Convert whole numbers between denary, binary and hexadecimal using place value, repeated division and nibble grouping

What this dot point is asking

SEAB wants you to convert whole numbers fluently between denary (base 10), binary (base 2) and hexadecimal (base 16). The central idea is that every base uses the same positional place-value rule, so converting is just a matter of deciding which powers (of two or of sixteen) you are counting in, and grouping binary into nibbles to reach hexadecimal.

The answer

Denary to binary

There are two reliable methods. Pick whichever you find quicker and check with the other.

Method 1: subtract descending powers of two. Write the place values $128, 64, 32, 16, 8, 4, 2, 1$. Put a $1$ under the largest power that fits, subtract it from the number, and repeat with the remainder. Put $0$ where a power does not fit.

Method 2: repeated division by two. Divide the number by $2$, writing the remainder ($0$ or $1$) each time, until you reach $0$. The remainders read from the bottom upwards give the binary digits.

46 / 2 = 23 remainder 0
23 / 2 = 11 remainder 1
11 / 2 =  5 remainder 1
 5 / 2 =  2 remainder 1
 2 / 2 =  1 remainder 0
 1 / 2 =  0 remainder 1
read up: 101110  ->  46 = 00101110

Binary to denary

Add the place values of every column that holds a $1$:

Binary to hexadecimal

Because one hex digit is four bits, group the binary into nibbles from the right, padding the left with zeros if needed, then convert each nibble:

binary:  1101 0110
hex:        D    6   -> D6

Hexadecimal to binary

Expand each hex digit to its 4-bit pattern and join them:

hex:        A    5
binary:  1010 0101  -> 10100101

Hexadecimal to denary

Multiply the left digit by $16$ and add the right digit (for a two-digit hex number):

Examples in context

Example 1. Reading a memory dump. A debugger shows a byte as 0x4D. A student converts it through binary: $\text{4} = 0100$, $\text{D} = 1101$, so the byte is $01001101_2 = 77$, which is the ASCII code for the letter M. Going through binary makes the value obvious.

Example 2. Setting a colour by hand. To make a medium grey, a designer wants each of red, green and blue at about half intensity. Half of $255$ is roughly $128 = 10000000_2 = \text{80}_{16}$, so the colour is #808080. The conversion turns an intensity into a hex pair.

Try this

Q1. Convert $89$ to 8-bit binary. [2 marks]

• Cue. $89 = 64 + 16 + 8 + 1 = 01011001_2$.

Q2. Convert $11110010_2$ to hexadecimal. [2 marks]

• Cue. $1111 = \text{F}$ and $0010 = 2$, so the answer is $\text{F2}_{16}$.

Q3. Convert $\text{6A}_{16}$ to denary. [2 marks]

• Cue. $6 \times 16 + 10 = 96 + 10 = 106$.