Binary shifts can be used to perform simple multiplication/division by…

The idea

Because each bit position represents a successive power of two, moving a pattern one place left doubles its value and moving it one place right halves it. Shifting left by n places multiplies by 2ⁿ, so shifting 0001 0100 (20) left two places gives 0101 0000 (80), which is 20 × 2². Shifting right by n places divides by 2ⁿ, so shifting 0011 0000 (48) right three places gives 0000 0110 (6), which is 48 ÷ 2³. This works only for powers of two; multiplying by 3 or 5 needs addition as well as shifting. Watch for overflow when left-shifting beyond the register width and for lost precision when right-shifting discards bits.

What earns the marks

• A left shift by n places multiplies the value by 2ⁿ.
• A right shift by n places divides the value by 2ⁿ.
• Shifting 0001 0100 (20) left two places gives 0101 0000 (80), i.e. 20 × 2².
• Shifting 0011 0000 (48) right three places gives 0000 0110 (6), i.e. 48 ÷ 2³.
• Shifts handle only powers of two; other multipliers need additional operations such as addition.
• Left shifts can overflow the register width and right shifts can lose precision.

Common mistakes and how to put them right

• Reversing the directions, treating left shift as division.

  Put it right by stating left shift multiplies and right shift divides.

• Using a shift to multiply by a non-power such as 3.

  Put it right by explaining shifts only scale by powers of two.

• Ignoring bits pushed off the end.

  Put it right by noting overflow on left shift and lost precision on right shift.

Exam tips

• Work out the decimal value before and after the shift to check your answer.
• Count the number of places shifted and raise 2 to that power to confirm the multiplier or divisor.
• State clearly that the method applies to powers of two only.