C1 · Number Systems

Spec reference: Section C1 - How Data is Represented Key idea: Understand binary, BCD, and how to perform binary arithmetic including negative and floating point numbers.

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Units of digital data

Unit	Size
Bit	Single binary digit: 0 or 1
Nibble	4 bits
Byte	8 bits
Kilobyte (KB)	1,024 bytes
Megabyte (MB)	1,024 KB
Gigabyte (GB)	1,024 MB
Terabyte (TB)	1,024 GB

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Number systems

Binary (Base 2)

Uses only 0 and 1. Each position is a power of 2.

128	64	32	16	8	4	2	1
0	1	0	1	1	0	0	1

This binary number = 64 + 16 + 8 + 1 = 89

Hexadecimal (Base 16)

Uses digits 0-9 and letters A-F. Each hex digit represents 4 bits (a nibble).

Hex	Binary	Decimal
0-9	0000-1001	0-9
A	1010	10
B	1011	11
C	1100	12
D	1101	13
E	1110	14
F	1111	15

Used in memory addresses, colour codes (HTML), and MAC addresses because it is more compact than binary.

Binary Coded Decimal (BCD)

Each decimal digit is stored as its 4-bit binary equivalent.

Example: 47 in BCD:

• 4 = 0100
• 7 = 0111
• BCD representation: 0100 0111

Used in systems where exact decimal representation is important, such as financial systems and digital clocks.

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Binary arithmetic

Addition

Rules: 0+0=0, 0+1=1, 1+0=1, 1+1=10 (write 0, carry 1), 1+1+1=11 (write 1, carry 1)

  0 1 1 0 1  (13)
+ 0 0 1 1 1  ( 7)
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  1 0 1 0 0  (20)

Subtraction

Using two's complement (see below) is the standard method.

Multiplication

Shift left and add. Multiplying by 2 is a single left shift.

Division

Shift right and subtract. Dividing by 2 is a single right shift.

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Representing negative numbers: Two's Complement

Two's complement is the standard way computers represent negative integers.

To convert a positive binary number to its negative equivalent:

1. Invert all bits (flip 0s to 1s and 1s to 0s).
2. Add 1 to the result.

Example: Represent -6 in 8-bit two's complement.

1. +6 in binary: 0000 0110
2. Invert: 1111 1001
3. Add 1: 1111 1010

So -6 = 1111 1010 in two's complement.

The most significant bit (MSB) acts as the sign bit: 0 = positive, 1 = negative.

| 8-bit two's complement range | -128 to +127 |

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Floating point numbers

Floating point allows computers to represent very large or very small numbers.

A floating point number has two parts:

• Mantissa: The significant digits.
• Exponent: The power of 2 by which the mantissa is multiplied.

Example (simplified): The number 0.001011 x 2^5 in normalised form:

• Mantissa: 0.101100
• Exponent: 5

Exam point

Normalised floating point means the mantissa is adjusted so the first significant digit is immediately after the binary point (for positive: 0.1..., for negative: 1.0...). This maximises precision.

Trade-offs:

• More bits for the mantissa: greater precision.
• More bits for the exponent: greater range.
• Rounding errors occur because not all decimal values can be represented exactly in binary floating point.

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Summary

Term	Meaning
Bit	The smallest unit of data, 0 or 1
Two's complement	Method for representing negative integers in binary
BCD	Each decimal digit stored as 4 binary bits
Mantissa	The significant digits in a floating point number
Exponent	The power applied to the mantissa in floating point
Overflow	When a result is too large to be stored in the available bits

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