Binary number system study guide

1. Identify the representation before calculating

Start by deciding whether the question specifies unsigned binary, two's complement, fixed point or floating point. The same bit pattern can have different meanings under different representations. For unsigned binary, add positive powers of two. For two's complement, give the most significant bit a negative place value. For fixed point, determine where the binary point is located. For floating point, separate the mantissa and exponent and apply the exponent as a binary-point shift.

2. Work systematically with ranges

For unsigned n-bit data, write the endpoints as 0 and 2^n - 1. For signed two's complement data, write -2^(n-1) to 2^(n-1)-1. These formulas provide a quick check on conversions. If a proposed answer lies outside the range, the representation cannot store it and overflow is relevant. Do not describe a value as negative merely because its first bit is 1 until you know that the representation is signed.

3. Use place values to check conversions

For an integer, make a place-value row before converting. For a binary fraction, continue the row to the right of the point: 2^-1, 2^-2, 2^-3 and so on. Add only the values associated with 1 bits. When converting in the other direction, select powers of two or negative powers of two whose sum gives the target value. If the required fraction does not terminate within the allowed number of bits, identify the need for rounding and explain that the stored value is not exact.

4. Two's complement subtraction

To subtract B from A, represent both values with the same number of bits, form the two's complement of B, and add it to A. Keep the calculation within the stated bit width and check whether the mathematical result is inside the signed range. A result outside that range indicates signed overflow; the bit pattern should not be interpreted as the intended mathematical answer.

5. Floating point reasoning

Treat a floating-point value as a mantissa and an exponent, not as two independent integers to add. Decode each field as two's complement, then apply the exponent's binary-point movement to the mantissa. For conversion in the other direction, first obtain the binary value, then choose a mantissa and exponent that represent it in the given field sizes. Normalisation shifts the binary point while adjusting the exponent so the value remains unchanged. Check that the normalised mantissa and exponent fit their fields.

6. Explain range and precision separately

Range concerns the smallest and largest magnitudes that can be represented. Precision concerns the detail or closeness of representable values. Fixed point allocates a fixed number of bits to the fractional part, giving predictable precision but a range constrained by the integer part. Floating point uses an exponent to extend range, but the finite mantissa means that not every value in that range is represented exactly. Calculation speed is also a comparison point: fixed point can be faster because its point position is fixed, whereas floating-point processing is more involved.

7. Error questions

For an intended value and a stored value, calculate absolute error as the magnitude of their difference. Calculate relative error by comparing the absolute error with the intended value, using the formula required by the question. Then interpret the result rather than listing numbers only. The same absolute error can be more significant for a small-magnitude value than for a large-magnitude value. Values close to one can also make relative-error comparisons useful because the scale of the intended value is clear. State whether the error is caused by limited available bits, an unrepresentable binary fraction, or a calculation exceeding the format's range when the question provides enough information.

8. Underflow and overflow checks

Ask whether the result is too large in magnitude or too small in magnitude for the representation. Too large means overflow; a non-zero value too small to represent means underflow. For floating point, inspect both the mantissa capacity and exponent range. For fixed point, inspect the fixed integer and fractional allocation. Do not use overflow to describe ordinary rounding merely because a value is slightly inaccurate.

Self-check checklist

Before finalising an answer, confirm: the representation has been identified; the number of bits has stayed constant; the binary point has been handled correctly; two's complement negative values were decoded correctly; the appropriate range formula was used; absolute and relative error were not confused; and any claim of overflow, underflow or rounding is supported by the available range or precision. For a longer response, show intermediate place values and explain why the final representation is valid.