Decimal To Hexadecimal

What Are Decimal and Hexadecimal Numbers?

Converting from decimal to hexadecimal means changing a number from base-10 to base-16. But what does that even mean? Let's break it down.

The decimal system, also called base-10, is the number system we use every day. It's called base-10 because it uses ten different digits to represent all possible numbers: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. When we count past 9, we add a new place value (the tens place), like in the number 10.

The hexadecimal system is a base-16 system. As you might guess, this means it uses sixteen different digits! It uses the same ten digits from the decimal system (0-9), but it needs six more to get to sixteen. Instead of inventing new symbols, we use the first six letters of the alphabet: A, B, C, D, E, and F.

So, the sixteen digits in hexadecimal are: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F.

Here's how the letters match up to decimal values:

• A = 10
• B = 11
• C = 12
• D = 13
• E = 14
• F = 15

Let's see how they compare side-by-side. Notice how hexadecimal uses a single digit for numbers where decimal needs two.

Look at that last row! The decimal number $16$ is written as $10$ in hexadecimal. This is because in base-16, the second place value represents groups of $16$, not groups of $10$. So, ${10}_{16}$ means one group of $16$ and zero ones.

Why Do We Need Another Number System?

You might be thinking, "The decimal system works just fine. Why learn a new one?" That's a great question! While we use base-10 for everyday counting, computers often use hexadecimal for very specific and important reasons.

Computers, at their core, only understand two things: ON and OFF. They use a base-2 system called binary, which only has two digits: $0$ (off) and $1$ (on). A simple decimal number like $200$ looks like this in binary: $11001000$. That's a long string of numbers and very hard for humans to read quickly and accurately!

This is where hexadecimal becomes a hero. Hexadecimal is like a shorthand for binary. Every four binary digits can be represented by exactly one hexadecimal digit.

For example:

• The binary $1100$ is $12$ in decimal, which is $C$ in hexadecimal.
• The binary $1000$ is $8$ in decimal, which is $8$ in hexadecimal.

So, the long binary number $11001000$ can be split into $1100$ and $1000$, which becomes the much shorter and easier-to-read hexadecimal number $C8$.

You see hexadecimal numbers all the time, even if you don't realize it:

• Web Colors: Have you ever seen a color code like #FF0000? That's a hexadecimal number telling the computer to show the color red.
• Error Codes: When a computer program crashes, it might show an error code full of letters and numbers. Those are often hexadecimal values that help programmers figure out what went wrong.
• Memory Addresses: Every piece of data in a computer's memory has an address, which is almost always shown in hexadecimal because the numbers can get incredibly large.

By learning hexadecimal, you're learning a language that helps people communicate with computers more efficiently.

How Do You Convert from Decimal to Hexadecimal?

The most common way to convert a decimal number to hexadecimal is the repeated division method. It might sound complicated, but it's just a simple process of dividing by $16$ over and over again. You just need to keep track of the remainders.

Here are the steps:

1. Divide by 16: Take your decimal number and divide it by $16$.
2. Record the Remainder: Write down the whole number remainder. This remainder is a hexadecimal digit! If the remainder is between $10$ and $15$, you must convert it to its letter equivalent (A, B, C, D, E, or F).
3. Use the Quotient: Take the whole number part of your answer (the quotient) and repeat the process. Divide it by $16$.
4. Keep Going: Continue dividing the new quotient by $16$ until you get a quotient of $0$.
5. Read Upwards: Once your quotient is $0$, your conversion is done! The hexadecimal number is the list of remainders you recorded, read from the bottom to the top.

Let's think about that last step. Why do we read the remainders from bottom to top? The first remainder you calculate is the value for the ones place (${16}^0$). The next remainder is for the sixteens place (${16}^1$), the next is for the two-hundred-fifty-sixes place (${16}^2$), and so on. Reading them in reverse order puts these digits in their correct place values. We'll see this in action in our examples.

Example 1: Converting a Two-Digit Decimal Number

Let's practice with a straightforward number. We'll convert the decimal number $75$ into hexadecimal.

Example 2: Converting a Larger Decimal Number

Now let's try a bigger number that requires a few more steps. This will help you get comfortable with the repeated division process.

Example 3: A Conversion with Multiple Letters

This final example will show how multiple letters can appear in a hexadecimal number. The process is exactly the same, so don't be intimidated!

What Are Some Common Mistakes to Avoid?

When you're first learning to convert from decimal to hexadecimal, it's easy to make a few common mistakes. Being aware of them is the best way to avoid them!

• Reading Remainders Top-to-Bottom: This is the most frequent error. Always remember to write down your remainders and then read them from the last one you found to the first one (bottom-up). If you read them in the order you found them, your answer will be backward.
• Forgetting to Convert to Letters: Any time you get a remainder that is $10,11,12,13,14,$ or $15$, you must change it to its hexadecimal letter (A, B, C, D, E, or F). Writing a remainder like "12" in your final answer is incorrect, as it would be read as the digits one and two. It must be written as "C".
• Simple Division Errors: The whole process relies on correct division. Work carefully and double-check your subtraction when finding remainders. Using a calculator to check your division (e.g., $75/16=4.6875$) can help. To find the remainder from this, you can calculate $0.6875\times 16=11$.
• Stopping Too Early: Don't stop dividing until the quotient is zero. Even if you get a quotient like $1$ or $5$ (which is less than $16$), you still need to do one last division step. For example, $5÷16$ gives a quotient of $0$ and a remainder of $5$. That $5$ is a critical part of your final answer.

Quick Summary and Reference Chart

Here is a quick summary of the steps to convert any decimal number to hexadecimal, along with the reference chart for digits $10$ through $15$.

Conversion Steps:

1. Take the decimal number and divide it by $16$.
2. Write down the remainder. Convert it to a letter (A-F) if it's $10$ or greater.
3. Take the whole number quotient from the previous step and divide it by $16$ again.
4. Repeat until the quotient is $0$.
5. Write the list of remainders in reverse order (from bottom to top).

Hexadecimal Digit Reference:

Keep this chart handy when you're working. With a little practice, you'll have these memorized in no time!