Number Theory • Discrete Mathematics Flagship

Base Converter

Synchronously convert numbers between Binary (Base 2), Decimal (Base 10), Hexadecimal (Base 16), Octal (Base 8), and arbitrary radix bases from 2 to 36 with step-by-step polynomial decompositions.

Reviewed by Sanjay Samanta & Academic Review Board

Verified IEEE-754 & Positional Radix Standards

Last Updated: September 2026

Quick-Select Computer Science Values Standard Benchmarks

Synchronous Multi-Base Inputs

Binary (Base 2) 0, 1

Decimal (Base 10) 0–9

Hexadecimal (Base 16) 0–9, A–F

Octal (Base 8) 0–7

8-Bit Binary Nibble Grouping

1111 1111

Formatted as 4-bit nibbles for clean hexadecimal byte alignment.

Bit Length

8 Bits

1 Byte

ASCII Character

ÿ (0xFF)

Latin Extended

Signed 8-Bit

-1

Two's Complement

Powers of 2 Exponent

2⁸ − 1

Mersenne number

2ⁿ

Step-by-Step Binary to Decimal Bit Expansion Proof

Direct Answer & Overview

Verified Educational Guide

Base Radix Conversion Definition & Formulas

A positional numeral base (radix b) represents quantities as polynomial sums of powers of b. Converting from base b to decimal uses polynomial expansion, while converting from decimal to base b uses successive Euclidean integer division with remainders.

Primary Mathematical Formula Standard Mathematical Model

Standard Equation

ƒ(x)

Q.E.D.

Base b to Decimal: N₁₀ = ∑ [dᵢ · bⁱ], Decimal to Base b: N = q · b + r (read remainders in reverse order)

Evaluated with exact mathematical formulation • Rigorously verified

Exact Formula

Input Parameters

Required

Input Value String (e.g. 11010110₂, 0xAF4E, 34710)

Source Base (Radix b₁ from 2 to 36)

Target Base (Radix b₂ from 2 to 36)

Expected Outputs

Calculated

Synchronous Binary (Base 2) Representation

Decimal (Base 10) Value

Hexadecimal (Base 16) Value

Octal (Base 8) Value

Step-by-Step Polynomial Expansion & Division Breakdown

Worked Numerical Example

Instant Verification

Convert hexadecimal 2A3₁₆ to decimal (Base 10)

→ 2A3₁₆ = (2 × 16²) + (10 × 16¹) + (3 × 16⁰) = (2 × 256) + (10 × 16) + (3 × 1) = 512 + 160 + 3 = 675.

2A3₁₆ = 675₁₀ | Binary: 0010 1010 0011₂ | Octal: 1243₈

Positional Radix Number Systems & Polynomial Expansion

In any positional number system with radix $b \ge 2$, an $n$-digit integer $(d_{n-1} d_{n-2} \dots d_1 d_0)_b$ represents the exact mathematical sum:

$$N_{10} = \sum_{i=0}^{n-1} d_i \cdot b^i = d_{n-1} b^{n-1} + \dots + d_1 b^1 + d_0 b^0$$

Where each digit $d_i \in \{0, 1, \dots, b-1\}$.

The Core Computer Science Bases (Binary, Octal, Decimal & Hex)

Binary (Base 2)

Uses digits $\{0, 1\}$. Represents the physical on/off states of silicon transistors. The fundamental language of modern digital computing.

Hexadecimal (Base 16)

Uses digits $\{0\text{–}9, \text{A}\text{–}\text{F}\}$. Exactly packs 4 binary bits per hex character, standard for memory addresses, UUIDs, and CSS colors.

Successive Division & Remainder Conversion Algorithm

To convert a decimal integer $N$ into target base $b$:

1. Divide $N$ by $b$: Quotient $q_0 = \lfloor N / b \rfloor$, Remainder $r_0 = N \bmod b$.

2. Set $N = q_0$ and repeat division until quotient $q_k = 0$.

3. The converted number is $(r_k r_{k-1} \dots r_1 r_0)_b$ (remainders written in reverse order).

Direct Bit Grouping (Octal 3-Bit & Hex 4-Bit Nibble Tables)

Decimal	Binary (4-bit Nibble)	Octal (Base 8)	Hex (Base 16)
0	0000	0	0
5	0101	5	5
10	1010	12	A
15	1111	17	F

Step-by-Step Worked Multi-Base Solutions

Example 1 • Decimal to Binary Conversion Standard Tier

Convert Decimal 43 into Binary (Base 2).

1. $43 \div 2 = 21$ remainder $1$

2. $21 \div 2 = 10$ remainder $1$

3. $10 \div 2 = 5$ remainder $0$

4. $5 \div 2 = 2$ remainder $1$

5. $2 \div 2 = 1$ remainder $0$

6. $1 \div 2 = 0$ remainder $1$

Read remainders from bottom to top: 43₁₀ = 101011₂.

Common Pitfalls & Overflow in Fixed-Bit Registers

Pitfall 1: Reading Remainders Forward Instead of in Reverse

When converting using successive division, the first remainder calculated is the least significant bit (LSB) on the right, not the leftmost digit. Always assemble remainders in reverse order.

Pitfall 2: Using Digits Equal to or Greater than the Radix Base

In base $b$, the maximum permissible digit is $b-1$. Writing '2' in binary ($102_2$) or '8' in octal ($78_8$) is invalid syntax and triggers parser errors.

Fact-Checked & Verified • Computational Accuracy Standards

Updated July 2026 • Editorial Policy

Authored By

Sanjay Samanta

Lead Developer & Founder of Basic Math Tools. Specializes in browser-native computational algorithms and applied mathematics.

Reviewed & Verified By

Academic Review Board

Mathematics & curriculum specialists. Audited against standard algebraic and arithmetic principles.

Found an error or have an improvement suggestion? Report a calculation issue

Frequently Asked Questions

What is a positional numeral system base (radix)?

A numeral system base (or radix b) is the number of unique digits (including zero) used to represent numbers. In any base b, the place value of digit dᵢ at index i is dᵢ × bⁱ. Decimal is Base 10, binary is Base 2, octal is Base 8, and hexadecimal is Base 16.

How do you convert any base b number into standard decimal (Base 10)?

Expand the number as a polynomial sum of its digits multiplied by successive powers of the base: Decimal = ∑_{i=0}^{n-1} (dᵢ × bⁱ), where index i starts at 0 from the rightmost digit.

How do you convert a decimal number into an arbitrary base b?

Perform repeated integer division by base b. Record the remainder at each step. Continue dividing until the quotient reaches 0. The digits of the new base are the remainders read in reverse order (from bottom to top / last remainder to first).

Why are binary, octal, and hexadecimal fundamental to computer science?

Computer hardware operates on binary digital logic gates (high/low voltages representing 0 and 1). Octal (Base 8 = 2³) and hexadecimal (Base 16 = 2⁴) group binary digits into exact 3-bit and 4-bit chunks (nibbles), allowing concise human inspection of memory addresses and machine instructions.

How do alphanumeric characters represent digits above 9 in bases up to 36?

In bases greater than 10, uppercase letters represent values 10 through 35: A = 10, B = 11, C = 12, D = 13, E = 14, F = 15, ..., Z = 35. For example, 0xFF in hexadecimal equals 15 × 16¹ + 15 × 16⁰ = 255 in decimal.

Base Converter

Synchronous Multi-Base Inputs

Step-by-Step Binary to Decimal Bit Expansion Proof

Base Radix Conversion Definition & Formulas

Positional Radix Number Systems & Polynomial Expansion

The Core Computer Science Bases (Binary, Octal, Decimal & Hex)

Binary (Base 2)

Hexadecimal (Base 16)

Successive Division & Remainder Conversion Algorithm

Direct Bit Grouping (Octal 3-Bit & Hex 4-Bit Nibble Tables)

Step-by-Step Worked Multi-Base Solutions

Convert Decimal 43 into Binary (Base 2).

Common Pitfalls & Overflow in Fixed-Bit Registers

Frequently Asked Questions

Related Tools

Modulo Calculator: Find a mod n

Prime Factorization Calculator

GCF Calculator

RGB to HEX Converter

HEX to RGB Converter

ASCII Converter

Cookie & Privacy Notice