Understand the limitations of binary representation of data when…

The idea

A fixed number of bits can store only a fixed set of values, so binary representation always has limits. With n bits there are 2ⁿ different patterns, so the range of unsigned integers runs from 0 to 2ⁿ − 1. For example, 8 bits give 256 patterns, covering 0 to 255, so the value 256 cannot be stored in 8 bits. If a result exceeds the largest storable value, overflow occurs and the extra carry has nowhere to go. Fewer bits also mean coarser resolution: a 4-bit sample has only 16 amplitude levels, while 8 bits give 256. Rounding or truncation happens when a value cannot be represented exactly, such as a fraction stored with a limited number of fractional bits. Adding bits widens the range and improves precision, but every extra bit doubles the number of patterns and increases storage.

What earns the marks

• With n bits there are 2ⁿ distinct bit patterns, so the number of representable values is limited by the number of available bits.
• For unsigned integers, n bits represent the range 0 to 2ⁿ − 1; for example, 8 bits represent 0 to 255.
• Overflow occurs when a calculation produces a value outside the range that the available bits can represent, so the result cannot be stored correctly.
• A smaller number of bits gives coarser resolution, so values such as measurements or sound amplitudes are rounded to the nearest representable level.
• Increasing the number of bits widens the range and improves precision, but it also increases the storage needed for each value.
• Limitations apply to all data types, including integers, characters and sampled sound, because each uses a fixed number of bits per value.

Common mistakes and how to put them right

• Thinking n bits store 2n values.

  Put it right: The number of patterns is 2ⁿ, so 8 bits give 256 patterns, not 16.

• Believing the largest unsigned value is 2ⁿ.

  Put it right: The range is 0 to 2ⁿ − 1, so 8 bits stop at 255.

• Assuming overflow changes the value to the correct larger number.

  Put it right: Overflow means the result cannot be represented in the available bits, so the stored pattern is wrong.

• Confusing range with precision.

  Put it right: Adding bits increases both the largest value and the number of levels, but these are separate effects.

Exam tips

• Calculate the number of patterns as 2ⁿ and the unsigned range as 0 to 2ⁿ − 1 before choosing an answer.
• Identify overflow by comparing the required value with the maximum representable value for the given number of bits.
• When a question mentions quality or accuracy, link the limitation to rounding or truncation caused by too few bits.