Understand that computers use binary to represent data (numbers, text, sound, graphics) and program instructions, and determine the maximum number of states that can be represented by a binary pattern of a given length.

What this dot point is asking

Edexcel wants you to explain why computers represent everything (numbers, text, sound, graphics and the program instructions themselves) in binary, and to calculate how many different states, or values, a binary pattern of a given number of bits can represent.

Why computers use binary

The deeper reason is reliability. A circuit only has to distinguish "voltage present" from "voltage absent", which is far more robust against noise and small errors than trying to distinguish ten different voltage levels for the ten decimal digits would be. This is why binary, rather than denary, is the natural language of digital hardware.

Crucially, everything is binary inside the machine: whole numbers, text (via character codes), images (via pixel colour values), sound (via samples) and even the program instructions the CPU runs are all stored as patterns of bits. The meaning of a pattern depends on how the program interprets it; the same bits could be a number or a character depending on context.

Bits and patterns

With one bit there are two possible patterns (0 and 1), so two states. Add a second bit and each of those can be followed by a 0 or a 1, giving four patterns (00, 01, 10, 11). Every extra bit doubles the number of patterns, which is why the count grows as a power of 2.

Calculating the number of states

This single rule answers a whole family of exam questions. "How many colours can 4 bits per pixel represent?" is $2^4 = 16$. "How many characters can a 7-bit code represent?" is $2^7 = 128$. "How many bits are needed for at least 100 states?" needs the smallest $n$ with $2^n \geq 100$, which is $n = 7$ because $2^6 = 64$ (too few) and $2^7 = 128$ (enough).

States versus largest value

A frequent source of lost marks is mixing up the number of states with the largest value. For an 8-bit unsigned number there are $2^8 = 256$ states, but the values run from 0 to 255, so the largest value is 255, one less than the number of states. Read each question carefully: "how many different values" or "how many states" wants $2^n$, while "the largest value that can be stored" wants $2^n - 1$.

Try this

Q1. Calculate the number of states a 7-bit pattern can represent. [1 mark]

• Cue. $2^7 = 128$.

Q2. State the largest unsigned value that can be stored in 8 bits. [1 mark]

• Cue. 255, because $2^8 - 1 = 255$.