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Learn How to Convert Binary Numbers: From Binary to Denary and Beyond!

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Reuben Cowell

13/02/2023

Computer Science

1.2.4 Data storage

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Bits and Bytes

Learn How to Convert Binary Numbers: From Binary to Denary and Beyond!

Converting numbers between different number systems is a fundamental concept in computer science and mathematics.

Binary to denary conversion involves translating numbers from base-2 (binary) to base-10 (decimal/denary). This process can be done manually using a binary table or through digital tools like a binary to decimal converter online. When converting manually, each binary digit (0 or 1) is multiplied by its corresponding power of 2, starting from the rightmost digit. For example, in binary to decimal conversion examples like "1011 binary to decimal", we calculate: (1×2³) + (0×2²) + (1×2¹) + (1×2⁰) = 8 + 0 + 2 + 1 = 11.

Binary shifting operations are essential in computer programming and digital electronics. A binary shift moves all digits in a binary number left or right by a specified number of positions. Binary shift multiplication occurs when shifting left (equivalent to multiplying by powers of 2), while binary shift division happens when shifting right (dividing by powers of 2). For instance, when you perform a binary shift of 3 places right on the binary number 10001110, the result demonstrates how binary shifting can efficiently perform mathematical operations. These concepts are often taught alongside truth tables, which show all possible combinations of inputs and outputs in logical operations. Students can practice these concepts through binary shift questions and explore resources like BBC Bitesize for additional learning materials. Understanding binary patterns and their manipulation is crucial for computer programming, particularly when working with Python or other programming languages that handle binary data.

Converting between different number systems, including how to convert binary to hexadecimal and how to convert denary to hexadecimal, builds upon these fundamental concepts. Common examples like 111 binary to decimal and 1010 binary to decimal help students grasp these conversion principles through practice. These skills are essential for anyone studying computer science, digital electronics, or programming, as they form the foundation for understanding how computers process and store information at the most basic level.

...

13/02/2023

317

00111110011111111011111100 11101111011111111111000111 11101100000010011001110111 00000111011111011101111101 00010010011111000100011000 10011

View

Understanding Binary Number Systems and Conversions

Binary numbers form the foundation of computer operations, using only 0s and 1s to represent all data. Understanding how to convert between binary, denary (decimal), and hexadecimal numbers is crucial for computer science students.

Definition: A binary pattern is a sequence of 1s and 0s that represents data in computer systems. Each position represents a power of 2, with values increasing from right to left.

The binary number system operates on base-2, unlike our everyday decimal system which uses base-10. When working with 8-bit binary numbers, you can represent values from 0 to 255, with each bit position having a specific value based on powers of 2.

Example: Binary to denary examples:

  • 1011 (binary) = 8 + 2 + 1 = 11 (decimal)
  • 1010 (binary) = 8 + 2 = 10 (decimal)
  • 111 (binary) = 4 + 2 + 1 = 7 (decimal)
00111110011111111011111100 11101111011111111111000111 11101100000010011001110111 00000111011111011101111101 00010010011111000100011000 10011

View

Binary Conversion Methods and Calculations

Converting between number systems requires understanding place values and systematic approaches. The Binary to denary table helps visualize these conversions:

Place Values: 128 64 32 16 8 4 2 1 Bit Position: 2⁷ 2⁶ 2⁵ 2⁴ 2³ 2² 2¹ 2⁰

Highlight: When converting binary to denary, multiply each 1 in the binary number by its corresponding place value and sum the results.

For those seeking practice, Binary to denary questions often appear in educational resources like BBC Bitesize. These resources provide structured learning paths and interactive exercises to master conversion techniques.

Example: To convert 10101010 to decimal: 1×128 + 0×64 + 1×32 + 0×16 + 1×8 + 0×4 + 1×2 + 0×1 = 170

00111110011111111011111100 11101111011111111111000111 11101100000010011001110111 00000111011111011101111101 00010010011111000100011000 10011

View

Binary Shifts and Mathematical Operations

Binary shift operations provide efficient ways to multiply or divide numbers by powers of 2. Understanding binary shift multiplication and binary shift division is essential for optimizing computer calculations.

Example: Binary shift example: Original number: 10001110 Perform a binary shift of 3 places right on the binary number 10001110 results in: 00010001 This effectively divides the number by 2³ (8)

A Right binary shift divides the number by 2 for each position shifted, while a left shift multiplies by 2. These operations are fundamental in computer arithmetic and data manipulation.

00111110011111111011111100 11101111011111111111000111 11101100000010011001110111 00000111011111011101111101 00010010011111000100011000 10011

View

Hexadecimal System and Conversions

Learning how to convert binary to hexadecimal provides a more compact way to represent binary numbers. Hexadecimal uses 16 digits (0-9 and A-F) and is commonly used in programming and memory addressing.

Vocabulary: Hexadecimal conversion rules:

  • Each hexadecimal digit represents 4 binary digits (nibble)
  • Values 10-15 are represented by letters A-F
  • How to convert denary to hexadecimal: First convert to binary, then group bits into sets of 4

The relationship between binary, decimal, and hexadecimal systems is crucial for computer science students. Many online tools like Binary to decimal converter online and Binary to decimal Converter with solution can help verify conversions while learning.

00111110011111111011111100 11101111011111111111000111 11101100000010011001110111 00000111011111011101111101 00010010011111000100011000 10011

View

Understanding Binary Conversions and Digital Data Representation

Binary numbers form the foundation of how computers process and store information. This comprehensive guide explores various aspects of binary conversions, character encoding, and digital data representation.

Binary to Denary Conversion Understanding how to convert between binary and denary (decimal) numbers is crucial for computer science. When converting binary to denary, each digit is multiplied by its corresponding power of 2 and then summed together. For example, the binary number 1011 converts to decimal by calculating (1×8) + (0×4) + (1×2) + (1×1) = 11.

Example: Converting binary 1010 to decimal: 1×8 + 0×4 + 1×2 + 0×1 = 8 + 0 + 2 + 0 = 10

Hexadecimal Conversions How to convert binary to hexadecimal involves grouping binary digits into sets of four (nibbles) and converting each group to its hex equivalent. For instance, binary 00011110 splits into 0001 (1) and 1110 (E), giving hexadecimal 1E. A binary to hexadecimal conversion table is essential for quick reference.

Definition: A nibble is a group of 4 binary digits, representing half a byte.

Character Encoding Computers use character sets to represent text as binary numbers. ASCII (American Standard Code for Information Interchange) uses 7 bits to represent 128 different characters, while Unicode expands this to multiple bytes to accommodate international character sets.

Vocabulary: Character sets are standardized collections of characters with their corresponding binary representations.

Digital Image Representation Images are stored as collections of pixels, with each pixel's color represented in binary. The image quality depends on two key factors:

  • Color depth: determines the number of possible colors (2^n where n is the bit depth)
  • Resolution: total number of pixels (width × height)

Highlight: File size (in bits) = resolution × color depth

Sound Digitization Digital sound recording involves converting analog sound waves into binary data through sampling. The quality and file size depend on:

  • Sample rate (measured in Hz)
  • Bit depth (number of bits per sample)
  • Duration of the recording

Example: A 44.1 kHz sample rate with 16-bit depth for 1 minute would result in: File size = 44,100 × 16 × 60 = 42,336,000 bits

This comprehensive understanding of binary systems and digital data representation is fundamental for anyone studying computer science or working with digital systems.

00111110011111111011111100 11101111011111111111000111 11101100000010011001110111 00000111011111011101111101 00010010011111000100011000 10011

View

Binary Number Systems and Conversions

Binary is the base-2 number system used by computers, consisting of only 0s and 1s. This section covers the fundamentals of binary numbers and how to convert between binary and denary (base-10) systems.

Definition: Binary is a base-2 number system used by computers, representing the flow of electricity with only two digits (0 and 1).

In binary, each digit represents a power of 2, increasing from right to left. Most commonly, 8-bit binary numbers are used, representing denary numbers from 0 to 255.

Example: An 8-bit binary number table:

| 2^7 | 2^6 | 2^5 | 2^4 | 2^3 | 2^2 | 2^1 | 2^0 | |-----|-----|-----|-----|-----|-----|-----|-----| | 128 | 64 | 32 | 16 | 8 | 4 | 2 | 1 |

To convert binary to denary, add up the place values where there is a 1 in the binary number. For instance, 10101010 in binary equals 170 in denary (128 + 32 + 8 + 2).

Highlight: The leftmost bit in an 8-bit binary number is the most significant bit, with a value of 128.

Converting from denary to binary involves subtracting the largest possible place value and placing a 1 in that position, repeating until reaching 0. Alternatively, you can divide the denary number by 2 repeatedly, noting the remainders.

Example: Converting 42 to binary: 42 - 32 = 10, 10 - 8 = 2, 2 - 2 = 0. This gives us 00101010 in 8-bit binary.

00111110011111111011111100 11101111011111111111000111 11101100000010011001110111 00000111011111011101111101 00010010011111000100011000 10011

View

00111110011111111011111100 11101111011111111111000111 11101100000010011001110111 00000111011111011101111101 00010010011111000100011000 10011

View

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Lena, iOS user

I love this app ❤️ I actually use it every time I study.

Subjects

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Computer Science

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Digital Information

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Bits and Bytes

Learn How to Convert Binary Numbers: From Binary to Denary and Beyond!

user profile picture

Reuben Cowell

@reubencowell

·

133 Followers

Follow

Converting numbers between different number systems is a fundamental concept in computer science and mathematics.

Binary to denary conversion involves translating numbers from base-2 (binary) to base-10 (decimal/denary). This process can be done manually using a binary table or through digital tools like a binary to decimal converter online. When converting manually, each binary digit (0 or 1) is multiplied by its corresponding power of 2, starting from the rightmost digit. For example, in binary to decimal conversion examples like "1011 binary to decimal", we calculate: (1×2³) + (0×2²) + (1×2¹) + (1×2⁰) = 8 + 0 + 2 + 1 = 11.

Binary shifting operations are essential in computer programming and digital electronics. A binary shift moves all digits in a binary number left or right by a specified number of positions. Binary shift multiplication occurs when shifting left (equivalent to multiplying by powers of 2), while binary shift division happens when shifting right (dividing by powers of 2). For instance, when you perform a binary shift of 3 places right on the binary number 10001110, the result demonstrates how binary shifting can efficiently perform mathematical operations. These concepts are often taught alongside truth tables, which show all possible combinations of inputs and outputs in logical operations. Students can practice these concepts through binary shift questions and explore resources like BBC Bitesize for additional learning materials. Understanding binary patterns and their manipulation is crucial for computer programming, particularly when working with Python or other programming languages that handle binary data.

Converting between different number systems, including how to convert binary to hexadecimal and how to convert denary to hexadecimal, builds upon these fundamental concepts. Common examples like 111 binary to decimal and 1010 binary to decimal help students grasp these conversion principles through practice. These skills are essential for anyone studying computer science, digital electronics, or programming, as they form the foundation for understanding how computers process and store information at the most basic level.

...

13/02/2023

317

 

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Computer Science

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00111110011111111011111100 11101111011111111111000111 11101100000010011001110111 00000111011111011101111101 00010010011111000100011000 10011

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Understanding Binary Number Systems and Conversions

Binary numbers form the foundation of computer operations, using only 0s and 1s to represent all data. Understanding how to convert between binary, denary (decimal), and hexadecimal numbers is crucial for computer science students.

Definition: A binary pattern is a sequence of 1s and 0s that represents data in computer systems. Each position represents a power of 2, with values increasing from right to left.

The binary number system operates on base-2, unlike our everyday decimal system which uses base-10. When working with 8-bit binary numbers, you can represent values from 0 to 255, with each bit position having a specific value based on powers of 2.

Example: Binary to denary examples:

  • 1011 (binary) = 8 + 2 + 1 = 11 (decimal)
  • 1010 (binary) = 8 + 2 = 10 (decimal)
  • 111 (binary) = 4 + 2 + 1 = 7 (decimal)
00111110011111111011111100 11101111011111111111000111 11101100000010011001110111 00000111011111011101111101 00010010011111000100011000 10011

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Binary Conversion Methods and Calculations

Converting between number systems requires understanding place values and systematic approaches. The Binary to denary table helps visualize these conversions:

Place Values: 128 64 32 16 8 4 2 1 Bit Position: 2⁷ 2⁶ 2⁵ 2⁴ 2³ 2² 2¹ 2⁰

Highlight: When converting binary to denary, multiply each 1 in the binary number by its corresponding place value and sum the results.

For those seeking practice, Binary to denary questions often appear in educational resources like BBC Bitesize. These resources provide structured learning paths and interactive exercises to master conversion techniques.

Example: To convert 10101010 to decimal: 1×128 + 0×64 + 1×32 + 0×16 + 1×8 + 0×4 + 1×2 + 0×1 = 170

00111110011111111011111100 11101111011111111111000111 11101100000010011001110111 00000111011111011101111101 00010010011111000100011000 10011

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Binary Shifts and Mathematical Operations

Binary shift operations provide efficient ways to multiply or divide numbers by powers of 2. Understanding binary shift multiplication and binary shift division is essential for optimizing computer calculations.

Example: Binary shift example: Original number: 10001110 Perform a binary shift of 3 places right on the binary number 10001110 results in: 00010001 This effectively divides the number by 2³ (8)

A Right binary shift divides the number by 2 for each position shifted, while a left shift multiplies by 2. These operations are fundamental in computer arithmetic and data manipulation.

00111110011111111011111100 11101111011111111111000111 11101100000010011001110111 00000111011111011101111101 00010010011111000100011000 10011

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Hexadecimal System and Conversions

Learning how to convert binary to hexadecimal provides a more compact way to represent binary numbers. Hexadecimal uses 16 digits (0-9 and A-F) and is commonly used in programming and memory addressing.

Vocabulary: Hexadecimal conversion rules:

  • Each hexadecimal digit represents 4 binary digits (nibble)
  • Values 10-15 are represented by letters A-F
  • How to convert denary to hexadecimal: First convert to binary, then group bits into sets of 4

The relationship between binary, decimal, and hexadecimal systems is crucial for computer science students. Many online tools like Binary to decimal converter online and Binary to decimal Converter with solution can help verify conversions while learning.

00111110011111111011111100 11101111011111111111000111 11101100000010011001110111 00000111011111011101111101 00010010011111000100011000 10011

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Understanding Binary Conversions and Digital Data Representation

Binary numbers form the foundation of how computers process and store information. This comprehensive guide explores various aspects of binary conversions, character encoding, and digital data representation.

Binary to Denary Conversion Understanding how to convert between binary and denary (decimal) numbers is crucial for computer science. When converting binary to denary, each digit is multiplied by its corresponding power of 2 and then summed together. For example, the binary number 1011 converts to decimal by calculating (1×8) + (0×4) + (1×2) + (1×1) = 11.

Example: Converting binary 1010 to decimal: 1×8 + 0×4 + 1×2 + 0×1 = 8 + 0 + 2 + 0 = 10

Hexadecimal Conversions How to convert binary to hexadecimal involves grouping binary digits into sets of four (nibbles) and converting each group to its hex equivalent. For instance, binary 00011110 splits into 0001 (1) and 1110 (E), giving hexadecimal 1E. A binary to hexadecimal conversion table is essential for quick reference.

Definition: A nibble is a group of 4 binary digits, representing half a byte.

Character Encoding Computers use character sets to represent text as binary numbers. ASCII (American Standard Code for Information Interchange) uses 7 bits to represent 128 different characters, while Unicode expands this to multiple bytes to accommodate international character sets.

Vocabulary: Character sets are standardized collections of characters with their corresponding binary representations.

Digital Image Representation Images are stored as collections of pixels, with each pixel's color represented in binary. The image quality depends on two key factors:

  • Color depth: determines the number of possible colors (2^n where n is the bit depth)
  • Resolution: total number of pixels (width × height)

Highlight: File size (in bits) = resolution × color depth

Sound Digitization Digital sound recording involves converting analog sound waves into binary data through sampling. The quality and file size depend on:

  • Sample rate (measured in Hz)
  • Bit depth (number of bits per sample)
  • Duration of the recording

Example: A 44.1 kHz sample rate with 16-bit depth for 1 minute would result in: File size = 44,100 × 16 × 60 = 42,336,000 bits

This comprehensive understanding of binary systems and digital data representation is fundamental for anyone studying computer science or working with digital systems.

00111110011111111011111100 11101111011111111111000111 11101100000010011001110111 00000111011111011101111101 00010010011111000100011000 10011

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Binary Number Systems and Conversions

Binary is the base-2 number system used by computers, consisting of only 0s and 1s. This section covers the fundamentals of binary numbers and how to convert between binary and denary (base-10) systems.

Definition: Binary is a base-2 number system used by computers, representing the flow of electricity with only two digits (0 and 1).

In binary, each digit represents a power of 2, increasing from right to left. Most commonly, 8-bit binary numbers are used, representing denary numbers from 0 to 255.

Example: An 8-bit binary number table:

| 2^7 | 2^6 | 2^5 | 2^4 | 2^3 | 2^2 | 2^1 | 2^0 | |-----|-----|-----|-----|-----|-----|-----|-----| | 128 | 64 | 32 | 16 | 8 | 4 | 2 | 1 |

To convert binary to denary, add up the place values where there is a 1 in the binary number. For instance, 10101010 in binary equals 170 in denary (128 + 32 + 8 + 2).

Highlight: The leftmost bit in an 8-bit binary number is the most significant bit, with a value of 128.

Converting from denary to binary involves subtracting the largest possible place value and placing a 1 in that position, repeating until reaching 0. Alternatively, you can divide the denary number by 2 repeatedly, noting the remainders.

Example: Converting 42 to binary: 42 - 32 = 10, 10 - 8 = 2, 2 - 2 = 0. This gives us 00101010 in 8-bit binary.

00111110011111111011111100 11101111011111111111000111 11101100000010011001110111 00000111011111011101111101 00010010011111000100011000 10011

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00111110011111111011111100 11101111011111111111000111 11101100000010011001110111 00000111011111011101111101 00010010011111000100011000 10011

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00111110011111111011111100 11101111011111111111000111 11101100000010011001110111 00000111011111011101111101 00010010011111000100011000 10011

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Can't find what you're looking for? Explore other subjects.

Knowunity is the #1 education app in five European countries

Knowunity has been named a featured story on Apple and has regularly topped the app store charts in the education category in Germany, Italy, Poland, Switzerland, and the United Kingdom. Join Knowunity today and help millions of students around the world.

Ranked #1 Education App

Download in

Google Play

Download in

App Store

Knowunity is the #1 education app in five European countries

4.9+

Average app rating

20 M

Pupils love Knowunity

#1

In education app charts in 17 countries

950 K+

Students have uploaded notes

Still not convinced? See what other students are saying...

iOS User

I love this app so much, I also use it daily. I recommend Knowunity to everyone!!! I went from a D to an A with it :D

Philip, iOS User

The app is very simple and well designed. So far I have always found everything I was looking for :D

Lena, iOS user

I love this app ❤️ I actually use it every time I study.