001010
Solution:
1 1 Borrow
0 1 0 0 0 1 0 = 3410
(-) 0 0 0 1 0 1 0 = 1010
——————
0 0 1 1 0 0 0 = 2410<br>
slide16. Binary subtraction subtract 1010101.10 from 1111011.11
1 borrow
1111011.11 – 1010101.10 ——————– 0100110.01 ——————–<br>
slide17. Binary subtraction using 1’s and 2’s complement method<br>
slide18. How to find 1’s Complement of given number 1’s complement of a number is found by changing all 1’s to 0’s and all 0’s to 1’s.
Ex: 1’s complement of a number 10111 is = 01000
Solve---- Find 1’s complement of
11010 = 00101
101101= 010010
1010= 0101
1111=0000
1011001=0100110<br>
slide19. How to find 2’s Complement of given number The 2’s complement of a number is obtained by adding 1 to the LSB of 1’s complement of that number
2’s complement = 1’s complement + 1
Ex: obtain 2’s complement of a number (10110010)2
Solution:<br>
slide20. Solve Find 2’s complement of following numbers.
(1101)2 0010=1=0011
(10111)2 01000+1=01001
(101101)2 010010+1=010011
(1011111)2 010000+1=010001
(101111101)2 010000010+1=010000011<br>
slide21. Subtraction using 1s complement A) For subtracting a smaller number from a larger number, the 1s complement method is as follows:
1. Determine the 1s complement of the smaller number.
2. Add the 1s complement to the larger number.
3. Remove the final carry and add it to the result. This is called the end-around carry.<br>
slide22. Binary subtraction using 1’s complement method To perform subtraction (A)2 - (B)2
Step 1: convert number to be subtracted (B)2 to its 1’s complement.
Step 2: Add first number (A)2 and 1’s complement of (B)2 using rules of binary addition.
Step 3: if final carry is 1 then add it to the result of addition obtained in step 2 to get final result.
**If final carry in step 2 is 1 then result obtained in step 2 is Positive and in its true form no conversion required.
Step 4: if final carry in step 2 is 0 then result obtained in step 2 is negative and in 1’s complement form. So convert it to its true form.<br>
slide23. Binary subtraction using 1’s complement 10---------------------------------- 10
-3 - 11-----1’complement of 11 +00
--- -----
-1 10 result
1’s complement of result 01 final result
carry 0 so result sign is negative 0<br>
slide24. 3 11 11
-2 10---------01
----------
1 0 0
1
------------
01<br>
slide25. Binary subtraction using 1’s complement 9 1001 1001
15 1111 1’s complement +0000
------- -------
-6 1001
1’s complement of result 0110 ---6
carry is 0 so result sign is negative 0<br>
slide26. Binary Subtraction Questions Using 1’s Complement Question 1:
(110101)2 – (100101)2
Solution:
(1 1 0 1 0 1)2 = 5310------- minuend.
(1 0 0 1 0 1)2 = 3710 – subtrahend
Now take the 1’s complement of the subtrahend and add with minuend.
1 carry
1 1 0 1 0 1
(+) 0 1 1 0 1 0
——————
0 0 1 1 1 1
+ 1 carry
——————
0 1 0 0 0 0
Therefore, the solution is 010000
(010000)2 = 1610 1<br>
slide27. Binary Subtraction Questions Using 1’s Complement Question 2:
(101011)2 – (111001)2 43-57= -14
Solution:
Take 1’s complement of the subtrahend
1 1 1
1 0 1 0 1 1
(+) 0 0 0 1 1 0 (1’s complement)
——————
1 1 0 0 0 1
Now take the 1’s complement of the resultant since it does not carry 1
The resultant becomes 0 0 1 1 1 0
Now, add the negative sign to the resultant value
Therefore the solution is – (001110)2.<br>
slide28. Binary subtraction using 2’s complement method To perform subtraction (A)2 - (B)2
Step 1: convert number to be subtracted (B)2 to its 2’s complement.
Step 2: Add first number (A)2 and 2’s complement of (B)2 using rules of binary addition.
Step 3: if final carry is 1 then the result Positive and in its true form no conversion required.
Step 4: if final carry in step 2 is 0 then result obtained in step 2 is negative and in 2’s complement form. So convert it to its true form.
** Carry always be discarded.<br>
slide29. Binary subtraction using 2’s complement 5 101
-7 111 1’s comple.---000
---- + 1
-2 ------
2’s complement 001
+101
-------
result 110
1’s complement 001
+ 1
--------
2’s complement 010 final result
Carry is 0 so result is negative 0<br>
slide30. 13 1101
-10 1010 0101
------- 1
03 ---------
0110
1101
--------------
1 0011<br>
slide31. Subtraction using 2’s complement 33 100001
-45 101101 1’s 010010
----- + 1
-12 ------------
2’s 010011
+100001
------------
110100 result
1’s 001011
+ 1
-------------
2’s 001100 final result 0<br>
slide32. Solve the following binary subtraction using 1’s and 2’s complement method 15-23
25-18
23-18
18-9<br>
slide34. 10111 10111
-10010 01101 1’s
1
---------
01110
10111
-----------
100101<br>
slide35. Binary Multiplication<br>
slide36. Multiplication (1 of 3) Decimal (just for fun) 35 x 105 175 000 35 3675<br>
slide37. Multiplication Binary, two 1-bit values<br>
slide38. Example Binary Multiplication A = 910 = 10012
A3A2A1A0
B = 810 = 10002
B3B2B1B0
1001
× 1000
------------------
0000 multiply by B0
+ 0000 multiply by B1
+ 0000 multiply by B2
+ 1001 multiply by B3
--------------
1001000
Cross check
1 0 0 1 0 0 0
64 32 16 8 4 2 1 place value of result bits
64+8=72<br>
slide39. Perform the following multiplication in binary number system: 10112 × 1012 1 0 1 1
× 1 0 1
---------
1 0 1 1
+ 0 0 0 0
+ 1 0 1 1
carry 1
-------------------
1 1 0 1 1 1<br>
slide40. Perform the following multiplication in binary number system: 101.12 × 11.12 1 0 1 .1
× 1 1 .1
----------------
1 0 1 1
+ 1 0 1 1
+ 1 0 1 1
Carry 1 1 1 1
---------------------
1 0 0 1 1. 0 1<br>
slide41. Multiplication Binary, two n-bit values
As with decimal values
E.g., 1110 x 1011 1110 1110 0000 1110 10011010<br>
slide42. Binary Multiplication Perform the following multiplication in binary number system: 1510 × 810
Perform the following multiplication in binary number system: 10012 × 11012
Perform the following multiplication in binary number system: 111.112 × 101.12<br>
slide43. Solve (205)10 x (3)10
(1110101)2 x (1001)2
(110)2 x (10)2
(1111101)2 x (101)2
(15)10 x (8)10<br>
slide44. Binary Division<br>
slide45. Binary Division:110 ÷10 10) 110 ( 11
- 10
----------
010
- 10
------------
000<br>
slide46. Binary division Ex: (25)10 ÷ (5)10<br>
slide47. Perform the division 111110.1 ÷101 1100.1 quotient
111110.1 5)62.5(12.5
-101 -5
-------------- ---------
0101 12.5
- 101 - 10
------------- --------
0000101 02.5
- 101 - 2.5
-------------- --------
0000000 remainder 000<br>
slide48. Solve (205)10 ÷ (3)10
(1110101)2 ÷ (1001)2
(110)2 ÷ (10)2
(1111101)2 ÷ (101)2<br>
slide50. 100)1100(11
100
---------------
0100
100
----------------
0000<br>
slide51. END<br>