Understand the binary, denary and hexadecimal number bases, why computers use binary, and convert between binary and denary.

What this dot point is asking

AQA wants you to know the three number bases (binary, denary and hexadecimal), explain clearly why computers use binary, and convert between binary and denary in both directions, showing working.

The three number bases

The key idea behind every number base is positional notation. In denary the number 154 means $1 \times 100 + 5 \times 10 + 4 \times 1$, that is the digit in each column is multiplied by a power of ten. Binary works identically but with powers of two, so $10011010$ means $1 \times 128 + 0 \times 64 + 0 \times 32 + 1 \times 16 + 1 \times 8 + 0 \times 4 + 1 \times 2 + 0 \times 1$. Understanding that the only thing that changes between bases is the base of the powers makes every conversion follow from one rule.

Why computers use binary

This reliability is the heart of the answer AQA wants. Storing ten distinguishable voltage levels in a circuit would make it sensitive to tiny fluctuations, so errors would be common; with only two levels there is a wide gap between "on" and "off" that noise cannot bridge. Binary also makes logic simple: AND, OR and NOT gates operate directly on 1s and 0s, so arithmetic and decision-making can be built from a few standard gates.

Binary to denary

The place values of an 8-bit binary number are 128, 64, 32, 16, 8, 4, 2, 1. Write them above the bits and add the place values where there is a 1.

Denary to binary

Start with the largest place value (128) and work down. If the value fits into what is left, write 1 and subtract it; otherwise write 0 and move on.

Checking your conversions

A quick way to catch errors is to convert back the other way. If you turn a denary number into binary, convert the binary straight back to denary by adding the place values; if it does not match, you have made a slip. Another useful check is the range: an 8-bit binary number can only represent 0 to 255, so any denary value above 255 cannot fit in eight bits, and any binary answer with a 1 in a place value beyond 128 has more than eight bits. Doing this check takes seconds and often saves a mark lost to a single misread digit.

Try this

Q1. Convert the binary number $00010110$ to denary. [2 marks]

• Cue. $16 + 4 + 2 = 22$.

Q2. Explain why computers use binary rather than denary. [2 marks]

• Cue. Components have two reliable states (on/off) that map to 1 and 0; a two-state system resists electrical noise, so storing data is reliable.