Converting between number bases

Welcome to the World of Number Bases!

Ever wondered how a computer, which is essentially just a bunch of tiny switches, can understand complex things like video games or homework? It all starts with how it counts. While we use ten fingers to count (Decimal), computers use electricity—either "on" or "off" (Binary). In this guide, we are going to learn how to translate between our human language of numbers and the computer's language.

Don't worry if this seems like a lot of math at first! It is actually more like learning a simple code. Once you know the "secret key" for each base, you'll be converting numbers like a pro.

1. The Basics: Our Three Main Bases

Before we start converting, let's meet the three systems you need to know for your AQA exam:

• Decimal (Base 10): This is what you use every day. It uses the digits \(0\) to \(9\).
• Binary (Base 2): The computer’s native language. It only uses \(0\) and \(1\).
• Hexadecimal (Base 16): A "shorthand" for binary. It uses \(0\) to \(9\) and then the letters A, B, C, D, E, and F.

Quick Review: Why do we use Hexadecimal? Imagine writing out a long phone number using only \(0\)s and \(1\)s. It would take forever! Hexadecimal is used by programmers because it is much shorter and easier for humans to read than long strings of binary.

2. Converting Binary to Decimal

To convert from Binary to Decimal, we use a simple place-value table. Since we are working with numbers up to \(255\), we always use \(8\) bits (one byte). Each position in binary is worth double the one to its right.

The Step-by-Step Method:

1. Draw a table with these eight numbers: 128, 64, 32, 16, 8, 4, 2, 1.
2. Write your binary number underneath the table.
3. Add up all the numbers from the top row that have a \(1\) underneath them.

Example: Convert \(10101000\) to Decimal.
\(128\) (Yes) + \(64\) (No) + \(32\) (Yes) + \(16\) (No) + \(8\) (Yes) + \(4\) (No) + \(2\) (No) + \(1\) (No)
Calculation: \(128 + 32 + 8 = 168\).
Answer: \(168\)

Key Takeaway: If there is a \(1\), the "light switch" is on—add that value! If there is a \(0\), skip it.

3. Converting Decimal to Binary

This is just the previous method in reverse! We call this the "Subtract and Move" method.

The Step-by-Step Method:

1. Start with your decimal number.
2. Look at your place-value table (starting at \(128\)).
3. Can you subtract the table number from your current number?
• If YES: Put a \(1\) in that column and subtract the value.
• If NO: Put a \(0\) in that column and move to the next number.

Example: Convert \(75\) to Binary.
• Can we take \(128\) from \(75\)? No (\(0\))
• Can we take \(64\) from \(75\)? Yes (\(1\)). Leftover: \(75 - 64 = 11\)
• Can we take \(32\) from \(11\)? No (\(0\))
• Can we take \(16\) from \(11\)? No (\(0\))
• Can we take \(8\) from \(11\)? Yes (\(1\)). Leftover: \(11 - 8 = 3\)
• Can we take \(4\) from \(3\)? No (\(0\))
• Can we take \(2\) from \(3\)? Yes (\(1\)). Leftover: \(3 - 2 = 1\)
• Can we take \(1\) from \(1\)? Yes (\(1\)). Leftover: \(0\)
Answer: \(01001011\)

Common Mistake: Students often stop early. Even if you reach \(0\), make sure you fill in the rest of the table with \(0\)s until you have all \(8\) bits!

4. Understanding Hexadecimal (Base 16)

Hexadecimal is a bit strange because we run out of numbers after \(9\). To keep each value to a single character, we use letters:

A = \(10\), B = \(11\), C = \(12\), D = \(13\), E = \(14\), F = \(15\)

Memory Aid: Just remember that A is the first letter and \(10\) is the first double-digit number. From there, you can just count on your fingers!

5. Binary to Hexadecimal (The "Nibble" Trick)

This is the easiest conversion of all! A "Nibble" is just half a byte (4 bits).

The Step-by-Step Method:

1. Split your \(8\)-bit binary number into two groups of \(4\).
2. Calculate the decimal value for each group separately (using the \(8, 4, 2, 1\) values).
3. Convert those values to Hex (using letters if the number is \(10\) or higher).
4. Put the two characters together.

Example: Convert \(11100011\) to Hex.
• Left side: \(1110\). (\(8+4+2 = 14\)). In Hex, \(14\) is E.
• Right side: \(0011\). (\(2+1 = 3\)). In Hex, \(3\) is 3.
Answer: E3

Did you know? The maximum value for an \(8\)-bit number is \(255\) in decimal, which is written as \(11111111\) in binary and FF in hexadecimal.

6. Hexadecimal to Decimal

To go from Hex to Decimal, we look at the two characters. The first character is in the "\(16\)s" column, and the second is in the "\(1\)s" column.

The Step-by-Step Method:

1. Convert the first Hex digit to its decimal value and multiply it by \(16\).
2. Convert the second Hex digit to its decimal value.
3. Add the two numbers together.

Example: Convert \(2A\) to Decimal.
• The first digit is \(2\). Multiply by \(16\): \(2 \times 16 = 32\).
• The second digit is A. In decimal, A is \(10\).
• Add them: \(32 + 10 = 42\).
Answer: \(42\)

Alternative Method: If multiplying by \(16\) is hard, you can use the "Bridge Method": Convert the Hex to Binary first (using the nibble trick), then convert that Binary to Decimal. It takes an extra step but is often more reliable!

7. Quick Review Box

• Maximums: Decimal \(255\) = Binary \(11111111\) = Hex FF.
• Binary Columns: \(128, 64, 32, 16, 8, 4, 2, 1\).
• Hex Letters: A(\(10\)) through F(\(15\)).
• The Nibble: To move between Binary and Hex, split the byte into two groups of \(4\).
• Precision: Always ensure your binary numbers have \(8\) digits (leading zeros are important!).

Final Summary

Converting between bases is just a matter of following the right map. Whether you are using the place-value table for binary or splitting bytes into nibbles for hex, the steps are always the same. Practice with small numbers first, and soon you'll be reading these "secret codes" as easily as your own name! You've got this!