Binary arithmetic: Be able to apply a binary shift to a binary number.

The idea

A binary shift moves every bit a fixed number of places left or right, filling the vacated positions with zeros and discarding bits that fall off the end. A left shift by n places multiplies the value by 2ⁿ; a right shift by n places divides by 2ⁿ, discarding any remainder. Worked method: shift 00001010 (denary 10) left by 2. Move each bit two places left, pad the right-hand end with two zeros, and drop bits pushed beyond the 8-bit width, giving 00101000 (denary 40, which is 10 × 2²). Shifting the same number right by 1 gives 00000101 (denary 5). Always state the direction, the number of places and the resulting 8-bit pattern, and check the arithmetic against the multiply-or-divide rule.

What earns the marks

• Identifies the direction of the shift (left or right) and the number of places moved.
• Moves every bit the stated number of positions and fills vacated positions with zeros.
• Discards bits shifted beyond the 8-bit width rather than wrapping them round.
• States the resulting 8-bit binary pattern correctly.
• Links a left shift by n places to multiplication by 2ⁿ and a right shift by n places to division by 2ⁿ, discarding any remainder.

Common mistakes and how to put them right

• Shifting in the wrong direction.

  Put it right: A left shift multiplies and a right shift divides, so check the direction against the intended effect before writing the answer.

• Forgetting to pad vacated positions with zeros, which changes the bit pattern and the value.

  Put it right: Every empty position must be filled with 0.

• Wrapping discarded bits round to the other end instead of dropping them.

  Put it right: Bits shifted past the 8-bit limit are lost.

• Treating a right shift as exact division when a remainder exists.

  Put it right: The remainder is discarded, so 00000111 shifted right by 1 gives 00000011.

Exam tips

• Write the original 8-bit pattern, then rewrite it with the bits physically moved before adding the padding zeros.
• Check your answer by converting both patterns to denary and confirming the multiply-or-divide relationship.
• State the direction and number of places explicitly so the method is visible, not just the final pattern.