Hexadecimal numbers

Binary numbers are a great way for computers to represent numbers. They're not as great for humans though—they're so very long, and it takes a while to count up all those $1$1s and $0$0s. When computer scientists deal with numbers, they often use either the decimal system or the hexadecimal system. Yes, another number system!

Fortunately, number systems are more alike than they are different, and now that you've mastered decimal and binary, hexadecimal will hopefully make sense.

In the decimal system, each digit represents a power of $10$10—the ones' place, the tens' place, etc. We call $10$10 the base of the decimal system.

In the hexadecimal system, each digit represents a power of $16$16.

Here's how to count to 10 in hexadecimal: $1$1, $2$2, $3$3, $4$4, $5$5, $6$6, $7$7, $8$8, $9$9, $\text{A}$start text, A, end text.

That looks very familiar until the final "number" $\text{A}$start text, A, end text. You see, in the hexadecimal system, each digit needs to represent the values $0$0-$15$15, but the decimal numbers $10$10-$15$15 don't fit into a single digit. To work around that, the hexadecimal system uses the letters $\text{A}$start text, A, end text-$\text{F}$start text, F, end text to represent the numbers $10$10-$15$15.

Here, let's count from 10 to 15: $\text{A}$start text, A, end text, $\text{B}$start text, B, end text, $\text{C}$start text, C, end text, $\text{D}$start text, D, end text, $\text{E}$start text, E, end text, $\text{F}$start text, F, end text. Strange but true!

There's actually been many different proposals over the years for how to represent the values 10-15, but $\text{A}$start text, A, end text-$\text{F}$start text, F, end text is the solution that won out.

Let's count higher now. Here's the decimal number $24$24, represented in hexadecimal as $18$18:

This number requires two digits, where the right digit represent the ones' place ($16^0$16, start superscript, 0, end superscript) and the left digit represents the sixteens' place ($16^1$16, start superscript, 1, end superscript). The same way that we added decimal and binary numbers, we can add up $(1 \times 16) + (8 \times 1)$left parenthesis, 1, times, 16, right parenthesis, plus, left parenthesis, 8, times, 1, right parenthesis and see that hexadecimal $18$18 equals the decimal value $24$24.

Let's try a number with a letter in it. Here's the decimal number $27$27, represented in hexadecimal as $1\text{B}$1, start text, B, end text:

To understand this one, we need to first remember that $\text{B}$start text, B, end text represents the value $11$11. I do that by counting the letters on my fingers, but you can count however you'd like, as long as you remember to start with $\text{A}$start text, A, end text representing $10$10.

Then we can add it up like before and see that $(1 \times 16) + (11 \times 1)$left parenthesis, 1, times, 16, right parenthesis, plus, left parenthesis, 11, times, 1, right parenthesis does equal the decimal value $27$27.

At this point, you might be wondering what it is that computer scientists like so much about the hexadecimal system. Why use a system where we have to use letters to represent numbers? A little detour into history will show us why...

Early computers used 4-bit architectures, which meant that they always processed bits in groups of 4. That's why we still write bits in groups of 4, like when we write $0111$0111 to represent decimal $7$7 even though we could just write $111$111.

How many values can 4 bits represent? The lowest value is $0$0 (all $0$0s, $0000$0000) and the highest value is $15$15 (all $1$1s, $1111$1111), so 4 bits can represent 16 unique values. Aha, there's that number 16!

Each group of 4 bits in binary is a single digit in the hexadecimal system. That makes it really easy to convert binary numbers to hexadecimal numbers, and it makes it a natural fit for computers to use as well.

Let's prove how well binary and hexadecimal get along by converting from binary to hexadecimal.

We'll start with a short binary number:

That's 4 bits long, which means it maps to a single hexadecimal digit. There's a $0$0 in every place except the twos' place, so it equals the decimal number $2$2. The decimal and hexadecimal systems both represent the numbers $0$0-$9$9 the same way, so $0010$0010 is simply $2$2 in hexadecimal.

Let's try a longer binary number:

One approach would be to figure out what decimal number is represented by that long string of $1$1s and $0$0s, and then convert that to hexadecimal. That approach would work, but gosh, it seems like a lot of work for this long of a number.

The easier approach is to convert each group of 4 bits, one at a time.

Starting with the left-most group of $1001$1001, that equals $(1 \times 8) + (1 \times 1)$left parenthesis, 1, times, 8, right parenthesis, plus, left parenthesis, 1, times, 1, right parenthesis, the decimal number $9$9. That number is less than $10$10, so $1001$1001 is simply $9$9 in hexadecimal.

The next group is $1010$1010, or $(1 \times 8) + (1 \times 2)$left parenthesis, 1, times, 8, right parenthesis, plus, left parenthesis, 1, times, 2, right parenthesis, the decimal value $10$10. That number requires a letter representation in hexadecimal, $\text{A}$start text, A, end text.

The next group is $0110$0110, or $(1 \times 4) + (1 \times 2)$left parenthesis, 1, times, 4, right parenthesis, plus, left parenthesis, 1, times, 2, right parenthesis, the decimal value $6$6. That number is the same in hexadecimal, $6$6.

The final group is $1100$1100, or $(1 \times 8) + (1 \times 4)$left parenthesis, 1, times, 8, right parenthesis, plus, left parenthesis, 1, times, 4, right parenthesis, the decimal value $12$12. That number requires a letter representation in hexadecimal, $\text{C}$start text, C, end text.

Our final hexadecimal result is $9\text{A}6\text{C}$9, start text, A, end text, 6, start text, C, end text. We did that conversion without ever understanding what decimal number is represented. I'll reveal now that both $1001\,1010 \,0110\,1100$1001101001101100 and $9\text{A}6\text{C}$9, start text, A, end text, 6, start text, C, end text equal the decimal number $39,532$39, comma, 532. That's an awfully big number, I'm glad we converted it one digit at a time.

Let's build some intuition around the hexadecimal system.

First: for a given number of digits, what does the biggest number look like for those digits? In the decimal system, that's all $9$9s, like $9,999$9, comma, 999. In the binary system, that's all $1$1s, like $1111$1111.

In the hexadecimal system, that's when every digit has an $\text{F}$start text, F, end text in it, $\text{FFFF}$start text, F, F, F, F, end text. When we see a number like that, we know that it represents the highest possible value for that number of digits. To represent any higher number, we'd need more digits.

Okay, now how much does a big number like that represent? We can use the same rule that we used for binary: the largest number that can be represented by a number of digits $n$n is the same as $16^n - 1$16, start superscript, n, end superscript, minus, 1. Now that we're dealing with base 16 instead of base 2, we might need to bring out a calculator to figure that out though.

Here's a table of the first four largest values:

In this unit on how computers work, we're mostly going to be dealing with binary numbers. It's important to understand the hexadecimal system as well though, because they'll pop up throughout later units and just generally, in the life of a programmer.

As one example from my life, web developers use hexadecimal numbers to represent colors. We describe colors as a combination of three components: red, green, and blue. Each of those components can vary from $0$0 to $255$255. A color like blue can be written as rgb(0, 0, 255) or the more concise hexadecimal version, #0000FF. Using that notation, we can describe $16^6$16, start superscript, 6, end superscript unique colors—more than 16 million colors!

Now that you know about hexadecimal numbers, keep your eyes out for them. You'll often see them written with an 0x in front, like 0x4F, or you might just recognize their distinctive mix of 0-9 with A-F. If you spot any interesting uses for them in the wild, share your discovery below in the Tips & Thanks.

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