What is 1010? If this is your first time learning about the binary number system, then this question may seem odd. Of course it’s ten, right?

Let’s try something different. Have you ever heard this joke?

There are 1010 types of people: those who understand binary and those who don’t.

Unless you’re familiar with binary numbers, this probably doesn’t make much sense. But by the end of this article, you’ll understand this awful joke!

In this beginner’s tutorial, we’ll look at everything you need to know about the binary number system, but we’ll also take a quick look at decimal and hexadecimal, as they’re closely related. I’ll include relevant bits of code and real-life examples to help you appreciate the beauty of binary.

Table of Contents

What Is a Number System?

Before we look at binary, let’s take a step back and discuss number systems more generally.

It may seem strange to think of number systems in the plural if this is your first time learning about them. That’s because the majority of the world is familiar with just one system: the decimal number system, also known as the Arabic number system. This number system uses the digits 090–9 to represent numbers symbolically, based on their position in a string.

For example, in the decimal number system, 579579 expands to this:

579=5(102)+7(101)+9(100)=500+70+9579 = 5(10^2) + 7(10^1) + 9(10^0) = 500 + 70 + 9

In school, you were taught that the 55 in 579579 is in the hundredths place, the 77 is in the tens place, and the 99 is in the ones place. Notice that the 55 is multiplied by one hundred (10210^2), the 77 by ten (10110^1), and the 99 by one (10010^0) to form the decimal number 579579. We say that the number 579579 is positional because the digits, from left to right, correspond to a specific power of ten based on the position of the digit in the number.

Here, the number 1010 is what we call the base (aka radix) of our number system. Notice the powers of 1010 in the expanded expression above: 10210^2, 10110^1, and 10010^0. For this reason, the terms decimal and base ten are interchangeable.

In the decimal number system, a number is represented by placing digits into “buckets” that represent increasing powers of ten, starting with 10010^0 in the rightmost “bucket,” followed by 10110^1 to its immediate left, and so on infinitely:

Increasing powers of ten from right to left, represented as square slots. From right to left, they are labeled: 10^0 (ones), 10^1 (tens), 10^2 (hundredths), and so on.

Any unused buckets to the far left have an implicit value of 00 in them. We usually trim leading zeros because there is no use in saying 0057900579 when that’s mathematically identical to 579579.

Why did humans pick 1010 to be the base of their preferred number system? Likely because most people are born with ten fingers and ten toes, and we’re used to counting with our fingers when we’re young. So it’s natural for us to have adopted ten as the base of our number system.

Bases, Exponents, and Digits

As I’ve already hinted, the decimal number system (base 1010) isn’t the only one in existence. Let’s use a more general notation to represent number systems beyond just our familiar one.

In a number system with a fixed base of bb, the available digits range from 00 to b1b - 1. For example, in the decimal number system (b=10b = 10), we can only use the digits 0,1,2,...,90, 1, 2, ..., 9. When you run out of digits to stuff into a single bucket, you carry over a one to the next power of the base. For example, to get to the number after 9999, you carry a one to the bucket representing the next power of ten (100100).

Now, suppose that we have a string of digits dn1dn2...d0d_{n-1} d_{n-2} ... d_0 (where nn is the number of digits). Maybe that’s d2d1d0=579d_2 d_1 d_0 = 579 from our earlier example. That string expands like this:

dn1bn1+dn2bn2+...+d0b0d_{n-1} b^{n-1} + d_{n-2} b^{n-2} + ... + d_{0} b^0

And you can visualize it like this:

Rectangles arranged side by side representing increasing powers of a generic base of b, with digits represented as d_0 through d_{n-1}.

Using our same example, dn1bn1+dn2bn2+...+d0b0=5(102)+7(101)+9(100)d_{n-1} b^{n-1} + d_{n-2} b^{n-2} + ... + d_{0} b^0 = 5(10^2) + 7(10^1) + 9(10^0). Again, we have buckets from right to left in increasing powers of our base (1010), as depicted below:

Expanding 579 in terms of powers of 10. The 5 goes in the hundredths bucket, the 7 in the tens bucket, and the 9 in the ones bucket.

Now, in reality, you can have a number system that uses a base of 22, 33, 44, 120120, and so on. Some of these have special names because they’re used more often than others:

Base Name Description
1 Unary Also known as tallying. A number n is represented by picking an arbitrary character and repeating it n times (e.g., xxxx would be 4).
2 Binary Only two digits: zero and one. Most commonly used in computing. Everything on a computer is, at the lowest possible level, stored using the binary number system.
8 Octal Only eight digits are available: 0–7.
16 Hexadecimal Fifteen digits: 0–9 and a–f. Often used to express binary strings more compactly.
60 Sexagesimal How many seconds are in a minute? How many minutes in an hour? This is the basis of the modern circular coordinate system (degrees, minutes, and seconds).

For this reason, when discussing number systems, we usually subscript a number with its base to clarify its value. Alternatively, you can prepend a number with a certain string (usually 0b for binary or 0x/# for hexadecimal). So we’d write 579579 as 57910579_{10}, or the binary number 10011001 as 100121001_2 (or 0b1001\text{0b}1001). Otherwise, if we were to merely write the number 10011001 without providing any context, nobody would know whether that’s in binary, octal, decimal, hexadecimal, and so on because the digits 00 and 11 are valid in all of those number systems, too!

The Binary Number System

We’re all familiar with decimal numbers because we use them everyday. But what about the binary number system?

By definition, the binary number system has a base of 22, and thus we can only work with two digits to compose numbers: 00 and 11. Technically speaking, we don’t call these digits—they’re called bits in binary lingo. Each “bucket” in a binary string represents an increasing power of two: 202^0, 212^1, 222^2, and so on.

Increasing powers of two from right to left, represented as square slots. From right to left, they are labeled: 2^0 (ones), 2^1 (twos), 2^2 (fours), and so on.

The leftmost bit is called the most significant bit (MSB), while the rightmost bit is called the least significant bit (LSB).

Here are some examples of representing decimal numbers in the binary number system:

  • Zero: 010=020_{10} = 0_2. Expansion: 0(20)0 (2^0)
  • One: 110=121_{10} = 1_2. Expansion: 1(20)1(2^0)
  • Two: 210=1022_{10} = 10_2. Expansion: 1(21)+0(20)1(2^1) + 0(2^0)
  • Three: 310=1123_{10} = 11_2. Expansion: 1(21)+1(20)1(2^1) + 1(2^0)
  • Four: 410=10024_{10} = 100_2. Expansion: 1(22)+0(21)+0(20)1(2^2) + 0(2^1) + 0(2^0)
  • Five: 510=10125_{10} = 101_2. Expansion: 1(22)+0(21)+1(20)1(2^2) + 0(2^1) + 1(2^0)

Having learned the binary number system, you should now understand the joke from earlier:

There are 1010 types of people: those who understand binary and those who don’t.

Here, we really mean the binary equivalent of two, which looks like ten to our eyes when it’s not properly subscripted: 102=1×21=21010_2 = 1 × 2^1 = 2_{10}.

Binary Is Close to the Hardware of a Computer

Why do we bother with using the binary number system in the first place? Doesn’t it seem like a whole lot of extra work to represent numbers in this manner when we could instead use the decimal number system? Well, yes—if you’re writing these out by hand, it’s certainly more work to represent (and manipulate) binary numbers.

You may not see any point in using binary if you haven’t learned about computer architecture at a low level. Internally, computers are nothing more than electrical circuits tied to hardware. Current either flows through a wire or doesn’t—a binary state. Likewise, computers use logic gates (AND/OR/NOR/XOR) to control the flow of a program’s execution, and these take binary inputs (true/false). The best way to represent these low-level interactions is to use the binary number system: 00 means “off” (or false in its boolean form) and 11 means “on” (true).

Everything on your computer—the files you save and the software you install—is represented as nothing more than zeros and ones. But how is this possible?

The Unicode Standard

Suppose you create a file on your computer and store some basic text in it:

echo Hello, Binary > file

At the end of the day, your computer can’t store a character like H, e, l, or o (or even the space between two words) literally. Computers only know how to work with binary. Thus, we need some way to convert these characters to numbers. And that’s why the Unicode standard was introduced.

Unicode is the most widely accepted character encoding standard: a method of representing human-readable characters like H, e, ,, ?, and 9 numerically so that computers can understand and use them like we do. Each character maps to a unique number known as a code point.

For example, the chart below shows a very limited subset of Unicode characters (known as the ASCII standard) and their corresponding code points:

An ASCII table showing characters and their numerical representations

For the sake of brevity, we’ll focus on just the ASCII standard for now, even though it doesn’t capture the full range of characters in the Unicode standard and the complexities that come with needing to support hundreds of thousands of characters.

The ASCII standard supports only 128 characters, each mapped to a unique number:

  • Arabic digits: 090-9 (10)
  • Uppercase Latin letters: AZA-Z (26)
  • Lowercase Latin letters: aza-z (26)
  • Punctuation and special characters (66)

Again, note that while the ASCII standard only allows us to represent a tiny fraction of Unicode characters, it’s simple enough that it can help us better understand how characters are stored on computers.

1 ASCII Character = 1 Byte

In the decimal number system, we’re used to working with digits. In binary, as we already saw, we’re used to working with bits. There’s another special group of digits in binary that’s worth mentioning: A sequence of eight bits is called a byte.

Here are some examples of valid bytes:

00000000
10000000
11101011
11111111

… and any other valid permutation of eight 00s and 11s that you can think of.

Why is this relevant? Because on modern computers, characters are represented using bytes.

Recall that the ASCII encoding format needs to support a total of 128 characters. So how many unique number can we represent with 88 bits (a byte)?

Well, using the product rule from combinatorics, we have eight “buckets,” each with two possible values: either a 00 or a 11. Thus, we have 2×2×...×2=282 × 2 × ... × 2 = 2^8 possible values.

In decimal, this is 28=2562^8 = 256 possible values. By comparison, 27=1282^7 = 128. And 128128 happens to be the number of characters that we want to represent.

So… That’s weird, and seemingly wasteful, right? Why do we use 88 bits (one byte) to represent a character when we could use 77 bits instead and meet the precise character count that we need?

Good question! We use bytes because it’s not possible to evenly divide a group of 77 bits, making certain low-level computations difficult if we decide to use 77 bits to represent a character. In contrast, a byte can be evenly split into powers of two:

11101011
[1110][1011]
[11][10][10][11]

The key takeaway here is that we only need one byte to store one character on a computer. This means that a string of five characters—like Hello—occupies five bytes of space, with each byte being the numerical representation of the corresponding character per the ASCII format.

Remember the file we created earlier? Let’s view its binary representation using the xxd Unix tool:

xxd -b file

The -b flag stands for binary. Here’s the output that you’ll get:

00000000: 01001000 01100101 01101100 01101100 01101111 00101100  Hello,
00000006: 00100000 01000010 01101001 01101110 01100001 01110010   Binar
0000000c: 01111001 00001010                                      y.

The first line shows a sequence of six bytes, each corresponding to one character in Hello,.

Let’s decode the first two bytes using our knowledge of the binary number system and ASCII:

  • 01001000=1(26)+1(23)=721001001000 = 1(2^6) + 1(2^3) = 72_{10}. Per our ASCII table, this corresponds to HH.
  • 01100101=1(26)+1(25)+1(22)+1(20)=1011001100101 = 1(2^6) + 1(2^5) + 1(2^2) + 1(2^0) = 101_{10}, which is ee in ASCII.

Cool! Looks like the logic pans out. You can repeat this for all of the other bytes as well. Notice that on the second line, we have a leading space (from Hello, Binary), represented as 25=32102^5 = 32_{10} in ASCII (which is indeed Space per the table).

By the way, what’s up with the numbers along the left-hand side of the output? What does 0000000c0000000c even mean? Time to explore another important number system!

The Hexademical Number System

As I mentioned in the table from earlier, the hexadecimal number system is closely related to binary because it’s often used to express binary numbers more compactly, instead of writing out a whole bunch of zeros and ones.

The hexadecimal number system has a base of 1616, meaning its digits range from 0150–15.

This is our first time encountering a number system whose digits are made up of more than two characters. How do we squeeze 1010, 1111, or 1515 into a single “bucket” or “slot” for a digit? To be clear, this is perfectly doable if you have clear delimiters between digits, like vertical lines—without which you wouldn’t know if 1515 is a one followed by a five or a single digit of 1515 in the ones place. But in reality, using delimiters isn’t practical.

Let’s take a step back and consider a simple hexadecimal number:

0x420x42

What does this mean to us humans in our decimal number system? Well, all we have to do is multiply each digit by its corresponding power of 1616:

0x42=4(161)+2(160)=6410+210=66100x42 = 4(16^1) + 2(16^0) = 64_{10} + 2_{10} = 66_{10}

Okay, so that’s a simple hex number. Back to the problem at hand: How do we represent the hex digits 1010, 1111, and so on? Here’s an example that’s pretty confusing unless we introduce some alternative notation:

0x150x15

Is this a 1515 in a single slot or a 11 and a 55 in two separate slots? One way to make this less ambiguous is to use some kind of delimiter between slots, but again, that’s not very practical:

0x8[15]290x8[15]29

The better solution that people came up with is to map 101510–15 to the the English letters afa–f. Note that we could’ve also used any other symbols to represent these digits. As long as we agree on a convention and stick with it, there’s no ambiguity as to what a number represents.

Here’s an example of a hexadecimal number that uses one of these digits:

0xf40xf4

And here’s its expansion:

0xf4=15(161)+4(160)=24010+410=244100xf4 = 15(16^1) + 4(16^0) = 240_{10} + 4_{10} = 244_{10}

There’s nothing magical about the hexadecimal number system—it works just like unary, binary, decimal, and others. All that’s different is the base!

Before we move on, let’s revisit the output from earlier when we used xxd on our sample file:

00000000: 01001000 01100101 01101100 01101100 01101111 00101100  Hello,
00000006: 00100000 01000010 01101001 01101110 01100001 01110010   Binar
0000000c: 01111001 00001010                                      y.

The numbers along the left-hand side mark the starting byte for each line of text on the far right. For example, the first line of text (Hello,) ranges from byte #0 (H) to byte #5 (,). The next line is marked as 0000000600000006, meaning we’re now looking at bytes #6 through 11 (B to r). Finally, the last label should make sense now that you know the hexadecimal number system: c maps to 1212, meaning the byte that follows corresponds to the twelfth character in our file.

How to Convert Between Binary and Hexadecimal

Now that we know a bit about binary and hexadecimal, let’s look at how we can convert between the two systems.

Binary to Hexadecimal

Say you’re given this binary string and you’d like to represent it in hexadecimal:

011011100101011011100101

While at first this may seem like a pretty difficult task, it’s actually straightforward!

Let’s do a bit of a thought exercise: In the hexadecimal number system, we have 1616 digits from 00 to 1515. Over in binary land, how many bits do we need to represent these 1616 values?

The answer is four because 24=162^4 = 16. With four “buckets,” we can create the numbers zero (00000000), one (00010001), ten (10101010), all the way up to fifteen (11111111). This means that when you’re given a binary string, all you have to do is split it into groups of four bits and evaluate them to convert binary to hexadecimal!

011011100101
[0110][1110][0101]
6 14 5

Now we just replace 101510–15 with afa-f and we’re done: 0x6e50x6e5.

Hexadecimal to Binary

What about the reverse process? How do you convert a hexadecimal number to binary? Say you’re given the hexadecimal number 0xad0xad. What do we know about each hexadecimal digit?

Well, from our earlier exercise, we know that four bits comprise one hex digit. So we can convert each individual hex digit to its 44-bit representation and then stick each group together!

a16=1010=10102d16=1310=11012ad16=101011012a_{16} = 10_{10} = 1010_{2} \\ d_{16} = 13_{10} = 1101_{2} \\ ad_{16} = 10101101_{2}

Real-World Application: Colors in RGB/Hex

While we’re on the topic of binary and hexadecimal, it’s worth taking a look at one real-world use case for the things we’ve learned so far: RGB and hex colors.

Colors have three components: red, green, and blue (RGB). With LED (light-emitting diode) displays, each pixel is really split into these three components using a color diode. If a color component is set to 00, then it’s effectively turned off. Otherwise, its intensity is modulated between 00 and 255255, giving us a color format like rgb(0-255, 0-255, 0-255).

Let’s consider this hex color: #4287f5. What is it in the RGB format?

Well, we need to split this hex string evenly between red, green, and blue. That’s two digits per color:

[42][87][f5][42][87][f5]

Now, we interpret the decimal equivalent for each part:

  • Red: 4216=4(161)+2(160)=6642_{16} = 4(16^1) + 2(16^0) = 66
  • Green: 8716=8(161)+7(160)=13587_{16} = 8(16^1) + 7(16^0) = 135
  • Blue: f516=15(161)+5(160)=245f5_{16} = 15(16^1) + 5(16^0) = 245

That means #4287f5 is really rgb(66, 135, 245)! You can verify this using a Color Converter:

A color converter verifying that #4287f5 is really rgb(66, 135, 245)

For practice, let’s convert this to binary as well. I’ll mark the groups of four bits to make it easier to see how I did this (you could also convert from the decimal RGB representation if you want to):

0x4287f5=0b[0100][0010][1000][0111][1111][0101]0x4287f5 = 0b[0100][0010][1000][0111][1111][0101]

Now, two groups of four bits will represent one component of the color (red/green/blue):

0b[01000010][10000111][11110101]0b[01000010][10000111][11110101]

Notice that each color component takes up a byte (88 bits) of space.

How Many Colors Are There?

As an additional exercise, how many unique colors can you possibly have in the modern RGB format?

We know that each component (red/green/blue) is represented using one byte (88 bits). So the colors we’re used to are really 2424-bit colors.

That means there are a whopping 224=16,777,2162^{24} = 16,777,216 possible unique colors that you can generate using hex/rgb! The 2424-bit color system is known as truecolor, and it’s capable of representing millions of colors.

Note that you could just as well have performed this calculation using hex: #4287f5. There are six slots, each capable of taking on a value from 00 to ff. That gives us a total of 16×16×...×16=166=16,777,21616 × 16 × ... × 16 = 16^6 = 16,777,216 values—the same result as before.

Or, if you’re using the decimal RGB format, the math still pans out:

256×256×256=16,777,216256 × 256 × 256 = 16,777,216

What Are 8-Bit Colors?

On older systems with limited memory, colors were represented using just eight bits (one byte). These 8-bit colors had a very limited palette, which meant that most computer graphics didn’t have gradual color transitions (so images looked very pixelated/grainy). With only 88 bits to work with, you are limited to just 28=2562^8 = 256 colors!

An 8-bit color palette

Naturally, you may be wondering: How did they split 88 bits evenly among red, green, and blue? After all, 88 isn’t divisible by three!

Well, the answer is that they didn’t. The process of splitting these bits among the color components is called color quantization, and the most common method (known as 8-bit truecolor) split the bits as 3-3-2 red-green-blue. Apparently, this is because the human eye is less sensitive to blue light than the other two, and thus it simply made sense to distribute the bits heavily in favor of red and green and leave blue with one less bit to work with.

Signed Binary Number System: Two’s Complement

Now that we’ve covered decimal, binary, and hexadecimal, I’d like us to revisit the binary number system and learn how to represent negative numbers. Because so far, we’ve only looked at positive numbers. How do we store the negative sign?

To give us some context, I’ll assume that we’re working with standard 3232-bit integers that most computers support. We could just as well look at 6464-bit or NN-bit integers, but it’s good to have a simple basis for a discussion.

If we have 3232 bits to fiddle with, that means we can represent a total of 232=4,294,967,2962^{32} = 4,294,967,296 (4 billion) numbers. More generally, if you have NN bits to work with, you can represent 2N2^N values. But we’d like to split this number range evenly between negatives and positives.

Positive or negative… positive or negative. One thing or another thing—ring a bell? That sounds like it’s binary in nature. And hey—we’re already using binary to store our numbers! Why not reserve just a single bit to represent the sign? We can have the most significant (leading) bit be a 00 when our number is positive and a 11 when it’s negative!

Earlier, when we were first looking at the binary number systems, I mentioned that you can strip leading zeros because they are meaningless. This is true except when you actually care about distinguishing between positive and negative numbers in binary. Now, we need to be careful—if you strip all leading zeros, you my be left with a leading 11, and that would imply that your number is negative (in a signed number system).

You can think of two’s complement as a new perspective or lens through which we look at binary numbers. The number 1002100_2 ordinarily means 4104_{10} if we don’t care about its sign (i.e., we assume it’s unsigned). But if we do care, then we have to ask ourselves (or whoever provided us this number) whether it’s a signed number.

How Does Two’s Complement Work?

What does a leading 11 actually represent when you expand a signed binary number, and how do we convert a positive number to a negative one, and vice versa? For example, suppose we’re looking at the number 221022_{10}, which is represented like this in unsigned binary:

10110210110_2

Since we’re looking at signed binary, we need to pad this number with an extra 00 out in front (or else a leading 11 would imply that it’s negative):

0101102010110_2

Okay, so this is positive 221022_{10}. How do we represent 2210-22_{10} in binary?

There are two ways we can do this: the intuitive (longer) approach and the “shortcut” approach. I’ll show you both, but I’ll start with the more intuitive one.

The Intuitive Approach: What Does a Leading 1 Denote?

Given an NN-bit binary string, a leading 11 in two’s complement represents 1-1 multiplied by its corresponding power of two (2n12^{n-1}). A digit of 11 in any other slot represents +1+1 times its corresponding power of two.

For example, the signed number 11010211010_2 has this expansion:

110102=1(24)+1(23)+1(21)=1610+810+210=61011010_2 = -1(2^4) + 1(2^3) + 1(2^1) = -16_{10} + 8_{10} + 2_{10} = -6_{10}

We simply treat the leading 11 as a negative, and that changes the resulting sum in our expansion.

Two’s Complement Shortcut: Flip the Bits and Add 1

To convert a number represented in two’s complement binary to its opposite sign, follow these two simple steps:

  1. Flip all of the bits (00 becomes 11 and vice versa).
  2. Add 11 to the result.

For example, let’s convert 431043_{10} to 4310-43_{10} in binary:

+43 in binary: 0101011
Flipped:       1010100
Add one:       1010101

What is this number? It should be 4310-43_{10}, so let’s expand it by hand to verify:

1(26)+1(24)+1(22)+1(20)=6410+1610+410+110=43-1(2^6) + 1(2^4) + 1(2^2) + 1(2^0) = -64_{10} + 16_{10} + 4_{10} + 1_{10} = -43

Sure enough, the process works!

How Many Signed Binary Numbers Are There?

We’ve seen that in a signed binary system, the most significant bit is reserved for the sign. What does this do to our number range? Effectively, it halves it!

Let’s consider 3232-bit integers again. Whereas before we had 3232 bits to work with for the magnitude of an unsigned number, we now have only 3131 for the magnitude of a signed number (because the 32nd bit is reserved for the sign):

Unsigned magnitude bits:  [31 30 29 ... 0]
Signed magnitude bits:    31 [30 29 ... 0]

We went from having 2322^{32} numbers to 2312^{31} positive and negative numbers, which is precisely half of what we started with (2322=231\frac{2^{32}}{2} = 2^{31}).

More generally, if you have an NN-bit signed binary string, there are going to be 2N2^N values, split evenly between 2n12^{n-1} positives and 2n12^{n-1} negatives.

Notice that the number zero gets bunched in with the positives and not the negatives:

Signed zero:  0  0  0  0 ... 0 0 0 0
Bits:        31 30 29 28 ... 3 2 1 0

As we’re about to see, this has an interesting consequence.

What Is the Largest Signed 32-bit Integer?

The largest signed 32-bit integer is positive, meaning its leading bit is a zero. So we just need to maximize the remaining bits to get the largest possible value:

Num:      0  1  1  1 ... 1
Bits:    31 30 29 28 ... 0

This is 23112^{31} - 1, which is 2,147,483,6472,147,483,647. In Java, this number is stored in Integer.MAX_VALUE, and in C++, it’s std::numeric_limits<int>::max().

More generally, for an NN-bit system, the largest signed integer is 2n112^{n-1}-1.

Why did we subtract a one at the end? Because we start counting at one, but computers start at zero. As I mentioned in the previous section, the number zero gets grouped along with the positives when we split our number range (by convention):

Signed zero:  0  0  0  0 ... 0 0 0 0
Bits:        31 30 29 28 ... 3 2 1 0

So to get the largest signed integer, we need to subtract one.

Real-World Application: Video Game Currency

In video games like RuneScape that use 3232-bit signed integers to represent in-game currency, the max “cash stack” that you can have caps out at exactly 23112^{31} - 1, which is roughly 2.1 billion.

The max cash stack you can have in Runescape is 2147m, or 2.1 billion.
Image source: YouTube

Now you know why! If you’re wondering why they don’t just use unsigned ints, it’s because RuneScape runs on Java, and Java doesn’t support unsigned ints (except in SE 8+).

What Is the Smallest Signed 32-bit Integer?

This occurs when we set the leading bit to be a 11 and set all remaining bits to be a 00:

Num:      1  0  0  0 ... 0
Bits:    31 30 29 28 ... 0

Why? Because recall that in the expansion of negative numbers in two’s complement binary, the leading 11 is a 1-1 times 2n12^{n-1}, and a 11 in any other position will be treated as +1+1 times its corresponding power of two. Since we want the smallest negative number, we don’t want any positive terms, as those take away from our magnitude. So we set all remaining bits to be 00.

Answer: 231-2^{31}

In Java, this value is stored in Integer.MIN_VALUE. In C++, it’s in std::numeric_limits<int>::min().

More generally, if we have an NN-bit system, the smallest representable signed int is 2n1-2^{n-1}.

Notice that the magnitude of the smallest signed 3232-bit integer is exactly one greater than the magnitude of the largest signed 3232-bit integer. As mentioned previously, this is because of where we chose to group the number zero itself, which “steals” one magnitude from that group’s available bits.

Binary Arithmetic

Spoiler: Adding, subtracting, multiplying, and dividing numbers in the binary number system is exactly the same as it is in decimal!

Adding Binary Numbers

We’ll first revisit what we learned in elementary school for decimal numbers and then look at how to add two binary numbers.

To add two numbers in the decimal number system, you stack them on top of one another visually and work your way from right to left, adding two digits and “carrying the one” as needed.

Now you should know what carrying the one really means: When you run out of digits to represent something in your fixed-base number system (e.g., 1313 isn’t a digit in base 1010), you represent the part that you can in the current digits place and move over to the next power of your base (the “column” to the left of your current one).

For example, let’s add 2424 and 1818 in decimal:

  24
+ 18
————
  42

We first add the 44 and 88 to get 1212, which is not a digit we support in the decimal number system. So we represent the part that we can (22) and carry the remaining value (ten) over to the next column as a 11 (1×101=10101 × 10^1 = 10_{10}). In that column, we have 110+210+110=4101_{10} + 2_{10} + 1_{10} = 4_{10}:

      1  <-- carried
      24
    + 18
————————
      42

Now, let’s add these same two numbers (241024_{10} and 181018_{10}) using the binary number system:

  11000
+ 10010
———————
 101010

We work from right to left:

  • Ones place: 0+0=00 + 0 = 0
  • Twos place: 0+1=10 + 1 = 1
  • Fours place: 0+0=00 + 0 = 0
  • Eighths place: 1+0=11 + 0 = 1
  • Sixteens place: 1+1=1021 + 1 = 10_2 (two)

That last step deserves some clarification: When we try to add the two ones, we get 12+12=1021_2 + 1_2 = 10_2 (two), so we put a 00 in the current column and carry over the 11 to the next power of two, where we have a bunch of implicit leading zeros:

               1      <-- carry bits
0000  ...    00011000
0000  ...  + 00010010
—————————————————————
0000  ...    00101010

In that column, 1(carried)+0(implicit)=11 (carried) + 0(implicit) = 1.

If we expand the result, we’ll find that it’s the same answer we got over in decimal:

1(25)+1(23)+1(21)=32+8+2=42101(2^5) + 1(2^3) + 1(2^1) = 32 + 8 + 2 = 42_{10}

Let’s look at one more example to get comfortable with carrying bits in binary addition: 2210+141022_{10} + 14_{10}, which we know to be 361036_{10}:

  10110
+ 01110
———————
 100100

Something interesting happens when we look at the twos place (the 212^1 column): We add 121_2 to 121_2, giving us two (10210_2), so we put a zero in the 212^1 column and carry the remaining one.

Now we have three ones in the 222^2 column: 12(carried)+12(operand1)+12(operand2)=1121_2(carried) + 1_2(operand1) + 1_2(operand2) = 11_2 (three). So we put a one in the 222^2 column and carry a one yet again. Rinse and repeat!

               1111    <-- carry bits
0000  ...    00010110
0000  ...  + 00001110
—————————————————————
0000  ...    00100100

Once again, it’s a good practice to expand the result so you can verify your work:

1(25)+1(22)=3210+410=36101(2^5) + 1(2^2) = 32_{10} + 4_{10} = 36_{10}

Subtracting Binary Numbers

Subtraction is addition with a negative operand: ab=a+(b)a - b = a + (-b). Now that we know how to represent negative numbers in the binary system thanks to two’s complement, this should be a piece of cake: negate the second operand and perform addition.

For example, what’s 1210261012_{10} - 26_{10}? In decimal, we know this to be 1410-14_{10}. Over in binary, we know that 121012_{10} is 0110001100. What about 2610-26_{10}? We’ll represent that using two’s complement.

We start by first representing 261026_{10} in binary:

+2610=0110102+26_{10} = 011010_2

Now we negate it by flipping the bits and adding one:

26 in binary: 011010
Flipped:      100101
Add one:      100110  = -26

Then, stack up the operands and add them like before:

     11    <-- carry bits
    001100
  + 100110
——————————
    110010

Notice that the result has a leading one, which we know denotes a negative number in signed binary. So we at least got the sign part right! Let’s check the magnitude:

1(25)+1(24)+1(21)=3210+1610+210=1410-1(2^5) + 1(2^4) + 1(2^1) = -32_{10} + 16_{10} + 2_{10} = -14_{10}

Adding and subtracting numbers in the binary number system is no different than in the decimal system! We’re just working with bits instead of digits.

Multiplying Binary Numbers

Let’s remind ourselves how we multiply numbers in decimal:

  21
x 12
————

Remember the process? We multiply the 22 by each digit in the first multiplicand and write out the result under the bar:

  21
x 12
————
  42

Then we move on to the 11 in 1212 and repeat the process, but adding a 00 in the right column of the result. Add the two intermediate products to get the answer:

   21
x  12
—————
   42
+ 210
—————
  252

Guess what? The process is exactly the same in the binary number system!

Let’s multiply these same two numbers in binary. They are 2110=01010121_{10} = 010101 and 1210=0110012_{10} = 01100:

   010101
x   01100
—————————

Obviously, this is going to be more involved in binary since we’re working with bits (and thus longer strings), but the logic is still the same. In fact, beyond having to write out so many intermediate results, we actually have it much easier over in binary. Whenever a digit is 11, you simply copy down the first multiplicand, padded with zeros. Whenever it’s a zero times the first multiplicand, the result is zero!

      010101
x      01100
————————————
      000000
     0000000
    01010100
   010101000
+ 0000000000
————————————
  0011111100

Expanding this in binary, we get:

00111111002=1(27)+1(26)+1(25)+1(24)+1(23)+1(22)=252100011111100_2 = 1(2^7) + 1(2^6) + 1(2^5) + 1(2^4) + 1(2^3) + 1(2^2) = 252_{10}

Easy peasy. The same process applies regardless of whether your multiplicands are signed or unsigned.

Dividing Binary Numbers

Let’s divide 12610126_{10} by 121012_{10} using long division:

    0 1 0 . 5
   _______
12 |1 2 6
  - 1 2
   ————
      0 6
    -   0
   ——————
        6 0
      - 6 0
      —————
          0

Answer: 10.510.5.

Now let’s repeat the process over in the binary number system. Note that I’m going to strip leading zeros to make my life easier since we’re working with two unsigned numbers:

      _______
1100 |1111110

Take things one digit at a time, and reference this useful YouTube video if you get stuck:

         0 0 0 1 0 1 0 . 1
        ______________
1 1 0 0 |1 1 1 1 1 1 0 . 0
        -0
        ——
         1 1
        -  0
        ————
         1 1 1
        -    0
        ——————
         1 1 1 1
       - 1 1 0 0
        ————————
             1 1 1
          -      0
        ——————————
             1 1 1 1
           - 1 1 0 0
           —————————
             0 0 1 1 0
             -       0
             —————————
                 1 1 0
                 -   0
                 —————
                 1 1 0 0
              -  1 1 0 0
                 ———————
                 0 0 0 0

Answer: 01010.101010.1.

What does the 11 to the right of the decimal point represent? Well, in the decimal number system, anything to the right of the decimal point represents a negative power of ten: 10110^{-1}, 10210^{-2}, and so on.

As you may have guessed, in the binary number system, these are 212^{-1}, 222^{-2}, and so on. So .1.1 above really means 1(21)1(2^{-1}), which is 12=0.510\frac{1}{2} = 0.5_{10} in decimal. And of course, the part in front of the decimal point evaluates to 101010_{10}.

That gives us 1010+0.510=10.510_{10} + 0.5_{10} = 10.5. So our answer using binary long division is exactly the same as the one we got over in decimal!

Integer Overflow and Underflow in Binary

What happens if you try to add one to the largest representable NN-bit signed integer?

For example, if N=32N = 32, we’re really asking what happens if we try adding one to the largest representable 3232-bit signed int.

Let’s give it a shot:

    0111...11111
  + 0000...00001
————————————————

In the rightmost column, we’ll get 12+12=1021_2 + 1_2 = 10_2, so that’s a zero carry a one. But as a result, all of the remaining additions will be 12+121_2 + 1_2 since we’ll always carry a one until we get to the leading bit:

    11111111111  <-- carry bits
    0111...11111     (2^{N-1} - 1)
  + 0000...00001     (1)
————————————————
    1000...00000     (-2^{N-1})

And what number is that in signed binary? Hmm… Looks like it’s the smallest representable negative number! What we’ve observed here is called integer overflow. When you try to go past the largest representable signed integer in a given NN-bit system, the result overflows or wraps around.

What if we try to subtract one from the smallest representable NN-bit signed integer? First, we’ll represent 110-1_{10} as a signed integer in binary:

1 in binary: 0000...00001
Flipped:     1111...11110
Add one:     1111...11111  <-- -1

Now let’s add this to the smallest representable signed integer:

   1             <-- carry bits
    1000...00000     (-2^{N-1})
  + 1111...11111     (-1)
————————————————
  1|0111...11111     (2^{N-1} - 1)

Notice that the result carries an additional bit over, yielding a result that has N+1N+1 bits. But our system only supports NN bits, so that leading 11 is actually discarded. The result is the largest representable NN-bit signed integer, and this is known as integer underflow.

Overflow and underflow are things you should be mindful of in programs that are performing lots of computations, as you may end up getting unexpected results.

The Binary Number System: Additional Topics for Exploration

That about does it for this introduction to the binary number system! We took a pretty in-depth look at decimal, binary, and hexadecimal, and I hope you now have a greater appreciation for the binary number system and the role that it plays in computing.

In reality, there’s much more to learn beyond what we covered here. If you’re curious, I encourage you to look into representing floating point numbers in binary using the IEE754 format.

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