Converting between number bases: decimal: 255

The idea

This row gives a decimal value, 255, as one half of a conversion task: you must be able to move between decimal and binary in both directions. To convert 255 to binary, use place values 128, 64, 32, 16, 8, 4, 2 and 1. Take 128 (remainder 127), then 64 (63), 32 (31), 16 (15), 8 (7), 4 (3), 2 (1) and 1 (0), so every place is set and the result is 1111 1111. Check by adding the place values back: 128+64+32+16+8+4+2+1 = 255. Recognise 255 as the largest 8-bit unsigned value, so all eight bits are 1. The reverse direction matters too: given 1111 1111, sum the place values to recover 255.

What earns the marks

• Recognises the 8-bit place values 128, 64, 32, 16, 8, 4, 2, 1 and that 255 requires all of them set.
• Correctly converts 255 to the 8-bit binary 1111 1111.
• Verifies the conversion by summing place values to return 255.
• States that 255 is the maximum value representable in 8 unsigned bits, so all bits are 1.

Common mistakes and how to put them right

• Writing the binary digits in the wrong order (least significant bit first).

  Put it right: Place 128 on the left and 1 on the right, giving 1111 1111.

• Assuming 255 needs nine bits.

  Put it right: Eight bits hold 0 to 255 inclusive, so eight 1s suffice.

• Confusing 255 with 256 and expecting a carry into a ninth bit.

  Put it right: 256 would require a ninth bit, but 255 does not.

Exam tips

• Write the place-value headings above the bits before converting, so each 1 or 0 is justified.
• Always check your answer by adding the place values back to the original decimal.
• Group binary digits in fours (1111 1111) to reduce reading errors.