Binary and data representation - EdexcelBinary shifts and negative numbers

There are two different types of binary shift which work differently for negative numbers. The two types are:

• arithmetic
• logical

Arithmetic shift left

To multiply a negative number, an arithmetic binary shift moves all the digits in the binary number along to the left and fills the gaps after the shift with 0:

• to multiply by two, all digits shift one place to the left
• to multiply by four, all digits shift two places to the left
• to multiply by eight, all digits shift three places to the left
• and so on

Example: 00011100 (denary -128 + 64 + 16 + 8 = -40) × 2

Result: shifting one place to the left gives 00111000 (-128 + 32 + 16 = denary -80)

Arithmetic shift right

To divide a negative number, an arithmetic binary shift moves all the digits in the binary number along to the right and fills the gaps after the shift with the previous MSB value:

• to divide by two, all digits shift one place to the right
• to divide by four, all digits shift two places to the right
• to divide by eight, all digits shift three places to the right
• and so on

Example: 10001100 (denary -116) ÷ 2

Result: (1 × -128) + (1 × 64) + (1 × 4) + (1 × 2) = denary -58

Two’s complement is therefore better for arithmetic shifts left since the sign is always retained.

Logical shifts

In logical shifts - left or right - a 0 is always copied in. This means that a logical shift is ideal for unsigned binary numbers.

Example: 10011000 (denary 152) ÷ 2

Result: (1 × 64) + (1 × 8) + (1 × 4) = denary 76