Binary to Decimal Calculator

Binary to Decimal Calculator: solve binary to decimal problems step-by-step. Clear formula, worked example, and FAQs included.

For nearby base-specific checks, compare the hexadecimal calculator while keeping the source and target bases visible.

Binary to decimal conversion translates a base-2 numeral into the familiar base-10 system. Binary uses only 0 and 1 because digital circuits can represent two stable states, but a binary string still follows the same positional rule as every other base: each place is a power of the base.

This page is intentionally focused on the binary-to-decimal question. Its source base starts at 2, its target base starts at 10, and the input accepts only 0 and 1. A reverse decimal-to-binary check remains available by swapping the bases after the result is shown. Binary conversion is useful when reading bit masks, network flags, byte values, digital electronics examples, and programming exercises where individual on/off positions matter.

Strict validation: A digit such as 2 is not a small binary number; it is an invalid binary digit. Rejecting it is safer than allowing a parser to interpret only the valid prefix.

Enter a whole number using the symbols allowed by the selected source base. Spaces are ignored, but prefixes such as 0b and 0x are not part of the input.
Choose the source base from 2 through 36. The source base controls how every character is interpreted.
Choose the target base. The result updates immediately and uses 0–9 followed by uppercase A–Z for values from 10 onward.
Open the place-value expansion when converting from a non-decimal base and check each digit contribution.
Use “Swap bases and convert reverse” to send the result back through the opposite direction and confirm the original value.
If an error appears, correct the complete input rather than relying on a partial parse. A rejected digit means the numeral is not valid in the selected source base.

Binary to Decimal Calculator formula and method

For a numeral with digits d and source base b, each position contributes d×bⁿ, where n starts at zero on the right. Add the contributions to obtain the quantity in decimal. The calculator uses exact integer arithmetic for this part, so long values are not first rounded to a JavaScript floating-point number.

Take 101101. From right to left the powers are 2⁰ through 2⁵. The expansion is 1×2⁵ + 0×2⁴ + 1×2³ + 1×2² + 0×2¹ + 1×2⁰ = 32 + 0 + 8 + 4 + 0 + 1 = 45.

To convert a decimal integer into another base, divide by the target base and keep each remainder. Continue with the quotient until it reaches zero, then read the remainders from the last one back to the first. The quotient-and-remainder method is the reverse of positional expansion.

Binary, octal, and hexadecimal have convenient shortcuts: one hex digit corresponds to four binary bits, and one octal digit corresponds to three. Those shortcuts are checks on the same place-value mathematics, not different definitions of the number.

Reading your binary to decimal calculator result

The base is part of the answer

A result without its base is incomplete. For example, “10” means two in binary, eight in octal, ten in decimal, and sixteen in hexadecimal. Always copy the target base with the displayed digits when moving the value into code, a worksheet, a shell command, or technical documentation.

This page is intentionally focused on the binary-to-decimal question. Its source base starts at 2, its target base starts at 10, and the input accepts only 0 and 1. A reverse decimal-to-binary check remains available by swapping the bases after the result is shown. Take 101101. From right to left the powers are 2⁰ through 2⁵. The expansion is 1×2⁵ + 0×2⁴ + 1×2³ + 1×2² + 0×2¹ + 1×2⁰ = 32 + 0 + 8 + 4 + 0 + 1 = 45. The decimal value is shown as a reference, while the selected target-base result is the answer to the conversion you requested.

For the reverse check, 45 divided repeatedly by 2 leaves remainders 1, 0, 1, 1, 0, 1 when read from bottom to top, returning 101101. The swap control performs this change without losing the selected bases. If the reverse result differs, check the source and target selectors, the direction of the copied value, and whether a leading zero is meaningful for the format you are using.

Math tips and best practices

Read the rightmost bit as 2⁰, which is 1.
Leading zeroes do not change the quantity, but they can matter when showing a fixed-width byte or word.
Group long strings into fours when checking a byte against hexadecimal.
Use the decimal result as a check, not as a replacement for understanding the active bit positions.
Remember that ordinary binary conversion is unsigned; signed two’s-complement interpretation needs a chosen bit width.

Eight binary bits form one byte and can represent 256 unsigned patterns.
A hexadecimal digit maps exactly to four binary bits.
The binary value 11111111 is 255 in decimal.
A zero bit contributes nothing but still identifies a place in a fixed-width representation.

Common mistakes to avoid

Adding the powers of 2 for zero bits as though they were active.
Reading the powers from left to right starting at 2⁰.
Confusing a binary numeral with a decimal number that happens to contain only 0 and 1.
Interpreting a leading 1 as a negative sign without a declared signed format.

Number-base conversion is educational mathematical information, not a security guarantee or a substitute for testing the syntax required by a particular programming language, operating system, protocol, or file format. Integer conversions on this page are exact when the input is valid. Check the receiving system's documentation for signed interpretation, fixed-width padding, prefixes, overflow rules, and permission semantics before using a result in a consequential technical setting.