Be able to convert between denary and 8-bit binary numbers (0 to 255, -128…

The idea

This topic teaches conversion between denary and 8-bit binary in two ranges: unsigned 0 to 255 and signed -128 to +127. For unsigned conversion, use place values 128, 64, 32, 16, 8, 4, 2, 1. To convert denary to binary, subtract the largest place value that fits, set that bit to 1, and repeat with the remainder until it is zero; fill remaining bits with 0. For example, 200 = 128 + 64 + 8, giving 11001000. To convert binary to denary, add the place values where the bit is 1. For signed numbers, the leftmost bit has place value -128; the other bits keep 64, 32, 16, 8, 4, 2, 1. For example, 10000001 = -128 + 1 = -127. Always state whether the pattern is unsigned or signed, because the same 8 bits can mean different values.

What earns the marks

• Unsigned denary-to-binary: subtract place values 128, 64, 32, 16, 8, 4, 2, 1 from the denary number, setting a 1 where each value fits and a 0 where it does not, until the remainder is zero.
• Unsigned binary-to-denary: add the place values of the bits that are 1; for example 11001000 = 128 + 64 + 8 = 200.
• Signed binary-to-denary: treat the leftmost bit as -128 and add the place values of any other 1 bits; for example 10000001 = -128 + 1 = -127.
• Range check: unsigned 8-bit patterns represent 0 to 255; signed 8-bit patterns represent -128 to +127; state which interpretation is being used.

Common mistakes and how to put them right

• Treating the leftmost bit as +128 in a signed pattern.

  Put it right: In signed 8-bit two's complement, the leftmost bit has place value -128, so 10000000 is -128, not 128.

• Forgetting to add 1 after inverting the bits when converting a negative denary number to binary.

  Put it right: The two's complement method is invert all bits, then add 1 to the least significant bit.

• Writing binary patterns with the wrong number of bits, such as 7 bits for 200.

  Put it right: Pad to exactly 8 bits, so 200 is 11001000, not 1100100.

• Confusing the unsigned and signed ranges.

  Put it right: Unsigned 8-bit values run from 0 to 255; signed 8-bit values run from -128 to +127.

Exam tips

• Show your place-value working, especially for signed conversions, so the examiner can see how you reached the pattern.
• Check the range in the question: if it says -128 to +127, use two's complement; if it says 0 to 255, use unsigned place values.
• After converting, verify by converting back: add the place values of the 1 bits and check you get the original denary number.