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18:["$","div",null,{"className":"mt-8 pt-8 border-t border-gray-200 dark:border-gray-700","children":["$","div",null,{"className":"flex flex-col md:flex-row justify-between items-center","children":[["$","div",null,{"className":"flex flex-wrap justify-center md:justify-start space-x-6 mb-4 md:mb-0","children":[["$","$Ld",null,{"href":"/privacy","className":"text-gray-500 dark:text-gray-400 hover:text-blue-600 dark:hover:text-blue-400 text-sm transition-colors","children":"Privacy Policy"}],["$","$Ld",null,{"href":"/terms","className":"text-gray-500 dark:text-gray-400 hover:text-blue-600 dark:hover:text-blue-400 text-sm transition-colors","children":"Terms of Service"}],["$","$Ld",null,{"href":"/sitemap.xml","className":"text-gray-500 dark:text-gray-400 hover:text-blue-600 dark:hover:text-blue-400 text-sm transition-colors","children":"Sitemap"}]]}],["$","div",null,{"className":"text-gray-500 dark:text-gray-400 text-sm text-center md:text-right","children":[["$","p",null,{"children":["© ",2025," UniversalCalc. All rights reserved."]}],["$","p",null,{"className":"mt-1","children":"Free online calculator tools for everyone."}]]}]]}]}]
1a:["$","h2",null,{"children":"1. Understanding Number Systems"}]
1b:["$","h3",null,{"children":"Decimal (Base 10) - Our Everyday System"}]
1c:["$","p",null,{"children":"Uses digits 0-9. Each position represents a power of 10:"}]
1d:["$","p",null,{"children":[["$","strong",null,{"children":"Example:"}]," 1234₁₀ = (1×10³) + (2×10²) + (3×10¹) + (4×10⁰) = 1000 + 200 + 30 + 4"]}]
1e:["$","h3",null,{"children":"Binary (Base 2) - Computer Language"}]
1f:["$","p",null,{"children":"Uses only digits 0 and 1. Each position represents a power of 2:"}]
20:["$","div",null,{"className":"bg-blue-50 dark:bg-blue-900 rounded-lg p-6 my-6","children":[["$","h4",null,{"className":"font-semibold text-blue-800 dark:text-blue-200 mb-4","children":"💻 Binary Example:"}],["$","p",null,{"className":"text-blue-700 dark:text-blue-300","children":["$","strong",null,{"children":"1101₂ = (1×2³) + (1×2²) + (0×2¹) + (1×2⁰) = 8 + 4 + 0 + 1 = 13₁₀"}]}]]}]
21:["$","h4",null,{"children":"Binary Conversion Tips"}]
22:["$","ul",null,{"children":[["$","li",null,{"children":[["$","strong",null,{"children":"Decimal to Binary:"}]," Repeatedly divide by 2, read remainders bottom-up"]}],["$","li",null,{"children":[["$","strong",null,{"children":"Binary to Decimal:"}]," Add powers of 2 for each '1' position"]}],["$","li",null,{"children":[["$","strong",null,{"children":"Quick powers of 2:"}]," 2⁰=1, 2¹=2, 2²=4, 2³=8, 2⁴=16, 2⁵=32, 2⁶=64, 2⁷=128, 2⁸=256"]}]]}]
23:["$","h3",null,{"children":"Hexadecimal (Base 16) - Programmer's Friend"}]
24:["$","p",null,{"children":"Uses digits 0-9 and letters A-F (A=10, B=11, C=12, D=13, E=14, F=15):"}]
25:["$","div",null,{"className":"bg-green-50 dark:bg-green-900 rounded-lg p-6 my-6","children":[["$","h4",null,{"className":"font-semibold text-green-800 dark:text-green-200 mb-4","children":"🔢 Hex Conversion:"}],["$","p",null,{"className":"text-green-700 dark:text-green-300","children":[["$","strong",null,{"children":"2A7₁₆ = (2×16²) + (10×16¹) + (7×16⁰) = 512 + 160 + 7 = 679₁₀"}],["$","br",null,{}],["$","strong",null,{"children":"Why hex?"}]," One hex digit = exactly 4 binary digits (bits)"]}]]}]
26:["$","h4",null,{"children":"Hex to Binary Conversion"}]
27:["$","p",null,{"children":"Each hex digit converts to exactly 4 binary digits:"}]
28:["$","ul",null,{"children":[["$","li",null,{"children":"0₁₆ = 0000₂, 1₁₆ = 0001₂, 2₁₆ = 0010₂, 3₁₆ = 0011₂"}],["$","li",null,{"children":"4₁₆ = 0100₂, 5₁₆ = 0101₂, 6₁₆ = 0110₂, 7₁₆ = 0111₂"}],["$","li",null,{"children":"8₁₆ = 1000₂, 9₁₆ = 1001₂, A₁₆ = 1010₂, B₁₆ = 1011₂"}],["$","li",null,{"children":"C₁₆ = 1100₂, D₁₆ = 1101₂, E₁₆ = 1110₂, F₁₆ = 1111₂"}]]}]
29:["$","h3",null,{"children":"Octal (Base 8) - Less Common but Important"}]
2a:["$","p",null,{"children":"Uses digits 0-7. Each octal digit represents exactly 3 binary digits:"}]
2b:["$","p",null,{"children":[["$","strong",null,{"children":"Example:"}]," 755₈ = (7×8²) + (5×8¹) + (5×8⁰) = 448 + 40 + 5 = 493₁₀"]}]
2c:["$","h2",null,{"children":"2. Bitwise Operations"}]
2d:["$","h3",null,{"children":"AND Operation (&)"}]
2e:["$","p",null,{"children":"Returns 1 only when both bits are 1:"}]
2f:["$","div",null,{"className":"bg-gray-50 dark:bg-gray-700 rounded-lg p-4 my-4 font-mono text-sm","children":[["$","p",null,{"children":"  1101 (13)"}],["$","p",null,{"children":"& 1010 (10)"}],["$","p",null,{"children":"------"}],["$","p",null,{"children":"  1000 (8)"}]]}]
30:["$","h4",null,{"children":"AND Applications"}]
31:["$","ul",null,{"children":[["$","li",null,{"children":[["$","strong",null,{"children":"Masking:"}]," Extract specific bits"]}],["$","li",null,{"children":[["$","strong",null,{"children":"Testing:"}]," Check if bit is set"]}],["$","li",null,{"children":[["$","strong",null,{"children":"Clearing:"}]," Set specific bits to 0"]}]]}]
32:["$","h3",null,{"children":"OR Operation (|)"}]
33:["$","p",null,{"children":"Returns 1 when at least one bit is 1:"}]
34:["$","div",null,{"className":"bg-gray-50 dark:bg-gray-700 rounded-lg p-4 my-4 font-mono text-sm","children":[["$","p",null,{"children":"  1101 (13)"}],["$","p",null,{"children":"| 1010 (10)"}],["$","p",null,{"children":"------"}],["$","p",null,{"children":"  1111 (15)"}]]}]
35:["$","h4",null,{"children":"OR Applications"}]
36:["$","ul",null,{"children":[["$","li",null,{"children":[["$","strong",null,{"children":"Setting bits:"}]," Turn specific bits on"]}],["$","li",null,{"children":[["$","strong",null,{"children":"Combining flags:"}]," Merge multiple boolean values"]}],["$","li",null,{"children":[["$","strong",null,{"children":"Default values:"}]," Provide fallback bits"]}]]}]
37:["$","h3",null,{"children":"XOR Operation (^)"}]
38:["$","p",null,{"children":"Returns 1 when bits are different:"}]
39:["$","div",null,{"className":"bg-gray-50 dark:bg-gray-700 rounded-lg p-4 my-4 font-mono text-sm","children":[["$","p",null,{"children":"  1101 (13)"}],["$","p",null,{"children":"^ 1010 (10)"}],["$","p",null,{"children":"------"}],["$","p",null,{"children":"  0111 (7)"}]]}]
3a:["$","h4",null,{"children":"XOR Applications"}]
3b:["$","ul",null,{"children":[["$","li",null,{"children":[["$","strong",null,{"children":"Toggling:"}]," Flip specific bits"]}],["$","li",null,{"children":[["$","strong",null,{"children":"Encryption:"}]," Simple cipher operations"]}],["$","li",null,{"children":[["$","strong",null,{"children":"Parity checking:"}]," Error detection"]}],["$","li",null,{"children":[["$","strong",null,{"children":"Swapping:"}]," Exchange values without temp variable"]}]]}]
3c:["$","h3",null,{"children":"NOT Operation (~)"}]
3d:["$","p",null,{"children":"Inverts all bits (1 becomes 0, 0 becomes 1):"}]
3e:["$","div",null,{"className":"bg-gray-50 dark:bg-gray-700 rounded-lg p-4 my-4 font-mono text-sm","children":["$","p",null,{"children":"~ 1101 = 0010 (in 4-bit representation)"}]}]
3f:["$","h2",null,{"children":"3. Bit Shifting Operations"}]
40:["$","h3",null,{"children":"Left Shift (<<)"}]
41:["$","p",null,{"children":"Moves bits to the left, filling with zeros:"}]
42:["$","div",null,{"className":"bg-yellow-50 dark:bg-yellow-900 rounded-lg p-6 my-6","children":[["$","h4",null,{"className":"font-semibold text-yellow-800 dark:text-yellow-200 mb-4","children":"⬅️ Left Shift Example:"}],["$","p",null,{"className":"text-yellow-700 dark:text-yellow-300","children":[["$","strong",null,{"children":"5 << 2"}],["$","br",null,{}],"101₂ << 2 = 10100₂ = 20₁₀",["$","br",null,{}],["$","strong",null,{"children":"Rule:"}]," Left shift by n positions = multiply by 2ⁿ"]}]]}]
43:["$","h3",null,{"children":"Right Shift (>>)"}]
44:["$","p",null,{"children":"Moves bits to the right:"}]
45:["$","div",null,{"className":"bg-purple-50 dark:bg-purple-900 rounded-lg p-6 my-6","children":[["$","h4",null,{"className":"font-semibold text-purple-800 dark:text-purple-200 mb-4","children":"➡️ Right Shift Example:"}],["$","p",null,{"className":"text-purple-700 dark:text-purple-300","children":[["$","strong",null,{"children":"20 >> 2"}],["$","br",null,{}],"10100₂ >> 2 = 101₂ = 5₁₀",["$","br",null,{}],["$","strong",null,{"children":"Rule:"}]," Right shift by n positions = divide by 2ⁿ (integer division)"]}]]}]
46:["$","h4",null,{"children":"Arithmetic vs Logical Right Shift"}]
47:["$","ul",null,{"children":[["$","li",null,{"children":[["$","strong",null,{"children":"Logical (>>>):"}]," Always fills with 0s"]}],["$","li",null,{"children":[["$","strong",null,{"children":"Arithmetic (>>):"}]," Preserves sign bit for negative numbers"]}]]}]
48:["$","h2",null,{"children":"4. Two's Complement (Negative Numbers)"}]
49:["$","h3",null,{"children":"Understanding Signed Binary"}]
4a:["$","p",null,{"children":"Computers represent negative numbers using two's complement:"}]
4b:["$","ol",null,{"children":[["$","li",null,{"children":[["$","strong",null,{"children":"Step 1:"}]," Write the positive number in binary"]}],["$","li",null,{"children":[["$","strong",null,{"children":"Step 2:"}]," Invert all bits (NOT operation)"]}],["$","li",null,{"children":[["$","strong",null,{"children":"Step 3:"}]," Add 1 to the result"]}]]}]
4c:["$","div",null,{"className":"bg-red-50 dark:bg-red-900 rounded-lg p-6 my-6","children":[["$","h4",null,{"className":"font-semibold text-red-800 dark:text-red-200 mb-4","children":"➖ Two's Complement Example:"}],["$","p",null,{"className":"text-red-700 dark:text-red-300","children":[["$","strong",null,{"children":"Find -5 in 8-bit two's complement:"}],["$","br",null,{}],"1. +5 = 00000101₂",["$","br",null,{}],"2. Invert: 11111010₂",["$","br",null,{}],"3. Add 1: 11111011₂ = -5₁₀"]}]]}]
4d:["$","h3",null,{"children":"Sign Bit"}]
4e:["$","p",null,{"children":"In signed binary, the leftmost bit indicates sign:"}]
4f:["$","ul",null,{"children":[["$","li",null,{"children":[["$","strong",null,{"children":"0:"}]," Positive number"]}],["$","li",null,{"children":[["$","strong",null,{"children":"1:"}]," Negative number"]}]]}]
50:["$","h2",null,{"children":"5. Practical Programming Applications"}]
51:["$","h3",null,{"children":"Bit Flags and Permissions"}]
52:["$","p",null,{"children":"Use individual bits to represent boolean flags:"}]
53:["$","div",null,{"className":"bg-indigo-50 dark:bg-indigo-900 rounded-lg p-6 my-6","children":[["$","h4",null,{"className":"font-semibold text-indigo-800 dark:text-indigo-200 mb-4","children":"🏴 File Permissions Example:"}],["$","div",null,{"className":"text-indigo-700 dark:text-indigo-300","children":[["$","p",null,{"children":["$","strong",null,{"children":"rwx permissions = 111₂ = 7₈"}]}],["$","ul",null,{"children":[["$","li",null,{"children":"r (read) = 100₂ = 4₈"}],["$","li",null,{"children":"w (write) = 010₂ = 2₈"}],["$","li",null,{"children":"x (execute) = 001₂ = 1₈"}]]}],["$","p",null,{"children":[["$","strong",null,{"children":"chmod 755:"}]," rwxr-xr-x = 111101101₂"]}]]}]]}]
54:["$","h3",null,{"children":"Color Values in Graphics"}]
55:["$","p",null,{"children":"RGB colors are often represented in hexadecimal:"}]
56:["$","ul",null,{"children":[["$","li",null,{"children":[["$","strong",null,{"children":"#FF0000:"}]," Pure red (255, 0, 0)"]}],["$","li",null,{"children":[["$","strong",null,{"children":"#00FF00:"}]," Pure green (0, 255, 0)"]}],["$","li",null,{"children":[["$","strong",null,{"children":"#0000FF:"}]," Pure blue (0, 0, 255)"]}],["$","li",null,{"children":[["$","strong",null,{"children":"#FFFFFF:"}]," White (255, 255, 255)"]}],["$","li",null,{"children":[["$","strong",null,{"children":"#000000:"}]," Black (0, 0, 0)"]}]]}]
57:["$","h3",null,{"children":"Memory Addresses"}]
58:["$","p",null,{"children":"Memory addresses are typically displayed in hexadecimal:"}]
59:["$","ul",null,{"children":[["$","li",null,{"children":[["$","strong",null,{"children":"32-bit systems:"}]," 0x00000000 to 0xFFFFFFFF"]}],["$","li",null,{"children":[["$","strong",null,{"children":"64-bit systems:"}]," 0x0000000000000000 to 0xFFFFFFFFFFFFFFFF"]}]]}]
5a:["$","h3",null,{"children":"Bit Manipulation Algorithms"}]
5b:["$","h4",null,{"children":"Check if Number is Power of 2"}]
5c:["$","p",null,{"children":"A number is a power of 2 if it has exactly one bit set:"}]
5d:["$","p",null,{"children":[["$","strong",null,{"children":"Method:"}]," n & (n-1) == 0 (for n > 0)"]}]
5e:["$","h4",null,{"children":"Count Set Bits"}]
5f:["$","p",null,{"children":"Count the number of 1s in binary representation:"}]
60:["$","div",null,{"className":"bg-gray-50 dark:bg-gray-700 rounded-lg p-4 my-4 font-mono text-sm","children":[["$","p",null,{"children":"// Brian Kernighan's algorithm"}],["$","p",null,{"children":"while (n) {"}],["$","p",null,{"children":"    count++;"}],["$","p",null,{"children":"    n = n & (n-1); // Remove rightmost set bit"}],["$","p",null,{"children":"}"}]]}]
61:["$","h4",null,{"children":"Isolate Rightmost Set Bit"}]
62:["$","p",null,{"children":"Extract the rightmost 1 bit:"}]
63:["$","p",null,{"children":[["$","strong",null,{"children":"Method:"}]," n & (-n)"]}]
64:["$","h2",null,{"children":"6. Debugging with Programmer Calculators"}]
65:["$","h3",null,{"children":"Common Programming Scenarios"}]
66:["$","ul",null,{"children":[["$","li",null,{"children":[["$","strong",null,{"children":"Buffer overflow:"}]," Check if values exceed bit width limits"]}],["$","li",null,{"children":[["$","strong",null,{"children":"Bit masking errors:"}]," Verify mask values in different bases"]}],["$","li",null,{"children":[["$","strong",null,{"children":"Signed/unsigned conversion:"}]," Understand value changes"]}],["$","li",null,{"children":[["$","strong",null,{"children":"Endianness issues:"}]," Check byte order in hex"]}]]}]
67:["$","h3",null,{"children":"Network Programming"}]
68:["$","ul",null,{"children":[["$","li",null,{"children":[["$","strong",null,{"children":"IP addresses:"}]," Convert between dotted decimal and hex"]}],["$","li",null,{"children":[["$","strong",null,{"children":"Subnet masks:"}]," Calculate network ranges"]}],["$","li",null,{"children":[["$","strong",null,{"children":"Port numbers:"}]," Understand 16-bit limits"]}],["$","li",null,{"children":[["$","strong",null,{"children":"Protocol headers:"}]," Parse binary packet data"]}]]}]
69:["$","h2",null,{"children":"7. Advanced Topics"}]
6a:["$","h3",null,{"children":"Floating Point Representation"}]
6b:["$","p",null,{"children":"IEEE 754 standard for floating point numbers:"}]
6c:["$","ul",null,{"children":[["$","li",null,{"children":[["$","strong",null,{"children":"32-bit float:"}]," 1 sign + 8 exponent + 23 mantissa bits"]}],["$","li",null,{"children":[["$","strong",null,{"children":"64-bit double:"}]," 1 sign + 11 exponent + 52 mantissa bits"]}]]}]
6d:["$","h3",null,{"children":"Bitwise Tricks for Optimization"}]
6e:["$","ul",null,{"children":[["$","li",null,{"children":[["$","strong",null,{"children":"Fast multiplication by powers of 2:"}]," Use left shift"]}],["$","li",null,{"children":[["$","strong",null,{"children":"Fast division by powers of 2:"}]," Use right shift"]}],["$","li",null,{"children":[["$","strong",null,{"children":"Fast modulo by powers of 2:"}]," Use AND with (2ⁿ - 1)"]}],["$","li",null,{"children":[["$","strong",null,{"children":"Toggle case of letters:"}]," XOR with 0x20"]}]]}]
6f:["$","h3",null,{"children":"Cryptographic Applications"}]
70:["$","ul",null,{"children":[["$","li",null,{"children":[["$","strong",null,{"children":"XOR ciphers:"}]," Simple encryption/decryption"]}],["$","li",null,{"children":[["$","strong",null,{"children":"Hash functions:"}]," Bit manipulation for mixing"]}],["$","li",null,{"children":[["$","strong",null,{"children":"Random number generation:"}]," Linear feedback shift registers"]}]]}]
71:["$","h2",null,{"children":"Common Mistakes and Pitfalls"}]
72:["$","div",null,{"className":"bg-yellow-50 dark:bg-yellow-900 border border-yellow-200 dark:border-yellow-700 rounded-lg p-6 my-6","children":[["$","h3",null,{"className":"text-yellow-800 dark:text-yellow-200 font-semibold mb-3","children":"⚠️ Watch Out For:"}],["$","ul",null,{"className":"text-yellow-700 dark:text-yellow-300 space-y-2","children":[["$","li",null,{"children":"Confusing logical vs arithmetic right shift"}],["$","li",null,{"children":"Integer overflow in calculations"}],["$","li",null,{"children":"Sign extension in negative numbers"}],["$","li",null,{"children":"Operator precedence in complex expressions"}],["$","li",null,{"children":"Language-specific behavior differences"}],["$","li",null,{"children":"Endianness assumptions across platforms"}]]}]]}]
73:["$","h2",null,{"children":"Practice Problems"}]
74:["$","h3",null,{"children":"Conversion Practice"}]
75:["$","ol",null,{"children":[["$","li",null,{"children":"Convert 255₁₀ to binary and hexadecimal"}],["$","li",null,{"children":"Convert 1010110₂ to decimal and hexadecimal"}],["$","li",null,{"children":"Convert 0x2F₁₆ to binary and decimal"}],["$","li",null,{"children":"Find -15 in 8-bit two's complement"}]]}]
76:["$","h3",null,{"children":"Bitwise Operation Practice"}]
77:["$","ol",null,{"children":[["$","li",null,{"children":"Calculate 0xA5 & 0x3C"}],["$","li",null,{"children":"Calculate 0xF0 | 0x0F"}],["$","li",null,{"children":"Calculate 0xFF ^ 0xAA"}],["$","li",null,{"children":"Calculate 12 << 3 and 96 >> 2"}]]}]
78:["$","h2",null,{"children":"Real-World Code Examples"}]
79:["$","h3",null,{"children":"Setting and Clearing Bits"}]
7a:["$","div",null,{"className":"bg-gray-50 dark:bg-gray-700 rounded-lg p-4 my-4 font-mono text-sm","children":[["$","p",null,{"children":"// Set bit n: number |= (1 << n)"}],["$","p",null,{"children":"// Clear bit n: number &= ~(1 << n)"}],["$","p",null,{"children":"// Toggle bit n: number ^= (1 << n)"}],["$","p",null,{"children":"// Check bit n: (number & (1 << n)) != 0"}]]}]
7b:["$","h3",null,{"children":"Fast Absolute Value"}]
7c:["$","div",null,{"className":"bg-gray-50 dark:bg-gray-700 rounded-lg p-4 my-4 font-mono text-sm","children":[["$","p",null,{"children":"// Without branching"}],["$","p",null,{"children":"int mask = x >> 31; // -1 if negative, 0 if positive"}],["$","p",null,{"children":"return (x + mask) ^ mask;"}]]}]
7d:["$","h2",null,{"children":"Conclusion"}]
7e:["$","p",null,{"children":"Programmer calculators and bit manipulation are essential skills for any serious programmer. Understanding number bases helps you read assembly code, debug low-level issues, and optimize performance-critical code. Bitwise operations provide elegant solutions to many algorithmic problems and are fundamental to systems programming, graphics, and cryptography."}]
7f:["$","p",null,{"children":"Practice these concepts regularly, especially when working on embedded systems, game development, or any performance-critical applications. The ability to think in binary and manipulate bits efficiently will make you a more versatile and effective programmer."}]
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