Hexadecimal Number System

What Is the Hexadecimal Number System?

The hexadecimal number system is a way of counting that uses a base of $16$ instead of the base-$10$ system we use every day. Our normal number system, called decimal or base-10, uses ten different digits to write any number: $0,1,2,3,4,5,6,7,8,$ and $9$. Once we count past $9$, we add a new place value and write $10$, which means 'one group of ten and zero ones'.

The hexadecimal system, or base-16, needs more symbols because it counts in groups of sixteen. It uses the same ten digits we're used to ($0$ through $9$), but then it needs six more symbols to represent the values for ten, eleven, twelve, thirteen, fourteen, and fifteen. For these, we use the first six letters of the alphabet: A, B, C, D, E, and F.

So, in hexadecimal (often shortened to just "hex"), the digits are:

$0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F$

It might seem strange to use letters as numbers, but once you get the hang of it, you'll see it's just another way of organizing numbers, one that is incredibly useful for computers.

Why Do We Use Hexadecimal Numbers?

If we have a perfectly good base-10 system, why bother with a complicated base-16 system that uses letters? The answer lies inside our computers. At their most basic level, computers don't think in base-10. They think in binary, which is base-2. Binary only uses two digits: $0$ and $1$.

A computer stores all information—from the words in this lesson to the colors on your screen—as incredibly long strings of zeros and ones, like $1101011010110010$. As you can imagine, it's very easy for a human to make a mistake when reading or writing such a long number. This is where hexadecimal becomes a hero!

The number $16$ is a power of $2$ ($16=2^4$). This creates a special relationship between hex and binary. It means that exactly one hexadecimal digit can be used to represent a group of four binary digits.

• For example, the binary string $1101$ is equal to $13$ in decimal, which is the digit $D$ in hexadecimal.
• The long binary number from before, $1101011010110010$, can be broken into four groups: $1101$ $0110$ $1011$ $0010$.
• In hex, this becomes $D6B{2}_{16}$.

Which one would you rather write down? ${1101011010110010}_2$ or $D6B{2}_{16}$? Hexadecimal acts as a much shorter, more human-friendly language to express the binary information that computers use. You see it in many places in technology:

• Web Colors: To tell a web browser exactly which shade of blue to show, you use a hex code like $\mathrm{\#}0000FF$.
• Memory Addresses: Computers use hex to label locations in their memory, like $0x7FFF$.
• Error Codes: When a program crashes, it might show an error code like $0x80070005$ that helps developers figure out what went wrong.

The Hexadecimal Digits: 0-9 and A-F

The key to understanding hexadecimal is to get comfortable with its sixteen digits. The first ten are easy, as they are the same as our decimal system. The next six are letters that represent the decimal values from $10$ to $15$. It's very important to remember that in this context, 'A' is not a letter; it is the number ten.

Here is a table that shows each hexadecimal digit and its decimal equivalent. This is a great chart to keep handy as you're learning!

When you're counting in hex, you go up the list. After $9$, the next number isn't $10$, it's $A$. Then $B$, $C$, $D$, $E$, and $F$. What comes after $F$? Just like in decimal where we run out of digits after $9$ and write $10$, in hex we run out of digits after $F$ and write ${10}_{16}$. This hex number ${10}_{16}$ means 'one group of sixteen and zero ones,' which is equal to $16$ in our decimal system.

How Do You Convert from Hexadecimal to Decimal?

Converting a hexadecimal number to a decimal number is all about understanding place value. In our familiar decimal system, each place represents a power of $10$. For the number $345$, we know it means:

Hexadecimal works the exact same way, but the place values are powers of $16$.

• The rightmost place is the ones place (${16}^0=1$).
• The next place to the left is the sixteens place (${16}^1=16$).
• The next place is the two-hundred-fifty-sixes place (${16}^2=256$), and so on.

To convert any hex number to decimal, you follow these steps:

1. Write down the hex number and identify the place value for each digit (${16}^0,{16}^1,{16}^2$, etc., from right to left).
2. If any digit is a letter (A-F), convert it to its decimal value ($10$-$15$).
3. Multiply each digit's decimal value by its corresponding place value.
4. Add all these products together. The sum is your answer in decimal.

How Do You Convert from Decimal to Hexadecimal?

Going the other way, from decimal to hexadecimal, uses a cool math trick involving repeated division. The goal is to figure out how many groups of $16$, $256$, and so on, fit into our decimal number. The remainders from our division will become the digits of our hex number.

Follow these steps:

1. Take your decimal number and divide it by $16$.
2. Write down the whole number part of the answer (the quotient) and the remainder. The remainder will be a number from $0$ to $15$.
3. Convert the remainder to its hexadecimal digit (for example, if the remainder is $10$, the digit is $A$). This is your first hex digit (the rightmost one).
4. Take the quotient from step 2 and divide it by $16$ again.
5. Repeat this process, writing down the new remainder's hex digit to the left of the previous one.
6. Continue until your quotient is $0$. The hex number is the sequence of remainders you found, read from the bottom up.

A Colorful Application: Hex Codes in Digital Design

One of the most common places you'll encounter hexadecimal numbers is in web design and digital art. Colors on a computer screen are created by mixing different amounts of red, green, and blue light. This is known as the RGB color model.

Each of these three colors (Red, Green, and Blue) is assigned an intensity value from $0$ (none of that color) to $255$ (the maximum amount). Why $255$? Because a two-digit hexadecimal number can represent exactly $256$ different values (from $0$ to $255$).

• ${00}_{16}=0_{10}$ (minimum intensity)
• $F{F}_{16}=(15\times 16)+(15\times 1)=240+15={255}_{10}$ (maximum intensity)

A hex color code is a six-digit hex number, usually starting with a hashtag ($\mathrm{\#}$). The six digits are actually three pairs of two-digit hex numbers, representing the amount of Red, Green, and Blue light:

$\mathrm{\#}RRGGBB$

• The first two digits (RR) are for Red.
• The middle two digits (GG) are for Green.
• The last two digits (BB) are for Blue.

Common Mistakes to Avoid

As you start working with hexadecimal numbers, there are a few common pitfalls to watch out for. Being aware of them can save you a lot of confusion!

• Forgetting Letters are Numbers: The most common mistake is seeing a letter like 'C' and not immediately thinking of it as the number $12$. When you are converting from hex to decimal, you must substitute the decimal value of the letter before you multiply by the place value.
• Place Value Errors: When converting from hex, always remember the place values are powers of $16$ ($1,16,256,...$), not powers of $10$ ($1,10,100,...$). It's easy to slip back into the habit of using the decimal system's place values.
• Reading Remainders Backwards: When converting from decimal to hex using the division method, the remainders must be read from the bottom up. The last remainder you calculate is the first (leftmost) digit of your hex number. Reading them top-down will give you the wrong answer.
• Mixing Up Conversion Methods: There are two different processes: one for hex-to-decimal (multiplication and addition) and one for decimal-to-hex (repeated division). Don't try to use division to convert from hex to decimal, or multiplication to go from decimal to hex. Keep the two methods separate in your mind.
• Forgetting a Zero Remainder: If you divide by $16$ and get a remainder of $0$, you must write down $0$ as that place's hex digit. It's a placeholder, just like in the number $205$, and is very important.

Quick Summary

Here are the most important points to remember about the hexadecimal number system:

• Base-16: The hexadecimal system is a base-16 system.
• 16 Symbols: It uses sixteen unique symbols for its digits: $0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F$.
• Letter Values: The letters A-F represent the decimal values $10$-$15$.
• Place Values: The place values are powers of $16$, such as ${16}^0=1$, ${16}^1=16$, ${16}^2=256$, and so on.
• Hex to Decimal Conversion: To convert a hex number to decimal, multiply each digit by its place value and then add all the results together.
• Decimal to Hex Conversion: To convert a decimal number to hex, use repeated division by $16$ and read the remainders from the bottom up.
• Computer Language: Hexadecimal is used in computing as a human-friendly shorthand for binary code, especially for things like colors, memory addresses, and error codes.