Topic 1: Data Representation — Part 1

Number Systems

Understanding number bases

Binary (Base 2)

Computers use binary because they are made of electronic components (like transistors) that have only two states: on (1) or off (0).

Denary (Base 10)

The standard system used by humans for counting.

Hexadecimal (Base 16)

Used by programmers as a shorthand representation of binary. It is more compact and easier for humans to read and type.

Common uses: MAC addresses, RGB colour codes, and identifying errors in assembly code.

Conversion Type	Method	Worked Example
Binary to Denary	Add place values (128, 64, 32—) where the bit is 1.	1011 = 8 + 2 + 1 = 11
Denary to Binary	Repeatedly divide by 2 and record remainders; read remainders from bottom to top.	13 ? 1101 = 1101
Hex to Binary	Convert each hex digit to a 4-bit nibble and combine.	B2 ? B(11) = 1011, 2 = 0010 ? 10110010
Denary to Hex	Convert denary to binary first, then split into nibbles.	47 ? 0010 1111 ? 2F

Addition rules: 0 + 0 = 0; 1 + 0 = 1; 1 + 1 = 0 (carry 1); 1 + 1 + 1 = 1 (carry 1).

Worked example: 0110 + 0011 = 1001 (denary 6 + 3 = 9).

Overflow error: Occurs when the result of an addition is too large to be represented by the available bits (e.g., a result greater than 255 in an 8-bit register).

Left shift: Moves all bits to the left, adding 0s on the right. This multiplies the positive integer by 2 for every place shifted.
Right shift: Moves all bits to the right, adding 0s on the left. This divides the positive integer by 2 for every place shifted.

Note: Bits shifted off the end of the register are lost.

Exam Traps

Logical shifts in these notes apply to positive integers — do not assume the same left/right rules work for negative two's complement values without a separate method.

Used to represent negative binary integers. The Most Significant Bit (MSB) has a negative value (e.g., -128 for 8-bit).

To convert negative denary to 8-bit binary:

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