Binary addition and binary shifts

What you'll learn

How to add two positive 8-bit binary patterns using carries.
How to recognise overflow when a result does not fit into 8 bits.
How logical shifts move bits and relate to multiplying or dividing by powers of 2.
How arithmetic shifts work with signed two's-complement values.

Before you start: 8-bit binary patterns

A bit is a binary digit: either 0 or 1. In GCSE questions, you will often work with an 8-bit pattern, such as 0011 0101. We usually group the bits into two nibbles of 4 bits to make them easier to read.

For an unsigned 8-bit integer, all bits represent positive place values:

Bit position	128	64	32	16	8	4	2	1

So unsigned 8-bit values can represent 0 to 255.

Definition

Unsigned integer

An unsigned integer is a whole number with no sign bit. In 8 bits, every bit contributes a positive place value, so the range is 0 to 255.

Example

Finding the value of an 8-bit pattern

Find the unsigned denary value of 0010 1101.

Match the 1 bits to their place values: 0010 1101 has 1s in the 32, 8, 4 and 1 columns.
Add those place values: $32 + 8 + 4 + 1 = 45$ .
Therefore, 0010 1101 represents denary 45 as an unsigned 8-bit integer.

Binary addition

Binary addition works like denary column addition, but each column can only contain 0 or 1. If a column adds up to 2 or 3, you write one bit in the answer and carry the extra value into the next column on the left.

Definition

Carry

A carry is a value passed into the next column during addition when the current column’s total is too large to fit in one bit.

The key rules are:

0 + 0 = 0
0 + 1 = 1
1 + 1 = 10, so write 0 and carry 1
1 + 1 + 1 = 11, so write 1 and carry 1

This diagram shows the column method and what overflow looks like when the result is too large for 8 bits.

8-bit binary addition showing carry bits and overflow

Column method for adding positive binary patterns

To add two positive binary patterns:

Line up the bits by place value.
Start at the right-hand side, in the 1 column.
Add the bits in that column, plus any carry from the previous column.
Write the result bit and carry if needed.
Continue left until all 8 columns have been added.

Example

Adding two positive binary patterns

Add 0011 0101 and 0001 1011.

Calculation:
0011 0101

0001 1011
= 0101 0000

Start from the right: in the 1 column, 1 + 1 gives binary 10, so write 0 and carry 1 into the 2 column.
Continue through the next columns, always adding the two bits plus the carry. Several columns also produce a carry because their total is 2 or 3.
After all 8 columns are added, the result is 0101 0000.
As a quick check, 0011 0101 is 53 and 0001 1011 is 27. $53 + 27 = 80$ , and 0101 0000 is 80.

Overflow

Because GCSE binary questions normally use fixed-size patterns, an 8-bit answer must still fit into 8 bits.

Definition

Overflow

Overflow happens when the true result of a calculation is outside the range that can be represented using the number of bits available.

For unsigned 8-bit integers, the largest value is 255, which is 1111 1111. If adding two positive binary patterns gives a carry out beyond the leftmost bit, the true answer needs more than 8 bits.

Example

Spotting overflow

Add 1111 0000 and 0011 0000 using 8 bits.

Check the denary size: 1111 0000 is 240 and 0011 0000 is 48, so $240 + 48 = 288$ .
Compare with the unsigned 8-bit maximum: $288 > 255$ , so the true result cannot fit into 8 bits.
In binary, the full result is 1 0010 0000. That leading 1 is a ninth bit, so an 8-bit register would only hold 0010 0000 and overflow has occurred.

Common Mistake

Ignoring the extra carry

If a carry goes past the leftmost bit in an 8-bit unsigned addition, do not just drop it and treat the answer as correct. That carry means overflow.

Binary shifts

A binary shift moves every bit in a pattern left or right by a fixed number of positions. Bits shifted out of the pattern are lost. The new empty positions are filled depending on the type of shift.

Logical and arithmetic 8-bit binary shifts

Definition

Binary shift

A binary shift moves all bits in a binary pattern left or right. In GCSE, you need to apply both logical shifts and arithmetic shifts.

Logical shifts

A logical shift treats the pattern simply as bits, usually as an unsigned value. Empty positions are always filled with 0s.

Logical left shift

In a logical left shift:

all bits move left
bits that fall off the left are lost
0s enter on the right

For unsigned values, shifting left by one position usually multiplies by 2. Shifting left by n positions usually multiplies by $2^n$ , as long as no important 1 bits are shifted out.

Example

Applying a logical left shift

Apply a logical left shift by 2 to 0001 0110.

Move every bit two positions to the left: 0001 0110 becomes 0101 10__ before filling the empty places.
Fill the two empty right-hand positions with 0s, giving 0101 1000.
Check the effect: 0001 0110 is 22, and $22 \times 2^2 = 88$ . 0101 1000 is 88, so the shift makes sense.

Logical right shift

In a logical right shift:

all bits move right
bits that fall off the right are lost
0s enter on the left

For unsigned values, shifting right by one position performs integer division by 2. Shifting right by n positions divides by $2^n$ , with any remainder discarded because lost bits cannot be recovered.

Example

Applying a logical right shift

Apply a logical right shift by 2 to 1011 0100.

Move every bit two positions to the right, so the two rightmost bits are shifted out and lost.
Fill the two empty left-hand positions with 0s, giving 0010 1101.
Check the effect: 1011 0100 is 180, and $180 \div 2^2 = 45$ . 0010 1101 is 45.

Common Mistake

The shortcut has limits

The multiply/divide shortcut is a useful check, but the bit movement is the real method. A left shift can overflow if 1 bits are shifted out of the left side.

Arithmetic shifts

Arithmetic shifts are used when the bit pattern represents a signed integer, usually using two's complement.

Definition

Sign bit

In an 8-bit two's-complement integer, the sign bit is the leftmost bit. A sign bit of 0 means non-negative; a sign bit of 1 means negative. The range is −128 to +127.

Arithmetic right shift

An arithmetic right shift preserves the sign of a two's-complement number.

If the sign bit is 0, fill new left positions with 0s.
If the sign bit is 1, fill new left positions with 1s.
Bits shifted out on the right are still lost.

Key Idea

Arithmetic right shifts copy the sign bit

For an arithmetic right shift, do not automatically fill with 0s. Copy the original leftmost bit into the new left-hand positions.

Example

Applying an arithmetic right shift to a negative value

Apply an arithmetic right shift by 2 to 1011 0000.

The leftmost bit is 1, so this is a negative two's-complement value. New left-hand positions must be filled with 1s.
Shift the bits two positions to the right and drop the two rightmost bits: 1011 0000 becomes __1011 00.
Fill the two empty left positions with 1s, giving 1110 1100.
Check the signed values: 1011 0000 is -80, and $−80 \div 2^2 = -20$ . 1110 1100 represents -20.

Arithmetic left shift

An arithmetic left shift uses the same bit movement as a logical left shift:

bits move left
0s enter on the right
bits shifted out on the left are lost

However, because this is a signed two's-complement value, you must be alert for overflow. If the expected result is outside −128 to +127, the shifted pattern is not a valid signed result.

Example

Checking an arithmetic left shift for overflow

Apply an arithmetic left shift by 2 to 0011 0000.

Move every bit two positions left and fill the right with 0s: 0011 0000 becomes 1100 0000.
Interpret the original as signed: 0011 0000 is +48, so a left shift by 2 should multiply by 4, giving $48 \times 2^2 = 192$ .
Compare with the signed 8-bit range. +192 is greater than +127, so overflow has occurred.
The result pattern 1100 0000 has sign bit 1, which would mean a negative value. That confirms the arithmetic left shift has not produced a valid signed answer.

Common Mistake

Using a logical right shift on a negative number

For a negative two's-complement value, a logical right shift fills with 0s and changes the sign. Use an arithmetic right shift when the number is signed and the sign must be preserved.

Tip

Quick sanity checks

Left shifts move 1s towards larger place values, so the number usually gets bigger. Right shifts move 1s towards smaller place values, so the number usually gets smaller. For arithmetic right shifts, always check what enters on the left.

Exam technique

In the exam

For addition, work from right to left and write carries clearly above the next column.
For unsigned 8-bit addition, check for overflow by looking for a carry beyond the leftmost bit or a denary result above 255.
For shifts, state whether the shift is logical or arithmetic before filling empty positions: logical fills with 0s; arithmetic right copies the sign bit.

Self review

Check yourself

What is the 8-bit result of adding 0100 1111 and 0001 0001?
Why does 1110 0000 + 0010 0000 cause overflow in unsigned 8-bit addition?
What is the difference between a logical right shift and an arithmetic right shift?

Binary addition and binary shifts

What you'll learn

How to add two positive 8-bit binary patterns using carries.
How to recognise overflow when a result does not fit into 8 bits.
How logical shifts move bits and relate to multiplying or dividing by powers of 2.
How arithmetic shifts work with signed two's-complement values.

Before you start: 8-bit binary patterns

For an unsigned 8-bit integer, all bits represent positive place values:

Bit position	128	64	32	16	8	4	2	1

So unsigned 8-bit values can represent 0 to 255.

Definition

Unsigned integer

An unsigned integer is a whole number with no sign bit. In 8 bits, every bit contributes a positive place value, so the range is 0 to 255.

Example

Finding the value of an 8-bit pattern

Find the unsigned denary value of 0010 1101.

Match the 1 bits to their place values: 0010 1101 has 1s in the 32, 8, 4 and 1 columns.
Add those place values: $32 + 8 + 4 + 1 = 45$ .
Therefore, 0010 1101 represents denary 45 as an unsigned 8-bit integer.

Binary addition

Definition

Carry

A carry is a value passed into the next column during addition when the current column’s total is too large to fit in one bit.

The key rules are:

0 + 0 = 0
0 + 1 = 1
1 + 1 = 10, so write 0 and carry 1
1 + 1 + 1 = 11, so write 1 and carry 1

This diagram shows the column method and what overflow looks like when the result is too large for 8 bits.

8-bit binary addition showing carry bits and overflow

Column method for adding positive binary patterns

To add two positive binary patterns:

Line up the bits by place value.
Start at the right-hand side, in the 1 column.
Add the bits in that column, plus any carry from the previous column.
Write the result bit and carry if needed.
Continue left until all 8 columns have been added.

Example

Adding two positive binary patterns

Add 0011 0101 and 0001 1011.

Calculation:
0011 0101

0001 1011
= 0101 0000

Start from the right: in the 1 column, 1 + 1 gives binary 10, so write 0 and carry 1 into the 2 column.
Continue through the next columns, always adding the two bits plus the carry. Several columns also produce a carry because their total is 2 or 3.
After all 8 columns are added, the result is 0101 0000.
As a quick check, 0011 0101 is 53 and 0001 1011 is 27. $53 + 27 = 80$ , and 0101 0000 is 80.

Overflow

Because GCSE binary questions normally use fixed-size patterns, an 8-bit answer must still fit into 8 bits.

Definition

Overflow

Overflow happens when the true result of a calculation is outside the range that can be represented using the number of bits available.

For unsigned 8-bit integers, the largest value is 255, which is 1111 1111. If adding two positive binary patterns gives a carry out beyond the leftmost bit, the true answer needs more than 8 bits.

Example

Spotting overflow

Add 1111 0000 and 0011 0000 using 8 bits.

Check the denary size: 1111 0000 is 240 and 0011 0000 is 48, so $240 + 48 = 288$ .
Compare with the unsigned 8-bit maximum: $288 > 255$ , so the true result cannot fit into 8 bits.
In binary, the full result is 1 0010 0000. That leading 1 is a ninth bit, so an 8-bit register would only hold 0010 0000 and overflow has occurred.

Common Mistake

Ignoring the extra carry

If a carry goes past the leftmost bit in an 8-bit unsigned addition, do not just drop it and treat the answer as correct. That carry means overflow.

Binary shifts

Logical and arithmetic 8-bit binary shifts

Definition

Binary shift

A binary shift moves all bits in a binary pattern left or right. In GCSE, you need to apply both logical shifts and arithmetic shifts.

Logical shifts

A logical shift treats the pattern simply as bits, usually as an unsigned value. Empty positions are always filled with 0s.

Logical left shift

In a logical left shift:

all bits move left
bits that fall off the left are lost
0s enter on the right

For unsigned values, shifting left by one position usually multiplies by 2. Shifting left by n positions usually multiplies by $2^n$ , as long as no important 1 bits are shifted out.

Example

Applying a logical left shift

Apply a logical left shift by 2 to 0001 0110.

Move every bit two positions to the left: 0001 0110 becomes 0101 10__ before filling the empty places.
Fill the two empty right-hand positions with 0s, giving 0101 1000.
Check the effect: 0001 0110 is 22, and $22 \times 2^2 = 88$ . 0101 1000 is 88, so the shift makes sense.

Logical right shift

In a logical right shift:

all bits move right
bits that fall off the right are lost
0s enter on the left

Example

Applying a logical right shift

Apply a logical right shift by 2 to 1011 0100.

Move every bit two positions to the right, so the two rightmost bits are shifted out and lost.
Fill the two empty left-hand positions with 0s, giving 0010 1101.
Check the effect: 1011 0100 is 180, and $180 \div 2^2 = 45$ . 0010 1101 is 45.

Common Mistake

The shortcut has limits

The multiply/divide shortcut is a useful check, but the bit movement is the real method. A left shift can overflow if 1 bits are shifted out of the left side.

Arithmetic shifts

Arithmetic shifts are used when the bit pattern represents a signed integer, usually using two's complement.

Definition

Sign bit

In an 8-bit two's-complement integer, the sign bit is the leftmost bit. A sign bit of 0 means non-negative; a sign bit of 1 means negative. The range is −128 to +127.

Arithmetic right shift

An arithmetic right shift preserves the sign of a two's-complement number.

If the sign bit is 0, fill new left positions with 0s.
If the sign bit is 1, fill new left positions with 1s.
Bits shifted out on the right are still lost.

Key Idea

Arithmetic right shifts copy the sign bit

For an arithmetic right shift, do not automatically fill with 0s. Copy the original leftmost bit into the new left-hand positions.

Example

Applying an arithmetic right shift to a negative value

Apply an arithmetic right shift by 2 to 1011 0000.

The leftmost bit is 1, so this is a negative two's-complement value. New left-hand positions must be filled with 1s.
Shift the bits two positions to the right and drop the two rightmost bits: 1011 0000 becomes __1011 00.
Fill the two empty left positions with 1s, giving 1110 1100.
Check the signed values: 1011 0000 is -80, and $−80 \div 2^2 = -20$ . 1110 1100 represents -20.

Arithmetic left shift

An arithmetic left shift uses the same bit movement as a logical left shift:

bits move left
0s enter on the right
bits shifted out on the left are lost

However, because this is a signed two's-complement value, you must be alert for overflow. If the expected result is outside −128 to +127, the shifted pattern is not a valid signed result.

Example

Checking an arithmetic left shift for overflow

Apply an arithmetic left shift by 2 to 0011 0000.

Move every bit two positions left and fill the right with 0s: 0011 0000 becomes 1100 0000.
Interpret the original as signed: 0011 0000 is +48, so a left shift by 2 should multiply by 4, giving $48 \times 2^2 = 192$ .
Compare with the signed 8-bit range. +192 is greater than +127, so overflow has occurred.
The result pattern 1100 0000 has sign bit 1, which would mean a negative value. That confirms the arithmetic left shift has not produced a valid signed answer.

Common Mistake

Using a logical right shift on a negative number

For a negative two's-complement value, a logical right shift fills with 0s and changes the sign. Use an arithmetic right shift when the number is signed and the sign must be preserved.

Tip

Quick sanity checks

Exam technique

In the exam

For addition, work from right to left and write carries clearly above the next column.
For unsigned 8-bit addition, check for overflow by looking for a carry beyond the leftmost bit or a denary result above 255.
For shifts, state whether the shift is logical or arithmetic before filling empty positions: logical fills with 0s; arithmetic right copies the sign bit.

Self review

Check yourself

What is the 8-bit result of adding 0100 1111 and 0001 0001?
Why does 1110 0000 + 0010 0000 cause overflow in unsigned 8-bit addition?
What is the difference between a logical right shift and an arithmetic right shift?

Binary addition and binary shifts

What you'll learn

Before you start: 8-bit binary patterns

Binary addition

Column method for adding positive binary patterns

Overflow

Binary shifts

Logical shifts

Logical left shift

Logical right shift

Arithmetic shifts

Arithmetic right shift

Arithmetic left shift

You've reached the end

Binary addition and binary shifts

What you'll learn

Before you start: 8-bit binary patterns

Binary addition

Column method for adding positive binary patterns

Overflow

Binary shifts

Logical shifts

Logical left shift

Logical right shift

Arithmetic shifts

Arithmetic right shift

Arithmetic left shift

You've reached the end

Binary addition and binary shifts

What you'll learn

Before you start: 8-bit binary patterns

Binary addition

Column method for adding positive binary patterns

Overflow

Binary shifts

Logical shifts

Logical left shift

Logical right shift

Arithmetic shifts

Arithmetic right shift

Arithmetic left shift

You've reached the end

Binary addition and binary shifts Revision Guide

Binary addition and binary shifts

What you'll learn

Before you start: 8-bit binary patterns

Binary addition

Column method for adding positive binary patterns

Overflow

Binary shifts

Logical shifts

Logical left shift

Logical right shift

Arithmetic shifts

Arithmetic right shift

Arithmetic left shift

You've reached the end

Binary addition and binary shifts Revision Guide