OCR OCR 2021-style practice: Convert the hexadecimal number 2F to denary, and the denary number 206 to hexadecimal, showing your method.

Hex to denary (2 marks): 2F has digits 2 and F (=15). Value = 2 × 16 + 15 = 32 + 15 = 47. Denary to hex (2 marks): 206 16 = 12 remainder 14. 12 = C and 14 = E, so reading the quotient then remainder gives CE. Check: 12 × 16 + 14 = 192 + 14 = 206. Markers reward the place-value method for hex to denary and the divide-by-16 (or four-bit grouping) method for denary to hex, with correct letter digits.

EnglandComputer ScienceSyllabus dot point

How are positive and negative whole numbers represented in binary, and how do we convert between binary, denary and hexadecimal?

Number systems: binary, denary and hexadecimal conversion, representing negative numbers with sign and magnitude and two's complement, binary addition and subtraction, fixed-point binary fractions, and the use of hexadecimal and bitwise masks.

An OCR H446 answer on number systems: converting between binary, denary and hexadecimal, representing negative numbers with sign and magnitude and two's complement, binary addition and subtraction, fixed-point binary fractions, and the use of hexadecimal and bitwise masks.

Generated by Claude Opus 4.814 min answerUpdated 2026-06-02

Reviewed by: AI editorial process; not yet individually human-reviewed

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Quick answer

A binary digit (bit) has place values that are powers of two. To convert binary to denary, add the place values where there is a 1. To convert denary to binary, subtract the largest fitting power of two repeatedly. Hexadecimal is base 16 (digits 0 to 9 then A to F), and one hex digit equals exactly four bits, so it is a compact way to write binary. Negative numbers use sign and magnitude (the most significant bit is the sign, giving two zeros and the range $-127$ to $+127$ in 8 bits) or two's complement (invert all bits and add 1, giving one zero and the range $-128$ to $+127$ in 8 bits), which is preferred because addition and subtraction work directly. Binary addition carries when a column sums to 2 or more; subtraction is done by adding the two's complement. Fixed-point binary places a binary point at a set position so fractions can be represented. A bitwise mask uses AND, OR or XOR to test, set or toggle individual bits.

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What this dot point is asking
The answer
Examples in context
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What this dot point is asking

OCR wants fluent conversion between binary, denary and hexadecimal, representation of negative numbers in sign and magnitude and two's complement, binary addition and subtraction, fixed-point binary fractions, and the use of hexadecimal and bitwise masks. This is a calculation-heavy dot point assessed without a calculator.

The answer

Binary, denary and hexadecimal

Key fact

In an 8-bit number the place values are $128, 64, 32, 16, 8, 4, 2, 1$ . Binary to denary: sum the place values where the bit is 1, so $0101\,1010 = 64 + 16 + 8 + 2 = 90$ . Denary to binary: repeatedly take the largest power of two that fits and set that bit. Hexadecimal is base 16 with digits $0$ to $9$ then $\text{A}=10$ through $\text{F}=15$ ; each hex digit maps to exactly four bits, so $1101\,0110$ groups into $1101 = \text{D}$ and $0110 = 6$ , giving $\text{D6}$ . Hex to denary uses place values that are powers of 16: $\text{D6} = 13 \times 16 + 6 = 214$ . Hexadecimal is used because it is far shorter to read and write than binary while mapping cleanly onto it.

Representing negative numbers

Key fact

Sign and magnitude uses the most significant bit as a sign ( $0$ positive, $1$ negative) and the rest as the magnitude, so in 8 bits $1000\,0101 = -5$ ; this is simple but has two representations of zero ( $+0$ and $-0$ ) and a range of $-127$ to $+127$ , and arithmetic is awkward. Two's complement makes the most significant bit a negative weight ( $-128$ in 8 bits): to negate a number, invert every bit and add 1. So $+5 = 0000\,0101 \to 1111\,1010 + 1 = 1111\,1011 = -5$ . Two's complement has a single zero, a range of $-128$ to $+127$ in 8 bits, and, crucially, addition and subtraction work with the ordinary binary rules, which is why processors use it.

Binary arithmetic, fixed point and masks

Convert and add in two's complement

Work out $0011\,1010$ in denary, then compute $58 - 22$ in 8-bit two's complement.

step 1: Binary to denary

$0011\,1010 = 32 + 16 + 8 + 2 = 58$ .

step 2: Represent $-22$ in two's complement

$+22 = 0001\,0110$ . Invert: $1110\,1001$ . Add 1: $1110\,1010$ . So $-22 = 1110\,1010$ .

step 3: Add $58 + (-22)$

  0011 1010   (+58)
+ 1110 1010   (-22)
-----------
  0010 0100   (final carry out of bit 8 discarded)

step 4: Interpret

$0010\,0100 = 32 + 4 = 36$ , and indeed $58 - 22 = 36$ . Subtraction was done as addition of the two's complement, and the carry out of the top bit was discarded.

Examples in context

Memory addresses and colour values are written in hexadecimal because it is compact (the colour #FF8000 is three bytes). Two's complement is used inside every processor so one adder circuit can do both addition and subtraction. Bitwise masks set or read individual flag bits in a status register or in file permissions. Fixed-point representation appears where a known range and precision are needed cheaply. OCR links this to floating-point representation and to the logic circuits (adders) that perform the arithmetic.

Try this

Q1. Convert $1011\,0011$ to denary. [1 mark]

Cue. $128 + 32 + 16 + 2 + 1 = 179$ .

Q2. Represent $-9$ in 8-bit two's complement. [2 marks]

Cue. $+9 = 0000\,1001$ ; invert to $1111\,0110$ ; add 1 to get $1111\,0111$ .

Q3. Convert the denary number $90$ to hexadecimal. [1 mark]

Cue. $90 = 0101\,1010 \to 5\text{A}$ (or $90 \div 16 = 5$ remainder $10 = \text{A}$ ).

Exam-style practice questions

Practice questions written in the style of OCR exam questions on this dot point, with worked answer explainers. The year tag is the paper they imitate, not the source.

OCR 20195 marksUsing 8-bit two's complement, represent

-20

in binary, and add it to the 8-bit two's complement representation of

+35

, showing your working and interpreting the result.

Show worked answer →

Represent $-20$ (up to 3): start with $+20 = 0001\,0100$ . Invert all bits: $1110\,1011$ . Add 1: $1110\,1100$ . So $-20 = 1110\,1100$ .

$+35 = 0010\,0011$ .

Add (up to 2):

  0010 0011   (+35)
+ 1110 1100   (-20)
-----------
  0000 1111   (carry out of the 8th bit is discarded)

The 8-bit result is $0000\,1111 = +15$ , which is correct since $35 + (-20) = 15$ . Markers reward the correct two's complement of $-20$ (invert and add 1), the binary addition, and discarding the final carry to read $+15$ .

OCR 20214 marksConvert the hexadecimal number

\text{2F}

to denary, and the denary number

206

to hexadecimal, showing your method.

Show worked answer →

Hex to denary (2 marks): $\text{2F}$ has digits $2$ and $\text{F}$ ( $=15$ ). Value $= 2 \times 16 + 15 = 32 + 15 = 47$ .

Denary to hex (2 marks): $206 \div 16 = 12$ remainder $14$ . $12 = \text{C}$ and $14 = \text{E}$ , so reading the quotient then remainder gives $\text{CE}$ . Check: $12 \times 16 + 14 = 192 + 14 = 206$ . Markers reward the place-value method for hex to denary and the divide-by-16 (or four-bit grouping) method for denary to hex, with correct letter digits.

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Sources & how we know this

OCR AS and A Level Computer Science (H046, H446) specification — OCR (2015)