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Binary Number System

What a number system is: A system that represents numbers in terms of symbols or digits.

A number system is a set of rules and symbols used to represent numbers. It serves as the basis for all mathematical computations. Decimal (base 10), binary (base 2), octal (base 8), and hexadecimal (base 16) are examples of common number systems. Every system has applications in a variety of domains, including digital electronics, computing, and everyday maths.

Table of Content:

Different Number Systems
Reasons Computers Utilize Binary Rather Than Decimal
Understanding Place Value in Binary
Conversion Between Decimal and Binary
Binary to Decimal Conversion
Binary Arithmetic
Binary Number Representation
Binary in Computing and Digital Systems
Binary Codes
Applications of Binary System
Common Mistakes to Avoid
Practice and Exercises
Conclusion
FAQs on Binary Number System

Different Number Systems

Decimal system (Base-10): Digits 0-9, what we all use in daily life
Binary system (Base-2): Digits 0 and 1, basis of computing
Octal system (Base-8): The digits 0-7, applied in some computing environments
Hexadecimal system (Base-16): The digits 0-9 and the letters A-F, utilized for concise digital notation

Reasons Computers Utilize Binary Rather Than Decimal

Electronic circuits conduct in two states (on/off, high/low voltage), making binary most suitable
Hardware design simplicity, reliability, and accuracy
The binary number system employs two digits only - 0 and 1
Binary digits are referred to as bits (short for Binary digit)

Each place value of the digit increases in powers of 2, from right to left:

For example, binary 1011 = 1×8 + 0×4 + 1×2 + 1×1 = 11 in decimal

Binary numbers are the basis of:

Digital electronics
Programming languages
Data storage
Network communication

Understanding Place Value in Binary

Place values in binary:

Rightmost bit → 2⁰ (1)
Next bit → 2¹ (2)
Next bit → 2² (4)
Continues increasing by powers of 2

Binary example:

Binary: 1101

1 × 2³ = 8
1 × 2² = 4
0 × 2¹ = 0
1 × 2⁰ = 1

Total = 8 + 4 + 0 + 1 = 13 in decimal

Conversion Between Decimal and Binary

Decimal to Binary

Method: Division by 2

Divide the decimal number by 2 over and over
Write the remainder each time
Read remainders from bottom to top

Example:

1. Binary for 18 is:

18 ÷ 2 = 9, remainder 0
9 ÷ 2 = 4, remainder 1
4 ÷ 2 = 2, remainder 0
2 ÷ 2 = 1, remainder 0
1 ÷ 2 = 0, remainder 1

Reading upwards → 10010

Binary to Decimal Conversion

Add up the product of each bit and its place value

Example:

1. Binary 10101:

(1 × 16) + (0 × 8) + (1 × 4) + (0 × 2) + (1 × 1)
16 + 0 + 4 + 0 + 1 = 21 in decimal

Binary Arithmetic

Binary Addition

Rules:

0 + 0 = 0
0 + 1 = 1
1 + 0 = 1
1 + 1 = 10 (0 with a carry of 1)

Example:

1. 1011 + 1101

Add each piece from right to left
Account for carries

Binary Subtraction

Borrow like in decimal subtraction

Binary subtraction rules:

0 − 0 = 0
1 − 0 = 1
1 − 1 = 0
0 − 1 = borrow from next higher bit

Example:

1001 − 0101 = 0100

Binary Multiplication

As decimal multiplication, but in a simpler way:

0 × any bit = 0
1 × any bit = that bit

Multiply and shift left for each new line, then add results

Binary Division

Same as long division in decimal

Work out how many times divisor will fit into the dividend

Subtract and bring down bits

Binary Number Representation

Signed and Unsigned Binary Numbers

Unsigned binary numbers only contain positive values
Signed binary numbers can have positive and negative values

Two's complement representation is widely used:

Reverse all bits and add 1 to represent negative values

Binary Fractions

Use binary digits after a "binary point" (similar to decimal point)

Every digit value to the right of the binary point is fractions:

2⁻¹ = 0.5
2⁻² = 0.25
2⁻³ = 0.125

Example:

0.101 in binary:

1 × 0.5 = 0.5
0 × 0.25 = 0
1 × 0.125 = 0.125

Total = 0.625

Binary in Computing and Digital Systems

Why computers keep data:

Every piece of data in a computer - text, pictures, music, videos - is translated into binary code

Examples:

The character 'A' in ASCII is 01000001
Pixel color values in a picture are binary
Binary signals (on/off) are used by digital circuits

Knowing about binary is important for:

Programmers
Computer engineers
Cybersecurity professionals
Electronics enthusiasts

Binary Codes

Binary codes are applied to effective data transfer and error detection

Typical binary codes:

BCD (Binary Coded Decimal): A decimal digit is expressed by 4 binary bits
Gray Code: One bit changes in successive numbers; employed in rotary encoders
ASCII Code: Expresses characters (letters, symbols) in computers
Parity bits: For data transmission error detection

Applications of Binary System

Design of digital electronics (microprocessors, logic gates)
Data encryption and cryptography
Data storage and memory devices
Compilers and programming languages
Networking protocols
Automation and robotics

Common Mistakes to Avoid

Binary 10 is decimal 2, not ten

Omitting reading remainders from bottom to top when converting
Confusing bit positions (little-endian vs big-endian representations)
Suspecting that binary fractions behave as decimal fractions (their place values are powers of 2, not 10)

Practice and Exercises

Convert the following decimals to binary:

Convert the following binaries to decimal:

11001
1010110

Add using binary:

1011 + 1101

Binary subtraction:

10001 - 1010

Attempt multiplying:

110 × 101

Investigate representing negative numbers with two's complement:

Represent -9 in an 8-bit system

Related Links

Prime Numbers: Unlock the secrets of prime numbers and understand why they are the building blocks of mathematics!

Co-prime Numbers:Learn how co-prime numbers work and why they matter in simplifying fractions and solving number puzzles.

Prime Numbers From 1 to 1000:Explore the complete list of prime numbers from 1 to 1000 and boost your number-crunching skills today!

Conclusion

The language of binary numbers is the basis of contemporary computing and electronics. Whether you are learning computer science, dealing with digital circuits, or venturing into coding, learning how binary works is imperative. It helps unlock knowledge about data storage, network protocols, and hardware design. Being proficient in binary gives you the ability to think like a computer-and that's a valuable asset in the digital world.