Binary Addition (Edexcel GCSE Computer Science)

Binary Addition

What is binary addition?

• Binary addition is the process of adding together two binary integers (up to and including 8 bits)

• To be successful there are 5 golden rules to apply:

• Like denary addition, start from the rightmost digit and move left

• Carrying over occurs when the sum of a column is greater than 1, passing the excess to the next left column

Example 1

• Add together the binary values 1001 and 0100

• Starting from right to left, add the two binary values together applying the 5 golden rules

• If your answer has 2 digits, place the rightmost digit in the column and carry the remaining digit to the next column on the left

• In this example, start with 1+0, 1+0 = 1, so place a 1 in the column

• Repeat until all columns have a value

• The sum of adding together binary 1001 (9) and 0100 (4) is 1101 (13)

Example 2

• Add together the binary values 00011001 and 10000100

• Starting from right to left, add the two binary values together applying the 5 golden rules

• If your answer has 2 digits, place the rightmost digit in the column and carry the remaining digit to the next column on the left

• In this example, start with 1+1, 1+1 = 10, so place a 0 in the column and carry the 1 to the next column

• Repeat until all columns have a value

• The sum of adding together binary 00011001 (25) and 10001001 (137) is 10100010 (162)

Overflows

What is an overflow error?

• An overflow error occurs when the result of a binary addition exceeds the available bits

• For example, if you took binary 11111111 (255) and tried to add 00000001 (1) this would cause an overflow error as the result would need a 9th bit to represent the answer (256)

Binary Shifts

What is a logical binary shift?

• A logical binary shift is how a computer system performs basic multiplication and division on unsigned binary integers

• Binary digits are moved left or right a set number of times

• A left shift multiplies a binary number by 2 (x2)

• A right shift divides a binary number by 2 (/2)

• A shift can move more than one place at a time, the principle remains the same

• A left shift of 2 places would multiply the original binary number by 4 (x4)

How do you perform a logical left shift of 1?

• Here is the binary representation of the denary number 40

• To perform a left logical binary shift of 1, we move each bit 1 place to the left

• The digit in the 128 column (MSB) will move left causing an overflow error

• The 1 column becomes empty so is filled with a 0

• The original binary representation of denary 40 (32+8)  has multiplied by 2 and became 80 (64+16)

How do you perform a logical left shift of 2?

• Here is the binary representation of the denary number 28

• To perform a left binary shift of 2, we move each bit 2 place to the left

• The digit in the 128 (MSB) and 64 column will move left causing an overflow error

• The 1 and 2 column become empty so are filled with a 0

• The original binary representation of denary 28 (16+8+4)  has multiplied by 4 and became 112 (64+32+16)

How do you perform a logical right shift of 1?

• Here is the binary representation of the denary number 40

• To perform a right binary shift of 1, we move each bit 1 place to the right

• The digit in the 1 column (LSB) will  move right causing an underflow error

• The 128 column becomes empty so is filled with a 0

• The original binary representation of denary 40 (32+8)  has divided by 2 and became 20 (16+4)

How do you perform a logical right shift of 2?

• Here is the binary representation of the denary number 200

• To perform a right binary shift of 2, we move each bit 2 places to the right

• The digits in the 1 (LSB) and 2 columns will move right causing an underflow error

• The 128 and 64 columns become empty so are filled with a 0

• The original binary representation of denary 200 (128+64+8) has divided by 4 and became 50 (32+16+2)

What is an arithmetic binary shift?

• An arithmetic binary shift is used to divide signed binary integers by moving bits right

• Multiplication on signed binary integers by moving bits left can not be completed because the MSB is not preserved

Example

• Take the 8 bit signed binary integer

• Using a logical binary shift of 1 place right would mean the MSB moves 1 place right, leaving an empty space to be filled with a 0, changing the value from negative to positive

• This results in a lack of precision and is why this method cannot be used

• In order to successfully complete the right shift, an arithmetic binary shift should be used

• As with a logical binary shift all bits move to the right, except, the MSB after moving is copied back to it's original state

• In this example an arithmetic binary shift of 1 place right would convert -125 to -63