0x means in representation

The Hexadecimal Number System Explained

Hexadecimal numbers, often shortened to “hex numbers” or “hex”, are numbers represented in base 16 as opposed to base 10 that we use for everyday arithmetic and counting.

In practical terms, this means that each column of a number written in hexadecimal can represent up to 16 values.

Digits in hexadecimal use the standard symbols 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9 to represent the corresponding value, and use the first six letters of the alphabet to represent the values 10 through 15 (E.G: A, B, C, D, E, F).

In programming, we prefix hexadecimal constants with 0x , with some exceptions.

Examples and explanation

In the standard base 10 system, each column represents increasing powers of 10, while in base 16 each column represents increasing powers of 16.

As seen in the table example above, with one hex digit we can represent numbers up to and including 15. Add another column and we can represent numbers up to 255, 4095 with another column, and so on.

Uses of Hexadecimal in Low Level Programming

Hexadecimal first found its use in Computer Science as a convenience feature.

Data in our computers has a lowest common storage unit, the Byte. Each byte contains 8 bits, and is able to store a number between 0 and 255 inclusive.

Hexadecimal has the advantage of being terse and having well defined boundaries.

A single byte is always represented by two hexadecimal digits from 0x00 to 0xFF, the latter being the largest per-byte value of 255.

The terseness and byte-aligned nature of hexadecimal numbers make them a popular choice for software engineers working on low-level code-bases or embedded software.

Uses of Hexadecimal Numbers in JavaScript

JavaScript supports the use of hexadecimal notation in place of any integer, but not decimals.

As an example, the number 2514 in hex is 0x9D2, but there is no language-supported way of representing 25.14 as a hex number.

Using hexadecimal in your code is a personal and stylistic choice, and has no effect on the underlying logic your code implements.

Uses of Hexadecimal Numbers in CSS

CSS has for a long time used hexadecimal notation to represent color values. Consider the following selector:

The background-color ’s value is in fact three hex bytes.

The CSS processor treats these as three individual bytes, representing Red, Green, and Blue.

In our example, 11 corresponds to the Red color component, 22 corresponds to the Green color component, and 33 to the Blue color component.

There is currently no way as of CSS3 to define a color with an alpha component using hex. The proposed CSS4 Draft 1 includes a proposal to allow for an extra byte to specify alpha values.

For now, use of the standard rgba() function is the recommended way to add an alpha value to your colors.

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• Hexadecimal Number System

Hexadecimal Number System is a base-16 numeral system used in diverse fields, especially in computing and digital electronics. It consists of 16 symbols, including numbers 0 to 9 and letters A to F, offering a compact way to represent binary-coded values. The hexadecimal number system is sometimes also represented as, ‘hex’.

Number Systems are various ways to use numbers to represent large numbers and information. The hexadecimal number system is introduced to students in class 9. In this article, we will learn about, the Hexadecimal Number System, Hexadecimal Number System Table, Hexadecimal Number System Examples, and Others in detail.

Before starting with the Hexadecimal Number System we first learn about the Number System.

Table of Content

What is Number System?

What is hexadecimal number system, hexadecimal number system table, hexadecimal numbers conversions, place value of digits in hexadecimal number system.

• Solved Examples

Practice Questions on Hexadecimal Number System

Number System is a method of representing Numbers and there are various types of the number systems in mathematics that are defined as,

A number system is a system for expressing numbers; it’s a mathematical notation for representing numbers of a given set, using digits or other symbols in a consistent manner.

Types of Number Systems

Based on the base value and the number of allowed digits, number systems are of many types. The four common types of Number System are: 

• Decimal Number System
• Binary Number System

Now let’s learn about, Hexadecimal Number in detail.

Hexadecimal is a number system combining “hexa” for 6 and “deci” for 10. It uses 16 digits: 0 to 9 and A to F, where A stands for 10, B for 11, and so on. Similar to the regular decimal system, it counts up to F instead of stopping at 9. Each digit in hexadecimal has a weight 16 times greater than the previous one, following a positional number system. When converting to another system, we multiply each digit by the power of 16 based on its position. For example, in the number 7B3, 7 is multiplied by 16 squared, B by 16 to the power of 1, and 3 by 16 to the power of 0.

Facts about Hexadecimal Numbers

• Hexadecimal is a number system with a base value of 16.
• Hexadecimal numbers use 16 symbols or digital values: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F.
• A, B, C, D, E, and F represent 10, 11, 12, 13, 14, and 15 in single-bit form.
• If you see an “0x” as Prefix, it indicates the number is in Hexadecimal. For example, 0x3A
• The position of each digit in a Hexadecimal number has a weight of 16 to the power of its position.

Following table represents the relationship between hexadecimal number system with decimal as well as binary number system.

The hexadecimal number can be easily converted to various other numbers such as, Binary Numbers, Octal Numbers, Decimal Numbers and vice-versa. Now let’s learn about them in detail.

Hexadecimal to Decimal Conversion

Converting hexadecimal to decimal follows a similar process as before, where each digit is multiplied by the respective power of 16.

Example: Convert (A7B) 16 to decimal.

(A7B) 16 = A × 16 3 + 7 × 16 2 + B × 16 1 ⇒ (A7B) 16 = 10 × 4096 + 7 × 256 + 11 × 16 (convert symbols A and B to their decimal equivalents; A = 10, B = 11) ⇒ (A7B) 16 = 40960 + 1792 + 176 ⇒ (A7B) 16 = 42828 Therefore, the decimal equivalent of (A7B) 16 is (42828) 10 .

Decimal to Hexadecimal Conversion

To convert a decimal number to hexadecimal, we use the base number 16. The process involves dividing the number by 16 repeatedly until the quotient becomes zero. The decimal to hexadecimal number system is shown in the image added below,

Example: Convert (92) 10 to hexadecimal.

Divide 92 by 16 Quotient: 5, Remainder: 12 (C in Hexadecimal) Divide 5 by 16 Quotient: 0, Remainder: 5 Write the remainders from bottom to top Therefore, (92) 10 is equivalent to (5C) 16 in hexadecimal.

Hexadecimal to Octal Conversion

To convert a hexadecimal number to octal, we follow a two-step process: first, convert the hexadecimal number to decimal, and then convert the decimal number to octal.

Example: Convert (1F7) 16 to Octal.

Step 1: Convert (1F7) 16 to decimal using the powers of 16:

(1F7) 16 = 1 × 16 2 + 15 × 16 1 + 7 × 16 0 ⇒ (1F7) 16 = 1 × 256 + 15 × 16 + 7 × 1 ⇒ (1F7) 16 = 256 + 240 + 7 ⇒ (1F7) 16 = (503) 10

Step 2: Convert the decimal number (503) 10 to octal by dividing it by 8 until the quotient is 0

503 ÷ 8 = 62 with a remainder of 7 62 ÷ 8 = 7 with a remainder of 6 7 ÷ 8 = 0 with a remainder of 7 Arrange the remainders from bottom to top Therefore, (1F7) 16 is equivalent to (767) 8 in octal

Octal to Hexadecimal Conversion

There is a two step process to convert an octal number into hexadecimal:

Step 1: Convert Octal to Binary

For each octal digit, replace it with its three-digit binary equivalent.

Example: Convert (345) 8 to binary.

3 in octal is 011 in binary 4 in octal is 100 in binary 5 in octal is 101 in binary Combine these binary equivalents: (345) 8 = (011100101) 2

Step 2: Convert Binary to Hexadecimal

Group the binary digits into sets of four, starting from the right, and convert each set to its hexadecimal equivalent.

Example: Convert (011100101) 2 to hexadecimal.

0111 in binary is 7 in hexadecimal 0010 in binary is 2 in hexadecimal 1101 in binary is D in hexadecimal Combine these hexadecimal equivalents: (011100101)2 = (72D) 16 Therefore, (345) 8 is equivalent to (72D) 16 in hexadecimal.

Hexadecimal to Binary Conversion

Converting hexadecimal to binary involves two methods: one with a conversion table and the other without a conversion table.

Method 1: Convert Hexadecimal to Binary with Conversion Table

To convert a hexadecimal number to binary using a conversion table, we follow these steps:

Example: Convert hexadecimal (4D) 16 to binary.

Look up Decimal Equivalent of each digit in the conversion table. 4 in decimal is (4) 10 , and D in decimal is (13) 10 Convert each decimal number to binary. (4) 10 is (0100) 2 , and (13) 10 is (1101) 2 Combine the binary numbers (4D) 16 is (01001101) 2

Method 2: Convert Hexadecimal to Binary without Conversion Table

This method involves multiplying each digit by 16 (n-1) to obtain the decimal number, and then dividing by 2 until the quotient is zero.

Example: Convert hexadecimal (A2) 16 to binary.

Convert (A2) 16 to decimal (A)₁₆ is (10) 10 , and (2) 16 is (2) 10 ⇒ (A2) 16 is 10 × 16 1 + 2 × 16 0 = 160 + 2 = 162 10 Convert the decimal number (162) 10 to binary Divide 162 by 2: Quotient = 81, Remainder = 0 Divide 81 by 2: Quotient = 40, Remainder = 1 Continue dividing until the quotient is zero: (10100010) 2 Therefore, (A2) 16 is (10100010)₂ in binary

Binary to Hexadecimal Conversion

To change binary to hexadecimal, we refer to a conversion table from the previous section.

Example: Convert (10111010101) 2 to hexadecimal.

In hexadecimal, every 4 binary digits represent one digit Group the binary number accordingly, and find their Hexadecimal equivalent using the hexadecimal table added above. 0010 = 2, 1011 = B, 1010 = A Combine these hexadecimal digits to get the final number. Therefore, (10111010101) 2 is equal to (2BA) 16

The numbers in the hexadecimal number system has weightage in powers of 16. The power of 16 increases as the digit is shift towards the left of the number. This is explained by the example as,

Example, (AB12) 16

Place value of each digit in (AB12) 16 is, = A×16 3 + B×16 2 + 1×16 1 + 2×16 0

• Binary Formula
• Difference Between Decimal and Binary Number Systems

Solved Examples on Hexadecimal Number System

Example 1: Convert Hexadecimal 1A5 to Decimal

Multiply First Digit (1) by 16 squared (256) 1×16 2 = 256 Multiply Second Digit (A, which is 10 in decimal) by 16 to the power of 1 (16) 10×16 1 = 160 Multiply Third Digit (5) by 16 to the power of 0 (1) 5×16 0 = 5 Adding the results, 1A5 = 1×16 2 + A×16 1 + 5×16 0 ⇒ 1A5 = 1×16 2 + 10×16 1 + 5×16 0 ⇒ 1A5 = 256 + 160 + 5 = 421 Decimal Equivalent of Hexadecimal number 1A5 is 421

Example 2: Convert Decimal 315 to Hexadecimal.

Divide Decimal Number by 16 315÷16 = 19 with Remainder 11 The remainder (11) is represented as B in hexadecimal Repeat the division with the quotient (19) 19÷16 = 1 with Remainder of 3 The remainder (3) is represented as 3 in hexadecimal Hexadecimal Equivalent of Decimal Number 315 is 13B

Problem 1: Convert the hexadecimal number 2A to binary.

Problem 2: Convert the binary number 110110 to hexadecimal.

Problem 3: Add the hexadecimal numbers 1F and A3. Provide the result in hexadecimal.

Problem 4: Subtract the hexadecimal number B6 from D9. Provide the result in hexadecimal.

Problem 5: Multiply the hexadecimal number 7E by 3. Provide the result in hexadecimal.

Hexadecimal Number System – FAQs

What is a hexadecimal number system.

The hexadecimal number system is a method of counting using 16 digits combining Numbers (0-9) Letters (A-F)

What are 16 Digits of Hexadecimal Number System?

The 16 digits of the hexadecimal number system are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, and the letters A, B, C, D, E, F.

What is the Use of Hexadecimal Number System?

Hexadecimal is handy in computers for expressing binary data more easily. It’s also used in colors on computers, making it simpler to show a wide range of shades.

What is 16 Called in Hexadecimal?

In hexadecimal, 16 is represented as 10.

What is 8 in Hexadecimal Number System?

8 in Hexadecimal Number System is also, called 8.

How Many Digits are Used in the Hexadecimal Number System?

The hexadecimal system uses sixteen distinct digits: 0-9 for values zero to nine, and the letters A-F for values ten to fifteen.

Where are Hexadecimal Numbers Used by Programmers?

Hexadecimal numbers are commonly used by programmers to define locations in memory units of computers because they can represent large numbers in a more compact form than binary or decimal systems.

How Do You Convert a Hexadecimal Number to Binary?

To convert a hexadecimal number to binary, replace each hexadecimal digit with its corresponding 4-bit binary sequence. For example, the hexadecimal number 2A converts to the binary number 00101010.

How Do You Convert Binary to Hexadecimal?

To convert binary to hexadecimal, group the binary digits into sets of four (starting from the right), and then replace each set with the corresponding hexadecimal digit. For example, the binary number 1101011011 converts to the hexadecimal number 2D6.

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Number - Hexadecimal notation (0x)

Hexadecimal is a numeral system in base 16 (0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F).

To see binary data in hexadecimal form, you use an hex editor .

Hexadecimal is more compact to write than binary. For example, the binary number 100110110100 is 9B4 in hexadecimal.

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Representation in Text

Hexadecimal numbers in text are specified with a prefix

Windows calculator, in Programmer mode, to see and convert:

Decimal to Binary to Hexadecimal

5.3 — Numeral systems (decimal, binary, hexadecimal, and octal)

Author’s note

This lesson is optional.

Future lessons reference hexadecimal numbers, so you should at least have a passing familiarity with the concept before proceeding.

In everyday life, we count using decimal numbers, where each numerical digit can be 0, 1, 2, 3, 4, 5, 6, 7, 8, or 9. Decimal is also called “base 10”, because there are 10 possible digits (0 through 9). In this system, we count like this: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, … By default, numbers in C++ programs are assumed to be decimal.

In binary , there are only 2 digits: 0 and 1, so it is called “base 2”. In binary, we count like this: 0, 1, 10, 11, 100, 101, 110, 111, …

Decimal and binary are two examples of numeral systems , which is a fancy name for a collection of symbols (e.g. digits) used to represent numbers. There are 4 main numeral systems available in C++. In order of popularity, these are: decimal (base 10), binary (base 2), hexadecimal (base 16), and octal (base 8).

Octal and hexadecimal literals

Octal is base 8 -- that is, the only digits available are: 0, 1, 2, 3, 4, 5, 6, and 7. In Octal, we count like this: 0, 1, 2, 3, 4, 5, 6, 7, 10, 11, 12, … (note: no 8 and 9, so we skip from 7 to 10).

To use an octal literal, prefix your literal with a 0 (zero):

This program prints:

Why 10 instead of 12? Because numbers are output in decimal by default, and 12 octal = 10 decimal.

Octal is hardly ever used, and we recommend you avoid it.

Hexadecimal is base 16. In hexadecimal, we count like this: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F, 10, 11, 12, …

You can also use lower case letters (though upper case is more common).

To use a hexadecimal literal, prefix your literal with 0x :

You can also use a 0X prefix, but 0x is conventional because its easier to read.

Using hexadecimal to represent binary

Because there are 16 different values for a hexadecimal digit, we can say that a single hexadecimal digit encompasses 4 bits. Consequently, a pair of hexadecimal digits can be used to exactly represent a full byte.

Consider a 32-bit integer with binary value 0011 1010 0111 1111 1001 1000 0010 0110. Because of the length and repetition of digits, that’s not easy to read. In hexadecimal, this same value would be: 3A7F 9826, which is much more concise. For this reason, hexadecimal values are often used to represent memory addresses or raw data in memory (whose type isn’t known).

Binary literals

Prior to C++14, there is no support for binary literals. However, hexadecimal literals provide us with a useful workaround (that you may still see in existing code bases):

In C++14 onward, we can use binary literals by using the 0b prefix:

Digit separators

Because long literals can be hard to read, C++14 also adds the ability to use a quotation mark (‘) as a digit separator.

Also note that the separator can not occur before the first digit of the value:

Digit separators are purely visual and do not impact the literal value in any way.

Outputting values in decimal, octal, or hexadecimal

By default, C++ outputs values in decimal. However, you can change the output format via use of the std::dec , std::oct , and std::hex I/O manipulators:

This prints:

Note that once applied, the I/O manipulator remains set for future output until it is changed again.

Outputting values in binary

Outputting values in binary is a little harder, as std::cout doesn’t come with this capability built-in. Fortunately, the C++ standard library includes a type called std::bitset that will do this for us (in the <bitset> header).

To use std::bitset , we can define a std::bitset variable and tell std::bitset how many bits we want to store. The number of bits must be a compile-time constant. std::bitset can be initialized with an integral value (in any format, including decimal, octal, hex, or binary).

In the above code, this line:

creates a temporary (unnamed) std::bitset object with 4 bits, initializes it with binary literal 0b1010 , prints the value in binary, and then discards the temporary object.

Related content

We cover std::bitset in more detail in lesson O.1 -- Bit flags and bit manipulation via std::bitset .

In C++20 and C++23, we have better options available via the new Format Library (C++20) and Print Library (C++23):

Numbers Representation Systems – Decimal, Binary, Octal and Hexadecimal

In this article we’ll discuss about different number representation systems, where they are used and why they are useful. Briefly we’ll go through decimal , binary , octal and hexadecimal number representation.

Decimal (base 10)

The most common system for number representation is the decimal . Everybody is using it. It’s so common than most people must believe that is the only one. It’s used in finances, engineering and biology, almost everywhere we see and use numbers.

If someone is asking you to think at a number for sure you’ll think at a decimal number. If you think at a binary or hexadecimal one, you must have an extreme passion for arithmetic or software/programming.

As the name is saying the decimal number system is using 10 symbols/characters. In Latin language 10 is “decem” so decimal might be linked to the Latin word.

As you can see there are 10 symbols from 0 to 9 . With these symbols we can construct all the numbers in the decimal system.

All the numbers in the decimal system can be constructed by using the above mentioned symbols ( 0 … 9 ) multiplied with the power of 10. The power of ten gives us ones, tens, hundreds, thousands and so on.

The example below breaks down the decimal number 67049 into powers of 10 multiplied with numbers between 0 and 9 . This is just to show that any number in the decimal system can be decomposed into a sum of terms made of from the product of the power of 10 and the symbols 0 … 9 .

The same technique is going to be applied to the binary, octal and hexadecimal systems, being in fact a method of converting a number from decimal system in another format (base).

We can keep in mind these characteristics of the decimal numbers system:

• it’s using 10 symbols
• can be decomposed in factors containing powers of 10
• it’s the most common number representation system

Binary (base 2)

Let’s step into the geek side now.

Another number representation system is the binary one. As the name suggests and by analogy with the decimal system we can say that the binary system is using only 2 symbols/characters:

In the binary representation we only use 0 (zeros) and 1 (ones) to represent numbers.

The binary system is used wherever you want to store information in electronic format. All the computers that you know, intelligent devices, everything that has to do with electronics and microcontrollers use the binary system.

In electronics (digital) all the operations are done using two levels of voltage: high and low. Each level of voltage is assigned to a value/symbol: HIGH for 1 and LOW for 0. For a microcontroller which is supplied with +5V the 1 (high) will be represented by +5 V and the 0 (low) by 0 V.

Roughly we can say that the binary system is used because it can be translated in electronic signal.

All the decimal numbers we can think of can be represented into binary symbols. We do this by using a sum between terms of the power of 2 multiplied with 0 or 1.

As example we’ll use the number 149 (decimal representation) and transform it into binary representation. We could use any number but if it’s too big it would end up into a long string of zeros and ones.

As you can see the decimal number 149 is represented in binary system by a series of zeros and ones ( 10010101 ). Usually to distinguish between a decimal or binary number we must specify the base to which we are referring to. The base is described as a subscript after the last character of the number

By specifying the base of the number we eliminate the probability of confusion, because the same representation (e.g. 11) can mean different things for different bases.

Another way to avoid confusion is to use a special notation (prefix) for binary numbers. This is because 1100 can represent eleven hundreds in decimal system or the decimal twelve represented in binary system. So if want to specify a binary number we use the prefix 0b . Example: 0b1100 .

Briefly the characteristics of a binary system are:

• it’s using 2 symbols
• can be decomposed in factors containing powers of 2
• it’s used in computers, microcontrollers

Octal (base 8)

All the numbers in the octal system are represented using 8 symbols/characters, from 0 to 7 . The reason of using the octal system instead of the decimal one can be various. One of them is that instead of using our fingers for counting, we use the spaces between fingers.

Humans have 4 spaces between the fingers of one hand; in total we’ll have 8 spaces, for both hands. In this case it makes sense to use an octal number representation system instead of a decimal one. The drawback is that higher numbers will require more characters compared to the decimal one.

To transform a decimal represented number into an octal system we split it into terms containing the power of 8:

As an example we are going to represent the decimal number 67049 in octal base:

Hexadecimal (base 16)

The hexadecimal number representation system is using 16 symbols/characters to define numbers. It’s used in computer science mostly because can represent bigger decimal numbers with fewer characters.

Compared with decimal system it’s also using numeric symbols from 0 to 9. Additionally it’s using alphanumeric characters from A to F for values between 10 and 15.

To represent a decimal number in hexadecimal format we split the decimal number into a sum of terms. Each term is a product between a hexadecimal symbol and a power of 16.

The representation of the decimal number 67049 in hexadecimal format is 105E9 . Similar to the binary system a common practice is to use the prefix “ 0x ” in order to distinguish from the decimal notation. Example: 0x105E9 .

Briefly the characteristics of a hexadecimal number representation system are:

• it’s using 16 symbols
• can be decomposed in factors containing powers of 16

The table below summaries the characteristics of the above mentioned number representation systems.

Both the octal and hexadecimal number representation systems are linked to the computer system, mainly with the kind of processors and microcontrollers. For example, if the microprocessor is using 8 bit data then the octal system is suitable to interface data. If the microprocessor is on 16 bits then the hexadecimal system is appropriate to represent data.

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Hexadecimal numbers

Before we can discuss memory, we must first learn hexadecimal counting.

We count with ten different digits, and a number represents coefficients multiplied by different powers of 10. For example, $543$ is equal to $5$ hundreds ($= 10^2$) plus $4$ tens ($= 10^1$) plus $3$ ones ($= 10^0$).

Nothing requires us to have 10 digits, and its almost the worst possible base other than perhaps base $7$, $11$ or $19$. One base that is more convenient for computers is base 16, so we now have sixteen different digits:

0 1 2 3 4 5 6 7 8 9 a b c d e f

Now, you could set up addition and multiplication tables in base 16, so $9 + 3 = c$, $f + 1 = 10$, $ff + 1 = 100$, $9 \cdot 4 = 1c$, and $d^2 = a9$, but this isn't necessary here.

Instead, all you need to do is understand how to count in hexadecimal.

First, we will represent hexadecimal numbers by prefixing them by 0x , so 0x32 is a hexadecimal number and not thirty-two. Instead, 0x32 means $3 \times 16 + 2 = 50$.

Next, the number that comes before 0x32 is 0x31 ($= 49$), and the number that comes after is 0x33 ($= 51$).

Similarly, the number that comes after 0x3d49 is 0x3d4a , and the number after that is 0x3d4b .

The number after 0x3d4f is 0x3d50 , the number after 0x3d9f is 0x3da0 , and the number after 0x32adfff is 0x32ae000 .

Addresses tend to be stored in multiples of 2, 4 or 8, so starting at 0xff8a3d0 and counting by twos:

Starting at the same number but counting by four:

Starting at the same number but counting by eight:

For humor, however, here are the words to a song written by Paul McCarthy rewritten for hexadecimal:

Applying this to memory

A 32-bit computer has $2^{32} = 2^{4 \times 8} = (2^4)^8 = 16^8$ possible different addresses. Each byte has a different address, and thus the maximum number of bytes that can have different addresses are $2^{32}$ or approximately four billion (hence four gigabytes). Thus, any address of a 32-bit computer can be written with exactly eight hexadecimal numbers.

On a 64-bit system, addresses can be up to 64 bits long, or 16 hexadecimal characters; however, currently no one on the planet has sufficient memory to allow each address to be used—this would 18 quintillion bytes, or almost 17 million terabytes of memory.

At the other end, microcontrollers may have address buses as small as 16 bits; such microcontrollers can only access 64 kilobytes of memory. One long-running microprocessor that used only 16 bit addresses is the Motorola 6800. First introduced in 1974, it was last produced in 2006.

We will be referring to memory locations in future lessons, but for the most part, everything discussed here should be sufficient.

As you are likely aware, memory is divided into bytes, and each byte has its own address. Each byte is 8 bits, and thus, there are $2^8$ different values a byte can store. Note that $2^8 = 2^{4 \times 2} = (2^4)^2 = 16^2$, so to represent all possible values of a byte, you need only two hexadecimal characters:

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Why is memory address 0x?

The prefix 0x at the beginning of a number indicates that the following digits are represented in hexadecimal, or base 16. You‘ll often see memory addresses written with the 0x prefix, such as 0x00400000 or 0x0000FFE0. But why is the hexadecimal numbering system used for memory addresses in the first place?

What does 0x mean?

The 0x prefix is used as a marker to denote hexadecimal numerals in programming languages like C, C++, Go, Rust, Java, JavaScript, and others. Hexadecimal is a base 16 numbering system that uses the digits 0-9 and the letters A-F to represent values. The 0x tells the compiler or interpreter that the number should be interpreted as a hexadecimal value instead of decimal (base 10).

Some examples:

• 0x1A = 26 in decimal
• 0x52 = 82 in decimal
• 0xFF = 255 in decimal

Without the 0x prefix, the number would be treated as a normal decimal number by default. The prefix removes any ambiguity.

Why are memory addresses represented in hex?

Hexadecimal is commonly used for memory addresses because it is a more compact and human-readable way to express binary numbers. Modern computers use binary numbering systems internally to store data. But binary strings like 01001001010010 are not as easy to read and write for humans.

Hexadecimal provides a nice shorthand abstraction while still mapping cleanly back to the underlying binary representation. Each hexadecimal digit corresponds to a 4-bit nibble:

A 32-bit memory address written in hexadecimal like 0xDEADBEEF is easier to read and document than writing it in binary (11011110101011011011111011101111). Hexadecimal is simply a more concise and programmer-friendly way to express memory addresses.

Structure of a memory address

A typical memory address specified in hexadecimal has the following general structure:

0xPREFIXADDRESS

• 0x – The prefix indicating hexadecimal
• PREFIX – Additional segment, section, or offset portion
• ADDRESS – The main memory location reference

For example, the address 0xC0004000 breaks down into:

• 0x – Hex prefix
• C000 – Segment or section
• 4000 – The offset within the section

The number of bits used for each component depends on the system architecture. The total address length reflects the processor‘s memory bus width. Common sizes are 32-bit and 64-bit addresses.

Mapping memory addresses

The memory address ultimately maps to a specific physical byte of memory on the hardware. The CPU generates memory addresses when accessing data. An address like 0x00400000 would refer to the 4,194,304th byte (0x400000th location) in memory.

On a 32-bit system, the full address space is 2^32 memory locations (4GB total). A 64-bit system increases that to a vast 16 exbibytes (2^64). The processor and memory management unit handle translating logical addresses to physical RAM locations.

Examples of memory addresses

Some common examples of hexadecimal memory addresses:

• 0x10000000 – Start of a memory region
• 0x2A4FFC – Address of a variable or data object
• 0xB8000 – Start of video memory in x86 systems
• 0x7FFFFFFF – The last memory address on a 32-bit system
• 0x00400000 – A library loaded at this base address

The actual memory contents stored at these locations would be the data values, instructions, variables, etc referenced by the program.

32-bit vs 64-bit addressing

The key difference between 32-bit and 64-bit memory addressing is the size of the address space:

• 32-bit – 2^32 addresses = 4 gigabytes maximum
• 64-bit – 2^64 addresses = 16 exbibytes maximum

Different CPU architectures support different address sizes. 32-bit CPUs were common but 64-bit is now standard for modern computers.

The width of a memory address determines the maximum amount of memory that can be referenced, which is why 64-bit systems can support much more RAM than 32-bit architectures.

Hexadecimal in other computing contexts

Hexadecimal and the 0x prefix are used in many computing areas besides memory addressing:

• MAC addresses – Ex: 0xAABBCCDDEEFF
• Color codes – Ex: 0xFF0000 = red
• Unicode code points – Ex: 0x2603 = snowman symbol
• Binary file hashes
• HTTP status codes
• Bit flags and bitmasks

Hex provides a compact and convenient way to express binary data across many domains.

Comparison to other prefixes

There are other numeric prefixes similar to 0x that denote different bases:

• 0x – Hexadecimal
• 0b – Binary
• 0 – Octal

For example:

0x specifically indicates hexadecimal numbers. Some languages also allow 0h as a prefix.

In summary, the 0x prefix denotes hexadecimal numbering, which provides a compact and readable way to represent memory addresses for humans while precisely mapping to the binary addresses used internally. The widespread use of 0x for memory addressing stems from the utility of hexadecimal as an important notation for expressing binary data in computing and programming.

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F-String Python Hex, Oct, and Bin: Efficient Number Conversions

As a Python developer, I’ve always been fascinated by the simplicity and versatility of the language. Python has always been my go-to language for all my programming needs, from its clean syntax to the vast array of libraries available. I recently had to work on a project requiring me to convert numbers to their hexadecimal, octal, and binary representations. While there are several ways to do this in Python, I discovered the f-string Python Hex, Oct, and Bin methods , which turned out to be the most efficient and straightforward approach. In this blog post, I’ll share my personal experience with this method and show you how to use it to convert numbers efficiently.

A Short Overview (Promised) 👇

Python’s f-strings , also known as “formatted string literals”, have been introduced in Python 3.6 and offer a more efficient and readable way to format strings. They improve upon the older string formatting methods, such as using % -formatting and the str.format() method.

One of the key advantages of f-strings is their ability to handle various numeric representations, such as hexadecimal (hex), octal (oct), and binary (bin).

Handling hex, oct, and bin values in Python with f-strings is quite simple, thanks to the built-in support for these number systems. These can easily be incorporated into f-strings by using specific format specifiers within curly braces that contain the expressions to be evaluated and formatted at runtime. This allows you to seamlessly integrate hexadecimal, octal, and binary values into their strings without resorting to clumsier methods or additional function calls.

When working with f-strings in Python, you can use {variable:x} for hex, {variable:o} for oct, and {variable:b} for bin.

F-String Basics

One powerful feature of f-strings is their ability to format numbers in different bases, such as hexadecimal (hex), octal (oct), and binary (bin). To represent numbers in these formats, you can use the appropriate format specifier :

• Hexadecimal: {x:X}
• Octal: {x:o}
• Binary: {x:b}

For example, to convert an integer into its hexadecimal representation, you can use the following f-string:

Similarly, for octal and binary representations, you can use the respective format specifiers within the f-string:

Using f-strings, you can create more readable and efficient code when working with different number systems in Python 🔥.

💡 Recommended : 7 Tips to Clean Code

Formatting Numerical Values with F-Strings

To format a number in hexadecimal with f-strings, you can use the format specifier {variable:x} . If you want zero padding, use the format specifier {variable:0Nx} , where N is the desired number of digit characters.

Similarly, to format a number in octal representation, use the format specifier {variable:o} .

For binary representation, you can use the format specifier {variable:b} .

💡 In addition to numerical formats, you can also use other format specifiers like:

• Decimal : {variable:d}
• Floating-point : {variable:.Nf} (where N is the number of decimal places to round)
• Percentage : {variable:.N%}

Hexadecimal Representation

You can easily represent numbers in hexadecimal notation using f-strings in Python. Hexadecimal is a base-16 numeric system that uses sixteen distinct symbols – 0-9 representing zero to nine and A-F for ten to fifteen. It is widely used in computer programming and digital electronics.

You can use the format specifier within curly braces to convert a number to its hexadecimal representation using f-strings. For example:

The same can be achieved using the format() function:

You can also include the 0x prefix by using the # flag in the format specifier:

Hexadecimal representation can be helpful in various situations, such as when working with colors in CSS or addressing memory locations in computer systems. 🖥️ 🎨

Octal Representation

In Python, working with octal representations is smooth and straightforward thanks to built-in functions and f-strings. An octal number system has a base of 8, which means it uses eight symbols: 0, 1, 2, 3, 4, 5, 6, and 7. Converting between decimal and octal values can be accomplished with ease.

💡 Recommended : Python Conversions: Decimal + Binary + Octal + Hex

To obtain the octal representation of an integer in Python, simply use the oct() function. This function takes an integer as input and returns its octal representation as a string:

With f-strings, you can elegantly include octal values directly in your strings. To do this, use the format specifier {value:#o} within an f-string:

Here are some essential points to remember when working with octal representations in Python:

• Octal literals can be defined by adding a 0o prefix to the number: octal_number = 0o52 .
• Using arithmetic operations on octal numbers works the same as on decimal numbers: result = 0o52 + 0o10 .
• Convert between integer and octal string representations with the int() and oct() functions, respectively.

Binary Representation

In Python, f-strings are a convenient option for formatting strings, which can be used for various purposes like binary, octal, and hexadecimal representations. In this section, we’ll focus on the binary representation using f-strings. 😄

To display a binary value of an integer using f-strings, you can use the formatting specifier b followed by the variable you want to convert inside the curly braces. The syntax looks like this: f"{variable:b}" .

As an example, let’s say we want to display the binary representation of the number 5:

Now, you can use a loop to display binary representations for a range of numbers. Let’s say we want to print binary values for numbers 0 to 5:

This will output the following:

Note the use of 08b within the f-string. The number 8 indicates the minimum width, and leading zeros are used to fill any remaining space. This ensures that binary values are displayed with a consistent width, making the output easier to read. 🤓

Conversion Functions

This section will focus on Python’s f-string capabilities for converting between hexadecimal, octal, and binary number systems. The primary goal is to make these conversions as simple as possible using f-string syntax.

Int to Hex, Oct, and Bin

When working with integers, you can easily convert them to their corresponding hexadecimal, octal, and binary representations using f-string formatting. Here are some examples:

This simple f-string syntax allows you to convert integers to their corresponding representation within an f-string using the 'x' , 'o' , and 'b' format specifiers.

String to Hex, Oct, and Bin

For converting strings to their hexadecimal, octal, and binary representations, you can use the following approach:

In this example, we first convert the string to bytes using the 'utf-8' encoding. Then we utilize the binas cii module to convert bytes to their hexadecimal representation. We use list comprehension and f-string formatting for octal and binary conversions to achieve the same result.

Incorporating f-strings allows you to conveniently and efficiently perform conversions between various number systems in Python. This technique can be particularly beneficial when dealing with data representation and manipulation tasks. 😊

Formatting Specifiers and F-Strings

The formatted string literals allow you to embed expressions inside the string, which are evaluated and formatted using the __format__ protocol.

Formatting numbers in hex, oct, and bin is quite easy with f-strings. Here are some examples:

• Hexadecimal: f"{number:x}"
• Octal: f"{number:o}"
• Binary: f"{number:b}"

You have the option to add formatting specifiers and padding to your f-Strings. For instance, you can center the output and add zero-padding:

This snippet first converts the data to hexadecimal using {data:#04x} and then centers the output with {data:^14s} (from Stack Overflow ).

Here’s a brief table that demonstrates how to format numbers using f-Strings:

Utilizing f-Strings and various formatting specifiers can help you efficiently manage complex string formatting tasks in your Python projects 😃.

To keep boosting your Python skills and stay on track with the emerging technologies of the exponential age (e.g., ChatGPT), check out our free cheat sheets and email academy here:

While working as a researcher in distributed systems, Dr. Christian Mayer found his love for teaching computer science students.

To help students reach higher levels of Python success, he founded the programming education website Finxter.com that has taught exponential skills to millions of coders worldwide. He’s the author of the best-selling programming books Python One-Liners (NoStarch 2020), The Art of Clean Code (NoStarch 2022), and The Book of Dash (NoStarch 2022). Chris also coauthored the Coffee Break Python series of self-published books. He’s a computer science enthusiast, freelancer , and owner of one of the top 10 largest Python blogs worldwide.

His passions are writing, reading, and coding. But his greatest passion is to serve aspiring coders through Finxter and help them to boost their skills. You can join his free email academy here.

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• Question - Binary to Decimal

Number System Conversions

When writing programs for microcontrollers we’re usually stuck dealing with 3 different number systems: decimal , binary and hexadecimal (or hex). We use decimal because it comes naturally; that’s the way we count. Unfortunately, it’s not how computers count. Since computers and microcontrollers are limited to 1’s and 0’s, they count using sequences of these numbers. This is the binary number system. Binary numbers are usually prefixed with the '0b' characters which are not part of the number. Sometimes they are also subdivided into groups of 4 digits to make them easier to read as well as easier to relate to the hexadecimal number system. An example of a binary number is 0b0100.1011. The periods in the number don't represent anything, they just make it easier to read the number.

The binary system is simple to understand, but it takes a lot of digits to use the binary system to represent large numbers. The hexadecimal system can represent much larger numbers using fewer characters, and it closely resembles binary numbers. Hexadecimal numbers are usually prefixed with the characters '0x' which are not part of the number. A single hexadecimal digit can represent four binary digits! Binary numbers can only consist of 1’s and 0’s; typically a binary number consists of 8 digits (or some multiple of 8) if it’s being used in some kind of a computer (or microcontroller). It’s useful to know how to convert a binary number into a decimal number and vice versa. So how do we convert between number systems? First consider how we determine the value of a decimal number. The number 268 can be broken down as 200 + 60 + 8, or 2 * (10^2) + 6 * (10^1) + 8 * (10^0).  There are two important numbers that we have to know to ‘deconstruct’ the number - the base of the number system and the location of the digit within the number. The base of a decimal number is 10. When we’re converting the number 268, 2 is the second digit, 6 is the first digit and 8 is the zero digit. Each digit has to be scaled according to its place within the number. The scale of the digit is the base of the number system raised to the power of the digit's location in the number. So each number is scaled, and then all of the scaled digits are added to find the total value of the number. The same method can be used to find the value of a binary number. For example, let’s look at the number 0b1011.0101. The base of the binary system is 2 (the prefix 0b is often used in code to indicate that the number is in the binary format). The value of our number is: 1*(2^7)+0*(2^6)+1*(2^5)+1*(2^4)+0*(2^3)+1*(2^2)+0*(2^1)+1*(2^0), which is equal to 181. 0b1011.0101. What a completely inefficient way of typing a number! But we can represent the same binary number using only 2 hexadecimal digits. First though, we'll start by converting a hexadecimal (hex) number to decimal like we did for a binary number. How about 0xB5? Wait, what?! The prefix 0x is used in code to indicate that the number is being written in hex. But what is ‘B’ doing in there? The hexadecimal format has a base of 16, which means that each digit can represent up to 16 different values. Unfortunately, we run out of numerical digits after ‘9,’ so we start using letters. The letter ‘A’ represents 10, ‘B’ is 11, ‘C’ is 12, ‘D’ is 13, ‘E’ is 14 and ‘F’ is 15. ‘F’ is the largest digit in the hex numbering system. We convert the number the same way as before. The value of 0xB5, then, is: B*(16^1)+5*(16^0) or 181. Knowing how to convert binary and hex to decimal is important, but the most useful number conversion is probably converting between hex and binary. These two numbering systems actually work pretty well together. The numbering systems happen to be related such that a single hex digit represents exactly 4 binary digits, and so 2 hex digits can represent 8 bits (or binary digits). Here’s a table that shows how each hex digit is related to the binary system:  

For example, to convert the hex number 0x1C to binary, we would find the corresponding binary value for 1 and C and combine them. So, 0x1C in binary is 0b0001.1100. If we wanted to figure out the hex value for a binary number we just go the other way. To find the hex representation of the binary number 0b0010.1011 we first find the hex value for 0010, then the hex value for 1011 and combine them; the hex value would be 0x2B.

There are many free tools available to help convert between these numbering systems, just google 'hex number conversion.' If you use Windows as an operating system you have a great tool built into the calculator. Just change the calculator to scientific mode and you can convert between number systems by typing a number then changing the format of the calculator!

Quiz Question - Binary to Decimal

What is the decimal representation of the binary number 0b10010100?

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