Number systems: binary, denary and hexadecimal, and how to convert between all three, including why hexadecimal is used as a shorthand for binary.

What this dot point is asking

Eduqas wants you to convert numbers between the three number systems used in computing: binary (base 2), denary (base 10, our everyday numbers) and hexadecimal (base 16). You must be able to go in every direction, show clear working, and explain why hexadecimal is used as a shorthand for binary. These conversions appear on almost every Component 1 paper.

The three number systems

In an 8-bit binary number the place values are powers of two: $128, 64, 32, 16, 8, 4, 2, 1$. In hexadecimal the place values are powers of sixteen: $\ldots, 256, 16, 1$. In denary they are powers of ten: $\ldots, 100, 10, 1$.

Converting denary to binary

Converting binary to denary

To convert binary to denary, add up the place values wherever there is a $1$. For example, $10110010$ has $1$s in the $128, 32, 16$ and $2$ columns, so it equals $128 + 32 + 16 + 2 = 178$.

Converting with hexadecimal

Why hexadecimal is used

Hexadecimal is a shorthand for binary. A single byte takes eight binary digits but only two hex digits, so hex is far shorter to write, read and copy, and people make fewer mistakes with it. Because each hex digit maps cleanly to four bits, converting between hex and binary is quick. This is why memory addresses, colour codes (such as #FF0000 for red) and error codes are usually written in hexadecimal.

Try this

Q1. Convert the denary number $45$ to 8-bit binary. [2 marks]

• Cue. $00101101$ (that is $32 + 8 + 4 + 1$).

Q2. Convert the binary number $10100110$ to denary. [2 marks]

• Cue. $128 + 32 + 4 + 2 = 166$.

Q3. State why hexadecimal is used instead of long binary numbers. [1 mark]

• Cue. It is much shorter and easier for people to read and copy, and each hex digit maps to exactly four bits.