Converting between number bases: binary: 1111 1111

The idea

This row gives an 8-bit binary value, 1111 1111, as the other half of a conversion task. To convert to decimal, label the columns 128, 64, 32, 16, 8, 4, 2 and 1 from left to right, then add the place values where a 1 appears. Here every column holds a 1, so 128+64+32+16+8+4+2+1 = 255. The reverse direction is equally required: to convert 255 to binary, subtract the largest place value that fits and repeat, which sets every bit and gives 1111 1111. Recognise that eight 1s represent the maximum 8-bit unsigned value, 255, and that adding one more would carry into a ninth bit.

What earns the marks

• Labels the eight bit positions with place values 128, 64, 32, 16, 8, 4, 2, 1.
• Adds the place values of the set bits to obtain 255.
• Correctly converts 255 back to 1111 1111 using the place-value method.
• Explains that 1111 1111 is the largest 8-bit unsigned value, equal to 255.

Common mistakes and how to put them right

• Reading the bits right-to-left and assigning 128 to the final bit.

  Put it right: The leftmost bit is the most significant, worth 128.

• Adding only some place values and reaching a total below 255.

  Put it right: All eight bits are 1, so all eight place values must be summed.

• Treating 1111 1111 as 256.

  Put it right: The sum of the eight place values is 255, and 256 would need a ninth bit.

Exam tips

• Write the place values above each bit before adding, to avoid misalignment.
• Show the addition of place values so partial credit can be awarded for method.
• Check the result is within the 8-bit unsigned range 0 to 255.