1.4.1 Data Types
1.4.1(a) Primitive Data Types
Understanding data types is fundamental to programming. Different types of data are stored and processed differently by computers.
Definition
Primitive data types are the basic building blocks for storing data in programs. They define what kind of data a variable can hold and what operations can be performed on it.
Integer
Definition
An integer is a whole number, positive or negative, with no decimal point.
Example
Valid integers: 42, -17, 0, 1000000
NOT integers: 3.14, "42", 5.0
NOT integers: 3.14, "42", 5.0
Real (Floating Point)
Definition
A real number (or float) is a number that can have a fractional part, represented with a decimal point.
Example
Valid reals: 3.14, -0.5, 100.0, 2.71828
Boolean
Definition
A Boolean can only have two values: True or False. Used for logical operations and conditions.
Example
isLoggedIn = True
hasPermission = False
isAdult = age >= 18 // Result is True or False
hasPermission = False
isAdult = age >= 18 // Result is True or False
Character
Definition
A character is a single symbol - a letter, digit, punctuation mark, or special symbol.
Example
Valid characters: 'A', 'z', '7', '!', ' '
Each character has a numeric code (ASCII/Unicode)
Each character has a numeric code (ASCII/Unicode)
String
Definition
A string is a sequence of characters, used to represent text.
Example
name = "Alice Smith"
empty = ""
number = "42" // This is a STRING, not an integer!
empty = ""
number = "42" // This is a STRING, not an integer!
Key Point
Integer
• Description: Whole numbers
• Example: 42, -7, 0
Real/Float
• Description: Decimal numbers
• Example: 3.14, -0.5
Boolean
• Description: True or False
• Example: True, False
Character
• Description: Single symbol
• Example: 'A', '7'
String
• Description: Text
• Example: "Hello"
• Description: Whole numbers
• Example: 42, -7, 0
Real/Float
• Description: Decimal numbers
• Example: 3.14, -0.5
Boolean
• Description: True or False
• Example: True, False
Character
• Description: Single symbol
• Example: 'A', '7'
String
• Description: Text
• Example: "Hello"
1.4.1(b) Binary Representation of Positive Integers
Computers store all data as binary (base-2) - patterns of 0s and 1s.
Definition
Binary is a number system using only two digits: 0 and 1. Each digit position represents a power of 2.
Key Point
Place Values (8-bit):
| Bit position | 7 | 6 | 5 | 4 | 3 | 2 | 1 | 0 |
|---|---|---|---|---|---|---|---|---|
| Value (2^n) | 128 | 64 | 32 | 16 | 8 | 4 | 2 | 1 |
Worked Example
Worked Example: Binary to Denary Conversion
Convert 10110101 to denary:
Calculation: 128 + 32 + 16 + 4 + 1 = 181
Answer: 10110101 in binary = 181 in denary
Convert 10110101 to denary:
| Value | 128 | 64 | 32 | 16 | 8 | 4 | 2 | 1 |
|---|---|---|---|---|---|---|---|---|
| Bit | 1 | 0 | 1 | 1 | 0 | 1 | 0 | 1 |
Answer: 10110101 in binary = 181 in denary
Worked Example
Worked Example: Denary to Binary Conversion
Convert 157 to binary:
Method: Repeatedly divide by 2, note remainders.
Read remainders bottom to top: 10011101
Answer: 157 in denary = 10011101 in binary
Convert 157 to binary:
Method: Repeatedly divide by 2, note remainders.
| Division step | Quotient | Remainder |
|---|---|---|
| 157 ÷ 2 | 78 | 1 |
| 78 ÷ 2 | 39 | 0 |
| 39 ÷ 2 | 19 | 1 |
| 19 ÷ 2 | 9 | 1 |
| 9 ÷ 2 | 4 | 1 |
| 4 ÷ 2 | 2 | 0 |
| 2 ÷ 2 | 1 | 0 |
| 1 ÷ 2 | 0 | 1 |
Answer: 157 in denary = 10011101 in binary
1.4.1(c) Binary Representation of Negative Numbers
Sign and Magnitude
Definition
In sign and magnitude, the leftmost bit indicates the sign (0 = positive, 1 = negative) and the remaining bits represent the value.
Example
8-bit sign and magnitude:
+5 = 00000101
-5 = 10000101
Problem: Two representations of zero (00000000 and 10000000)
+5 = 00000101
-5 = 10000101
Problem: Two representations of zero (00000000 and 10000000)
Two's Complement
Definition
Two's complement is the standard method for representing negative numbers in binary. To negate a number: starting from the right, keep everything up to and including the first 1 the same, then flip all the remaining bits to the left.
Worked Example
Worked Example: Converting to Two's Complement
Represent -42 in 8-bit two's complement:
Step 1: Write +42 in binary
42 = 00101010
Step 2: Starting from the right, find the first 1 — keep everything from there rightward the same, flip everything to the left.
The first 1 from the right is at position 1 (the "10" at the end). Keep "10", flip the rest.
Answer: -42 in two's complement = 11010110
Represent -42 in 8-bit two's complement:
Step 1: Write +42 in binary
42 = 00101010
Step 2: Starting from the right, find the first 1 — keep everything from there rightward the same, flip everything to the left.
| Original | 0 | 0 | 1 | 0 | 1 | 0 | 1 | 0 |
|---|---|---|---|---|---|---|---|---|
| Action | flip | flip | flip | flip | flip | flip | keep | keep |
| Result | 1 | 1 | 0 | 1 | 0 | 1 | 1 | 0 |
Answer: -42 in two's complement = 11010110
Worked Example
Worked Example: Reading Two's Complement
What decimal number does 11110011 represent?
Step 1: MSB is 1, so it's negative.
Step 2: To find the magnitude, convert back — keep everything up to and including the first 1 from the right, flip the rest.
00001101 = 13
Answer: 11110011 = -13
What decimal number does 11110011 represent?
Step 1: MSB is 1, so it's negative.
Step 2: To find the magnitude, convert back — keep everything up to and including the first 1 from the right, flip the rest.
| Original | 1 | 1 | 1 | 1 | 0 | 0 | 1 | 1 |
|---|---|---|---|---|---|---|---|---|
| Action | flip | flip | flip | flip | flip | flip | keep | keep |
| Result | 0 | 0 | 0 | 0 | 1 | 1 | 0 | 1 |
Answer: 11110011 = -13
Key Point
Two's Complement Range (8-bit):
• Minimum: -128 (10000000)
• Maximum: +127 (01111111)
• One more negative number than positive!
• Minimum: -128 (10000000)
• Maximum: +127 (01111111)
• One more negative number than positive!
1.4.1(d) Binary Addition and Subtraction
Binary Addition Rules
Key Point
Binary Addition Rules:
| Sum | Result bit | Carry out |
|---|---|---|
| 0 + 0 | 0 | 0 |
| 0 + 1 | 1 | 0 |
| 1 + 0 | 1 | 0 |
| 1 + 1 | 0 | 1 |
| 1 + 1 + 1 | 1 | 1 |
Worked Example
Worked Example: Binary Addition
Add 01101011 + 00110101
Working right to left:
• Bit 0: 1+1 = 10 → write 0, carry 1
• Bit 1: 1+0+1(carry) = 10 → write 0, carry 1
• Bit 2: 0+1+1(carry) = 10 → write 0, carry 1
• Bit 3: 1+0+0 = 1 → write 1, no carry
• Bit 4: 0+1+0 = 1 → write 0... (continue)
• Bit 5: 1+1+0 = 10 → write 0, carry 1
• Bit 6: 1+0+1(carry) = 10 → write 0, carry 1
• Bit 7: 0+0+1(carry) = 1 → write 1
Answer: 01101011 + 00110101 = 10100000 (107 + 53 = 160)
Add 01101011 + 00110101
| Decimal check | Value | |||||||
|---|---|---|---|---|---|---|---|---|
| 01101011 | 107 | |||||||
| 00110101 | 53 | |||||||
| Total | 160 | |||||||
| Row | 7 | 6 | 5 | 4 | 3 | 2 | 1 | 0 |
| --- | --- | --- | --- | --- | --- | --- | --- | --- |
| Carry | 1 | 1 | 1 | 0 | 0 | 1 | 1 | |
| A | 0 | 1 | 1 | 0 | 1 | 0 | 1 | 1 |
| B | 0 | 0 | 1 | 1 | 0 | 1 | 0 | 1 |
| Sum | 1 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
• Bit 0: 1+1 = 10 → write 0, carry 1
• Bit 1: 1+0+1(carry) = 10 → write 0, carry 1
• Bit 2: 0+1+1(carry) = 10 → write 0, carry 1
• Bit 3: 1+0+0 = 1 → write 1, no carry
• Bit 4: 0+1+0 = 1 → write 0... (continue)
• Bit 5: 1+1+0 = 10 → write 0, carry 1
• Bit 6: 1+0+1(carry) = 10 → write 0, carry 1
• Bit 7: 0+0+1(carry) = 1 → write 1
Answer: 01101011 + 00110101 = 10100000 (107 + 53 = 160)
Binary Subtraction Using Two's Complement
Worked Example
Worked Example: Subtraction
Calculate 50 - 30 using two's complement (i.e. 50 + (-30)).
Now add 50 + (-30):
Ignore the overflow (bit 8): 00010100 = 20
Answer: 50 - 30 = 20 ✓
Calculate 50 - 30 using two's complement (i.e. 50 + (-30)).
| Step | Working |
|---|---|
| Convert 50 to 8-bit binary | 00110010 |
| Convert 30 to 8-bit binary | 00011110 |
| Two's complement of 30 | Keep from right up to and including first 1 ("10"), flip the rest → 11100010 |
| Row | 8 | 7 | 6 | 5 | 4 | 3 | 2 | 1 | 0 |
|---|---|---|---|---|---|---|---|---|---|
| Carry | 1 | 1 | 1 | 0 | 0 | 0 | 1 | ||
| A | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 0 | |
| B | 1 | 1 | 1 | 0 | 0 | 0 | 1 | 0 | |
| Sum | 1 | 0 | 0 | 0 | 1 | 0 | 1 | 0 | 0 |
Answer: 50 - 30 = 20 ✓
1.4.1(e-f) Hexadecimal
Definition
Hexadecimal (hex) is a base-16 number system using digits 0-9 and letters A-F (representing 10-15). It provides a more compact way to represent binary.
Key Point
| Hex | Decimal | 4-bit Binary |
|---|---|---|
| 0 | 0 | 0000 |
| 1 | 1 | 0001 |
| 2 | 2 | 0010 |
| 3 | 3 | 0011 |
| 4 | 4 | 0100 |
| 5 | 5 | 0101 |
| 6 | 6 | 0110 |
| 7 | 7 | 0111 |
| 8 | 8 | 1000 |
| 9 | 9 | 1001 |
| A | 10 | 1010 |
| B | 11 | 1011 |
| C | 12 | 1100 |
| D | 13 | 1101 |
| E | 14 | 1110 |
| F | 15 | 1111 |
Worked Example
Worked Example: Binary ↔ Hexadecimal
| Conversion | Working | Answer |
|---|---|---|
| Binary to Hex | 11010110 → 1101 0110 → D 6 | D6 |
| Hex to Binary | 4A → 4 = 0100, A = 1010 | 01001010 |
Worked Example
Worked Example: Hex ↔ Denary
| Conversion | Working | Answer |
|---|---|---|
| Hex to Denary | 2F = (2 × 16^1) + (15 × 16^0) = 32 + 15 | 47 |
| Denary to Hex | 200 ÷ 16 = 12 remainder 8; 12 = C | C8 |
1.4.1(g) Floating Point Numbers
Definition
Floating point is a method of representing real numbers in binary using a mantissa (significant digits) and an exponent (scale factor).
Key Point
Floating Point Structure:
• Sign bit: 0 = positive, 1 = negative
• Mantissa: The significant digits
• Exponent: Power of 2 to multiply by
Trade-off:
• More mantissa bits = more precision
• More exponent bits = larger range
• Sign bit: 0 = positive, 1 = negative
• Mantissa: The significant digits
• Exponent: Power of 2 to multiply by
Trade-off:
• More mantissa bits = more precision
• More exponent bits = larger range
Key Point
Normalisation:
A normalised floating point number has its mantissa starting with 01 (positive) or 10 (negative). This ensures maximum precision.
A normalised floating point number has its mantissa starting with 01 (positive) or 10 (negative). This ensures maximum precision.
1.4.1(h) Character Sets
Definition
A character set is a mapping between characters and their numeric codes. It defines which numbers represent which characters.
ASCII
Key Point
ASCII (American Standard Code for Information Interchange):
• 7 bits = 128 characters
• Covers English letters, digits, punctuation, control characters
• 'A' = 65, 'a' = 97, '0' = 48
• Limited - only English characters
• 7 bits = 128 characters
• Covers English letters, digits, punctuation, control characters
• 'A' = 65, 'a' = 97, '0' = 48
• Limited - only English characters
Unicode
Key Point
Unicode:
• Supports over 140,000 characters
• Covers virtually all world languages
• Includes emojis, mathematical symbols, historic scripts
• UTF-8: Variable length (1-4 bytes per character)
• Backward compatible with ASCII
• Supports over 140,000 characters
• Covers virtually all world languages
• Includes emojis, mathematical symbols, historic scripts
• UTF-8: Variable length (1-4 bytes per character)
• Backward compatible with ASCII
Key Point
| Character set | Bits | Characters | Scope |
|---|---|---|---|
| ASCII | 7 | 128 | English only |
| Extended ASCII | 8 | 256 | English + European |
| Unicode (UTF-8) | 8-32 | 140,000+ | All languages + emoji |
Exam Tip
Why Unicode over ASCII?
• Global internet requires all languages
• ASCII cannot represent Chinese, Arabic, Hindi, etc.
• Emojis need Unicode
• UTF-8 is efficient - uses 1 byte for ASCII characters
• Global internet requires all languages
• ASCII cannot represent Chinese, Arabic, Hindi, etc.
• Emojis need Unicode
• UTF-8 is efficient - uses 1 byte for ASCII characters