Number Bases (AQA A-Level Computer Science): Revision Notes

Number Bases

Introduction to digital data representation

Computers work with data in a digital format, which means they represent everything as distinct values using zeros and ones. This is called binary data, and it forms the foundation of how computers process information. In this topic, you'll explore how binary works and how it connects to other number systems such as decimal and hexadecimal.

The bit

A bit is the smallest unit of data in computing. It stands for binary digit and can have only one of two values: 0 or 1. Think of a bit as an on/off switch - the computer either receives an electrical signal (1) or doesn't receive one (0).

Computers process data using microprocessors (also called chips or silicon chips). These chips contain millions of tiny electronic circuits. When electricity flows through these circuits, the processor interprets this as different signals. The processor can recognise whether electricity is flowing (on) or not flowing (off). This simple two-state system is what we call binary.

Each binary digit is either:

• 0 representing "off" (no signal)
• 1 representing "on" (signal present)

By combining these zeros and ones at very high speeds, computers can represent text, numbers, sounds, images and everything else you use your computer for. Modern processors work at speeds measured in gigahertz (GHz). For example, a 2 GHz processor receives 2 billion electrical pulses every second, allowing it to process vast amounts of binary data incredibly quickly.

The byte

While a single bit can only represent two values, computers need to represent much more information than just on or off. This is where the byte comes in.

A byte is a group of bits, typically 8 bits grouped together. This group of 8 bits forms a useful unit because it provides enough different combinations to represent all the characters you can type on a keyboard.

The number of different values you can represent with bits follows a pattern based on powers of 2:

• One bit gives 2 possibilities: 0 and 1 (written as $2^1$)
• Two bits give 4 possibilities: 00, 01, 10, 11 (written as $2^2$)
• Three bits give 8 possibilities ($2^3 = 2 \times 2 \times 2$)
• Four bits give 16 possibilities ($2^4 = 2 \times 2 \times 2 \times 2$)
• Eight bits give 256 possibilities ($2^8$)

Eight bits (one byte) provides $2^8 = 256$ different combinations. This is sufficient to represent every letter of the alphabet (both uppercase and lowercase), all digits, punctuation marks, and other keyboard characters. That's why one byte typically represents one character.

Units of data storage

When measuring computer memory and storage capacity, we combine bytes into larger units. However, there's an important distinction between two types of units: binary prefixes and decimal prefixes.

Binary prefixes are based on powers of 2 and are more accurate for describing computer memory. These use the terms kibibyte, mebibyte, gibibyte, and tebibyte:

• Kibibyte (Ki) = $2^{10}$ bytes = 1,024 bytes
• Mebibyte (Mi) = $2^{20}$ bytes = 1,048,576 bytes
• Gibibyte (Gi) = $2^{30}$ bytes
• Tebibyte (Ti) = $2^{40}$ bytes

Decimal prefixes are based on powers of 10 and are often used in marketing for storage devices. These use the familiar terms kilobyte, megabyte, gigabyte, and terabyte:

• Kilobyte (KB) = $10^3$ bytes = 1,000 bytes
• Megabyte (MB) = $10^6$ bytes = 1,000,000 bytes
• Gigabyte (GB) = $10^9$ bytes
• Terabyte (TB) = $10^{12}$ bytes

The table below shows the comparison:

Understanding number bases

A number base (also called a radix) indicates how many different digits are available in a particular number system. The base determines which digits can be used and how place values work in that system.

Humans typically use decimal, which is base 10. This means it uses 10 different digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. We likely developed this system because we have ten fingers to count on.

Computers use binary, which is base 2. This means it uses only 2 different digits: 0 and 1. Binary works perfectly for computers because it matches their two-state electrical system (on or off).

Computing also uses hexadecimal (often shortened to "hex"), which is base 16. This system uses 16 different digits: 0-9 (just like decimal) plus the letters A, B, C, D, E, and F to represent the values 10-15.

The number base tells you how many digits are needed to represent a value. For instance, the decimal number 98 requires two digits in base 10, but the same value written in binary ($1100010_2$) requires seven digits. This is why longer codes are often needed in binary - there are fewer digits available, so more positions are needed to represent larger numbers.

Hexadecimal

Hexadecimal is particularly useful in computing because it provides a convenient shorthand for binary. Since binary numbers can become very long, hex offers a more compact way to represent the same values.

Hexadecimal uses 16 different symbols:

• 0-9 represent the values 0 to 9 (same as decimal)
• A represents 10
• B represents 11
• C represents 12
• D represents 13
• E represents 14
• F represents 15

Here's a lookup table showing decimal numbers and their hexadecimal equivalents:

Hexadecimal is commonly used in several computing contexts:

• Memory addresses are displayed in hex
• Colour codes in web design use hex (e.g., #FF0000 for red)
• Machine code and assembly language often display values in hex
• Error codes and debugging information frequently use hex

Working with number bases

To understand how different number bases work, it helps to think about how we use decimal in everyday life.

How decimal works

When you write the number 2098 in decimal, each digit has a different place value based on powers of 10:

• The rightmost digit (8) is worth $8 \times 1$ ($10^0$)
• The next digit (9) is worth $9 \times 10$ ($10^1$)
• The next digit (0) is worth $0 \times 100$ ($10^2$)
• The leftmost digit (2) is worth $2 \times 1000$ ($10^3$)

Adding these together: $(2 \times 1000) + (0 \times 100) + (9 \times 10) + (8 \times 1) = 2098$

Each position is worth ten times more than the position to its right. This is why we call it base 10.

How binary works

Binary follows the same principle, but each position is worth two times more than the position to its right because it's base 2.

Consider the 8-bit binary number $10000111_2$:

Reading from right to left, each bit position represents a power of 2:

• Position 1 (rightmost): $2^0 = 1$
• Position 2: $2^1 = 2$
• Position 3: $2^2 = 4$
• Position 4: $2^3 = 8$
• Position 5: $2^4 = 16$
• Position 6: $2^5 = 32$
• Position 7: $2^6 = 64$
• Position 8 (leftmost): $2^7 = 128$

To find the decimal value, add up the place values where there's a 1:

$(1 \times 128) + (0 \times 64) + (0 \times 32) + (0 \times 16) + (0 \times 8) + (1 \times 4) + (1 \times 2) + (1 \times 1) = 135$

Therefore, $10000111_2 = 135_{10}$

Converting between number bases

Being able to convert between binary, decimal, and hexadecimal is an essential skill in computer science. Here are the methods for each type of conversion.

Binary to decimal conversion

To convert a binary number to decimal, follow these steps:

1. Write down the binary number
2. Above each bit, write the place value (starting from 1 on the right and doubling each time as you move left)
3. Wherever there's a 1, add that place value to your total
4. Wherever there's a 0, ignore that place value

Decimal to binary conversion

There are two common methods for converting decimal to binary. Both give the same result.

Method 1: Using a place value table

1. Write down the powers of 2 (place values) from right to left
2. Starting from the left (MSB), put a 1 or 0 in each position to make the sum equal your target number
3. Work through each place value, asking "Can I include this without going over?"

Method 2: Repeated division by 2

1. Divide the decimal number by 2
2. Write down the remainder (0 or 1)
3. Divide the quotient by 2 again
4. Keep repeating until the quotient is 0
5. Read the remainders from bottom to top

Decimal to hexadecimal conversion

The most common approach for converting decimal to hex is to first convert the decimal to binary, then convert the binary to hex.

Hexadecimal to decimal conversion

To convert hex to decimal, first convert each hex digit to binary (using 4 bits per hex digit), then convert the complete binary number to decimal.

Remember!