1.1.2 - Binary Addition and Subtraction
Aims
Show how to carry out binary addition on two binary numbers
Show how to carry out binary subtraction on two binary numbers
Be able to describe what is meant by the term overflow error and when they occur.
Computers built from electronics combinations called “Logic Gates”
Most basic operation a computer processer can perform is adding values
Binary values input into processer and logic circuits determine the result
Denary - addition result can have a value and a carry when the result is too large (more than 9)
Working right to left: | 1 | ||
|---|---|---|---|
1st, Add the units | 2 | 6 | |
2nd If > 9 Carry Tens | + | 1 | 8 |
3rd Add Tens | = | 4 | 4 |
when adding in Binary, the same rules applies but in Base-2
0+0=0
0+1=1
1+0=1
1+1=10 (Carry the 1)
1+1+1=11 (Carry the 1)
e.g.
0 | 0 | 1 | 1 | 1 | 1 | 1 | ||
|---|---|---|---|---|---|---|---|---|
0 | 0 | 0 | 1 | 1 | 1 | 0 | 1 | |
+ | 1 | 0 | 0 | 1 | 1 | 0 | 1 | 1 |
= | 1 | 0 | 1 | 1 | 1 | 0 | 0 | 0 |
Overflow Errors
Overflow is the 9th bit which occurs when the result of binary addition has a carry value on the far left column and nowhere to carry it to, making it too large for the number of bits the computer is working with
One’s and Two’s complements are used to represent negative binary numbers
To show negative binary numbers in one’s complements the bits need to be flipped (1’s to 0’s and vice versa) e.g -5 is 11010 and 5 00101
Becomes complex when used in arithmetic so Two’s complements is preferred
For two’s complements, all bits are flipped and then 1 is added
7 in binary is 00000111
bits are flipped 11111000
then +1 - 11111001
Convert number to be subtracted into negative two’s complements and then add together
To show a negative number in binary the sign (left most bit) becomes negative e.g. -128 (shown as 1)
the magnitude (other values) don’t show as negative
e.g. 1010101-0101100=0101001
1.Take the two binary numbers to be subtracted
1 | 0 | 1 | 0 | 1 | 0 | 1 |
0 | 1 | 0 | 1 | 1 | 0 | 0 |
Find the two’s complement of the second binary number
1 | 0 | 1 | 0 | 1 | 0 | 1 |
0 | 1 | 0 | 1 | 1 | 0 | 0 |
1 | 0 | 1 | 0 | 1 | 0 | 0 |
Add both binary numbers
1 | 0 | 1 | 0 | 1 | 0 | 1 | |
0 | 1 | 0 | 1 | 1 | 0 | 0 | |
Convert | 1 | 0 | 1 | 0 | 1 | 0 | 0 |
Add | 0 | 1 | 0 | 1 | 0 | 0 | 1 |
1 | 1 | 1 |
When subtracting two numbers, borrow a 1 from the next column and it will add 2 to the column
e.g. 1010101-0101100
0 | 2 | 0 | 2 | ||||
1 | 0 | 1 | 0 | 1 | 0 | 1 | |
- | 0 | 1 | 0 | 1 | 1 | 0 | 0 |
= | 0 | 1 | 0 | 1 | 0 | 0 | 1 |
0-1= borrow a one from the next left column making the current column 2 becomes 2-1
Aims
Show how to carry out binary addition on two binary numbers
Show how to carry out binary subtraction on two binary numbers
Be able to describe what is meant by the term overflow error and when they occur.
Computers built from electronics combinations called “Logic Gates”
Most basic operation a computer processer can perform is adding values
Binary values input into processer and logic circuits determine the result
Denary - addition result can have a value and a carry when the result is too large (more than 9)
Working right to left: | 1 | ||
|---|---|---|---|
1st, Add the units | 2 | 6 | |
2nd If > 9 Carry Tens | + | 1 | 8 |
3rd Add Tens | = | 4 | 4 |
when adding in Binary, the same rules applies but in Base-2
0+0=0
0+1=1
1+0=1
1+1=10 (Carry the 1)
1+1+1=11 (Carry the 1)
e.g.
0 | 0 | 1 | 1 | 1 | 1 | 1 | ||
|---|---|---|---|---|---|---|---|---|
0 | 0 | 0 | 1 | 1 | 1 | 0 | 1 | |
+ | 1 | 0 | 0 | 1 | 1 | 0 | 1 | 1 |
= | 1 | 0 | 1 | 1 | 1 | 0 | 0 | 0 |
Overflow Errors
Overflow is the 9th bit which occurs when the result of binary addition has a carry value on the far left column and nowhere to carry it to, making it too large for the number of bits the computer is working with
One’s and Two’s complements are used to represent negative binary numbers
To show negative binary numbers in one’s complements the bits need to be flipped (1’s to 0’s and vice versa) e.g -5 is 11010 and 5 00101
Becomes complex when used in arithmetic so Two’s complements is preferred
For two’s complements, all bits are flipped and then 1 is added
7 in binary is 00000111
bits are flipped 11111000
then +1 - 11111001
Convert number to be subtracted into negative two’s complements and then add together
To show a negative number in binary the sign (left most bit) becomes negative e.g. -128 (shown as 1)
the magnitude (other values) don’t show as negative
e.g. 1010101-0101100=0101001
1.Take the two binary numbers to be subtracted
1 | 0 | 1 | 0 | 1 | 0 | 1 |
0 | 1 | 0 | 1 | 1 | 0 | 0 |
Find the two’s complement of the second binary number
1 | 0 | 1 | 0 | 1 | 0 | 1 |
0 | 1 | 0 | 1 | 1 | 0 | 0 |
1 | 0 | 1 | 0 | 1 | 0 | 0 |
Add both binary numbers
1 | 0 | 1 | 0 | 1 | 0 | 1 | |
0 | 1 | 0 | 1 | 1 | 0 | 0 | |
Convert | 1 | 0 | 1 | 0 | 1 | 0 | 0 |
Add | 0 | 1 | 0 | 1 | 0 | 0 | 1 |
1 | 1 | 1 |
When subtracting two numbers, borrow a 1 from the next column and it will add 2 to the column
e.g. 1010101-0101100
0 | 2 | 0 | 2 | ||||
1 | 0 | 1 | 0 | 1 | 0 | 1 | |
- | 0 | 1 | 0 | 1 | 1 | 0 | 0 |
= | 0 | 1 | 0 | 1 | 0 | 0 | 1 |
0-1= borrow a one from the next left column making the current column 2 becomes 2-1
