1The Positional Radix Value Theorem

In a positional numeral system with base radix b, any non-negative integer N is uniquely represented by an ordered sequence of digits (dₙ dₙ₋₁ ... d₁ d₀) where each digit satisfies 0 ≤ dᵢ < b. The numerical value is determined by the polynomial expansion:

When the radix b exceeds 10, alphabetical characters are assigned to represent values from 10 onward: A=10, B=11, C=12, D=13, E=14, F=15, continuing up to Z=35 for Base 36.

2Power-of-Two Bit Grouping Shortcuts (2ⁿ)

Because 8 = 2³ and 16 = 2⁴, converting between Binary (Base 2), Octal (Base 8), and Hexadecimal (Base 16) does not require converting to intermediate Decimal!

• Binary to Octal (3-bit Nibbles): Group binary bits in sets of 3 from right to left. Each 3-bit group maps directly to an octal digit (000₂ = 0, 111₂ = 7). For example, 11010110₂ = (011)(010)(110) = 326₈.
• Binary to Hexadecimal (4-bit Nibbles): Group binary bits in sets of 4 from right to left. Each 4-bit group maps directly to a hex digit (0000₂ = 0x0, 1111₂ = 0xF). For example, 11010110₂ = (1101)(0110) = 0xD6.

This one-to-one mapping is why hexadecimal is universally chosen in computer systems architecture for memory addresses, color codes (#FFFFFF), and machine code opcodes: exactly two hex digits represent a full 8-bit byte.

3Successive Modulo-Division Algorithm

To convert any positive decimal integer into an arbitrary target base b, the successive modulo-division algorithm is applied:

1. Divide the decimal integer N by the target base b.
2. Record the integer quotient Q and the remainder R (where 0 ≤ R < b).
3. The remainder R becomes the next least significant digit (LSB).
4. Repeat the division using quotient Q until Q reaches 0.
5. Read all collected remainders in reverse order (from bottom to top) to obtain the final base-b string.

1The Positional Radix Value Theorem

In a positional numeral system with base radix b, any non-negative integer N is uniquely represented by an ordered sequence of digits (dₙ dₙ₋₁ ... d₁ d₀) where each digit satisfies 0 ≤ dᵢ < b. The numerical value is determined by the polynomial expansion:

When the radix b exceeds 10, alphabetical characters are assigned to represent values from 10 onward: A=10, B=11, C=12, D=13, E=14, F=15, continuing up to Z=35 for Base 36.

2Power-of-Two Bit Grouping Shortcuts (2ⁿ)

Because 8 = 2³ and 16 = 2⁴, converting between Binary (Base 2), Octal (Base 8), and Hexadecimal (Base 16) does not require converting to intermediate Decimal!

• Binary to Octal (3-bit Nibbles): Group binary bits in sets of 3 from right to left. Each 3-bit group maps directly to an octal digit (000₂ = 0, 111₂ = 7). For example, 11010110₂ = (011)(010)(110) = 326₈.
• Binary to Hexadecimal (4-bit Nibbles): Group binary bits in sets of 4 from right to left. Each 4-bit group maps directly to a hex digit (0000₂ = 0x0, 1111₂ = 0xF). For example, 11010110₂ = (1101)(0110) = 0xD6.

This one-to-one mapping is why hexadecimal is universally chosen in computer systems architecture for memory addresses, color codes (#FFFFFF), and machine code opcodes: exactly two hex digits represent a full 8-bit byte.

3Successive Modulo-Division Algorithm

To convert any positive decimal integer into an arbitrary target base b, the successive modulo-division algorithm is applied:

1. Divide the decimal integer N by the target base b.
2. Record the integer quotient Q and the remainder R (where 0 ≤ R < b).
3. The remainder R becomes the next least significant digit (LSB).
4. Repeat the division using quotient Q until Q reaches 0.
5. Read all collected remainders in reverse order (from bottom to top) to obtain the final base-b string.