Binary to Decimal Converter

How the Binary to Decimal Converter Works

A binary number is written in base 2, using only the digits 0 and 1. The decimal system we use every day is base 10. This converter takes a binary string, validates that it contains only 0s and 1s, and returns the equivalent base-10 value plus the hexadecimal form.

The Method

Each digit in a binary number has a place value that is a power of 2. Reading from right to left, the positions are worth 2⁰ (1), 2¹ (2), 2² (4), 2³ (8), and so on. To convert:

Decimal = (digit × 2ᵖᵒˢⁱᵗⁱᵒⁿ) summed across every digit

You multiply each binary digit by the power of 2 for its position, then add all the products together.

Worked Example

Convert binary 101101 to decimal:

Digit	Position	Power of 2	Value
1	5	2⁵ = 32	32
0	4	2⁴ = 16	0
1	3	2³ = 8	8
1	2	2² = 4	4
0	1	2¹ = 2	0
1	0	2⁰ = 1	1

Adding the values: 32 + 0 + 8 + 4 + 0 + 1 = 45. So binary 101101 equals decimal 45.

Why Binary Matters

Binary is the native language of computers. Every number, character, image, and instruction is ultimately stored as a pattern of 1s and 0s. Understanding how binary maps to decimal helps with:

• Programming and debugging — reading memory addresses, flags, and bit masks.
• Networking — interpreting IP addresses and subnet masks, which are based on 8-bit binary groups.
• Digital electronics — working with logic gates, registers, and microcontrollers.

The hexadecimal output is a handy shortcut because each hex digit represents exactly four binary digits, making long binary values much easier to read.