Understand unsigned and signed binary using two's complement, binary addition and subtraction, fixed point and floating point representation of real numbers, and the effects of overflow and rounding.

What this dot point is asking

AQA wants you to represent unsigned and signed integers (two's complement), add and subtract in binary, represent real numbers in fixed and floating point, and explain overflow and rounding errors with worked arithmetic.

Unsigned and signed (two's complement) integers

To find the two's complement (the negative) of a number: invert all the bits and add 1. So $+5 = 0000\,0101$, inverted is $1111\,1010$, add 1 gives $1111\,1011 = -5$. A quick check: a two's complement number is negative exactly when its most significant bit is 1, and the range of an $n$-bit two's complement value is $-2^{n-1}$ to $2^{n-1} - 1$, so 8 bits cover $-128$ to $+127$. There is one more negative value than positive because zero takes a positive-looking pattern.

Binary addition and subtraction

Binary addition follows the carry rules ($1 + 1 = 10$, carry the 1; $1 + 1 + 1 = 11$). Subtraction is performed by adding the two's complement of the number being subtracted, so $A - B = A + (-B)$, reusing the adder rather than building separate subtraction hardware.

  0011 (3)
+ 1110 (-2 in two's complement)
------
  0001 (1)   the final carry out is discarded

The carry out of the most significant column is simply discarded in two's complement subtraction; it is not an error. An error (overflow) is instead signalled when adding two numbers of the same sign yields a result of the opposite sign.

Fixed point, floating point, overflow and rounding

Overflow occurs when an arithmetic result is too large to fit in the available bits, producing an incorrect (often wrong-signed) answer. Underflow is the related problem when a floating point value is too small (too close to zero) to represent. Rounding errors occur because many decimal fractions (like $0.1$) cannot be represented exactly in binary, so the stored value is slightly off, and these errors can accumulate over many calculations, which matters in long financial or scientific computations.