Data storage: Binary shifts

The idea

A binary shift moves every bit in a binary pattern a stated number of places left or right. A left shift by n places multiplies the value by 2ⁿ, because each bit moves to a place value n positions higher. A right shift by n places divides the value by 2ⁿ, discarding any bits shifted past the right-hand end and losing fractional parts. For example, 00010110 (22) shifted left by 2 gives 01011000 (88), which is 22 × 2². The same pattern shifted right by 1 gives 00001011 (11), which is 22 ÷ 2. Bits shifted out of the fixed-width register are lost, and zeros fill the vacated positions. This means a left shift can cause an overflow error if significant bits are lost, changing the value unexpectedly, while a right shift causes a loss of precision.

What earns the marks

• Moving every bit the stated number of positions in the stated direction, keeping the remaining bits in their relative order.
• Filling vacated positions with zeros rather than copying existing bits or leaving gaps.
• Recognising that a left shift by n places multiplies the original value by 2ⁿ, for example a left shift by 3 multiplies by 8.
• Recognising that a right shift by n places divides the original value by 2ⁿ, discarding any remainder, for example a right shift by 2 divides by 4.
• Identifying bits shifted beyond the available width as lost, which can cause overflow on a left shift or loss of precision on a right shift.
• Calculating the new denary value by applying the multiplication or division, or by re-evaluating the place values of the shifted pattern.

Common mistakes and how to put them right

• Shifting in the wrong direction, which changes multiplication into division.

  Put it right by checking whether the value should increase (left) or decrease (right).

• Forgetting to fill empty positions with zeros, which produces a pattern with missing bits.

  Put it right by writing the full-width result with zeros in every vacated position.

• Assuming a right shift rounds to the nearest integer, when it truncates the fractional part.

  Put it right by discarding the remainder, so 00000101 (5) shifted right by 1 gives 00000010 (2).

Exam tips

• Write the original pattern with place values above the bits so you can track each bit as it moves.
• State the arithmetic effect as well as the new pattern, for example left shift by 2 multiplies by 4, to show full understanding.
• Check for overflow by comparing the number of bits needed for the new value with the fixed width of the register.