1. Data Units, Binary & Hexadecimal

Every digital system, from smartphones to supercomputers, relies on binary – a language made entirely of 1s and 0s. This chapter lays the foundation for understanding how computers represent and store all kinds of data. Whether it’s a number, a word, or a file, it must be converted into binary before it can be processed or saved.

You’ll explore the different units used to measure data – from bits to petabytes – and how to calculate the size of files using common units. You’ll also learn to convert between number systems: denary, binary, and hexadecimal. These skills are vital for understanding how memory works and for solving common computing problems like overflow errors.

All modern computers rely on binary – a number system made up of just two digits: 1 and 0. This is because the internal components of a computer, such as circuits and transistors, are designed to be in one of two states: on (represented by a 1) or off (represented by a 0). These two states can be used to encode all forms of data, from numbers and letters to images, videos, and software instructions.

Before any type of data can be stored, processed, or transmitted by a computer, it must first be converted into binary format. This means that a photograph, for example, becomes a long string of 1s and 0s, where each group of digits might represent the colour of a single pixel. Similarly, the letter “A” is stored as 01000001 using an agreed character encoding system (ASCII). Music, videos, programs, websites – they are all ultimately represented in binary when inside a computer system.

Binary is a practical choice for computers because it’s both simple and reliable. There’s no confusion in digital electronics between just two voltage levels – either something is there or it isn’t. This makes processing fast, storage efficient, and error detection easier to manage. Without binary, digital computing as we know it would not be possible.

When you save a file or install a program, the data takes up space in the computer’s memory or storage device. To measure this space, computers use a system of data units, starting from the smallest possible value – the bit. A bit (short for binary digit) is either a 0 or a 1. By itself, a single bit can only store two possible values, which isn’t very useful on its own. That’s why bits are grouped together to form larger units.

A byte is a group of 8 bits and is enough to store a single character, such as the letter "A". Two bytes can store a short word like "Hi". Four bits make up a nibble, which is rarely used on its own but appears in some technical contexts. As data becomes larger and more complex, we need much larger units to measure it:

• 1 kilobyte (KB) = 1,000 bytes

• 1 megabyte (MB) = 1,000 KB

• 1 gigabyte (GB) = 1,000 MB

• 1 terabyte (TB) = 1,000 GB

• 1 petabyte (PB) = 1,000 TB

For example, a single page of text might take around 30 KB, while a photo from a modern phone might be 2–5 MB. A film could take up 2–4 GB depending on its quality. Large online services like Google or Netflix deal with data on the petabyte scale. Understanding these units helps us manage storage and transfer data more efficiently.

When working with digital storage, it's important to understand how to convert between units like kilobytes (KB), megabytes (MB), and gigabytes (GB). In the OCR specification, conversions are based on base-10, which means every step up or down involves multiplying or dividing by 1,000. For example, 1 MB = 1,000 KB, and 1 GB = 1,000 MB.

To convert a large unit into a smaller one, multiply by 1,000. For example, if you have 3.5 GB, you multiply by 1,000 to get 3,500 MB. To go the other way and convert a smaller unit into a larger one, divide by 1,000. So, 1,200 KB divided by 1,000 equals 1.2 MB. For more detailed measurements, you may also need to convert bytes: 1 MB = 1,000,000 bytes, so 5 MB = 5,000,000 bytes.

Understanding these conversions helps when comparing storage devices, estimating how many files will fit on a USB stick, or choosing the right size memory card for a phone or camera. It's also essential when calculating how much space will be needed to download or back up data.

Different types of files require different amounts of storage space, depending on what kind of data they contain. To estimate how much space is needed, we use simple formulas that vary depending on the file type. For text files, the calculation is straightforward: each character typically takes up 1 byte (in ASCII), so a 1,000-character document would require 1,000 bytes, or 1 KB.

For images, the amount of data depends on the resolution (width × height in pixels) and the colour depth (the number of bits used to store the colour of each pixel). To calculate image file size: multiply the width by the height by the colour depth. For example, a 200 × 300 image using 24-bit colour (which allows over 16 million colours) would be:
200 × 300 × 24 = 1,440,000 bits = 180,000 bytes = 180 KB.

Audio files require three key pieces of information to calculate size: the sample rate (how many samples per second), bit depth (bits used to store each sample), and the number of channels (e.g., 1 for mono, 2 for stereo), along with the duration in seconds. For example, an audio file recorded at 44,100 samples/second, 16-bit depth, 2 channels, and 60 seconds long would be: 44,100 × 16 × 2 × 60 = 84,480,000 bits = 10.56 MB.

Being able to calculate data capacity helps you choose appropriate storage devices, plan for downloads, and understand why some file types take up more space than others.

Binary is a number system based on powers of 2. In an 8-bit binary number, each position has a specific value from left to right: 128, 64, 32, 16, 8, 4, 2, 1. These are known as place values. Converting a denary (decimal) number to binary means working out which of these values can be added together to make the number, and marking each with a 1 (used) or a 0 (not used).

For example, to convert the denary number 156 into binary, start with the largest place value that fits into it – 128. Subtracting 128 leaves 28, then take 16 (leaving 12), then 8 (leaving 4), then 4 (leaving 0). So 156 = 128 + 16 + 8 + 4 → the binary result is 10011100.
To convert binary to denary, reverse the process: look at which positions have a 1, and add those values. So 01101001 becomes 64 + 32 + 8 + 1 = 105.

All numbers in this course are stored using 8-bit binary, which allows values from 0 to 255. But what happens if we go beyond that? This is called an overflow error. For example, 255 + 1 = 256, but 256 cannot be stored in 8 bits, because 256 in binary is 100000000, which is 9 bits. In this case, the extra leftmost bit is lost, and the result becomes 00000000, causing incorrect results. Overflow is a common issue in computing and must be carefully managed.

Hexadecimal (often shortened to hex) is a base-16 number system used as a simpler way to represent binary. Instead of using just digits 0–9, hexadecimal also uses letters A to F, where A = 10, B = 11, up to F = 15. Hex is helpful in computing because it's shorter and easier for humans to read than long binary strings, but still maps directly to binary values.

To convert a denary number (0–255) into a 2-digit hexadecimal number, divide the number by 16. The quotient becomes the first hex digit, and the remainder becomes the second. For example:
200 ÷ 16 = 12 remainder 8 → the hex value is C8 (since 12 = C and 8 = 8).

To convert a hexadecimal number to denary, do the reverse: multiply the first digit by 16 and add the second. So for 1A, we convert it as:
(1 × 16) + 10 = 26. Another example: 3F = (3 × 16) + 15 = 63.

Hex is especially useful for representing values in memory addresses, colours in HTML (like #FF0000 for red), and simplifying binary code.

You can find more information about converting between hex at this page in the Tutorials section of the site.

Converting between binary and hexadecimal is fast and simple because both number systems are closely related. Each hexadecimal digit can be directly matched with a group of 4 binary digits (called a nibble). This makes hex a convenient shorthand for long binary numbers, which are harder to read and write.

To convert binary to hex, start by splitting the binary number into groups of 4 bits from right to left. Each nibble is then converted into a single hex digit. For example, the binary number 10101100 splits into 1010 and 1100. Looking up the values, 1010 = A and 1100 = C, so the final hex result is AC.

To go the other way – hex to binary – simply convert each hex digit into its 4-bit binary equivalent. For instance, B7 becomes B = 1011 and 7 = 0111, so the full binary number is 10110111.

This method is widely used when viewing memory addresses, setting colours in design tools, or debugging code. Because of the clean 4-to-1 relationship, hex is the most compact and readable way to work with binary.