CONVERTER · MATH · NUMBER SYSTEMS

Binary to Decimal Converter: Hex, Octal and Bit Table

101101 becomes 45 — plus its hexadecimal (2D) and octal (55) forms, the bit-by-bit breakdown, and the byte size, all in one result. Flip the switch to go the other way: 42 becomes 101010. The powers-of-two table below covers 2⁰ through 2¹⁶ for mental conversions.

483

Binary converter

Pick the direction, enter the number, and get the decimal, binary, hexadecimal, and octal forms together — with a positional breakdown showing exactly which powers of two add up to your answer. Presets: the default 101101, 11111111 (all bits set), 100000000, and decimal 42 going the other way. Leading zeros are accepted; anything that isn’t a 0 or 1 gets a clear error naming the bad character.

Conversion direction

Which way the conversion runs. Binary → decimal reads 0s and 1s and gives you 45; decimal → binary takes 42 and gives you 101010. Hex and octal come along for free in both directions.

Binary number

101101 at the defaults. Digits 0 and 1 only — spaces and underscores are ignored, so 1010_1010 is accepted. Up to 53 bits; anything longer exceeds exact-integer range and is rejected.

Decimal number

Used in decimal → binary mode: whole, non-negative numbers only (this converter is unsigned — negatives need a sign convention, see the FAQ). Try 42 → 101010, or 255 → 11111111.

Result—Enter values above

Enter values to see the result.

The 60-second rule

Binary is positional notation with only two digits, and the positions are powers of two. The default 101101 reads right to left: 1×32 + 0×16 + 1×8 + 1×4 + 0×2 + 1×1 = 45. The same 45 is 2D in hexadecimal (two groups of four bits: 0010 = 2, 1101 = D) and 55 in octal (groups of three: 101 = 5, 101 = 5). Six bits fit in 1 byte, and the largest 6-bit value is 63 (2⁶ − 1).

The worked example, line by line

Setup: binary 101101, binary → decimal mode.

Line 1 — write the positions. Six digits, so the positions are 2⁵ down to 2⁰: 32, 16, 8, 4, 2, 1.

Line 2 — keep the 1s, drop the 0s. 1×32 + 0×16 + 1×8 + 1×4 + 0×2 + 1×1 = 32 + 8 + 4 + 1 = 45.

Line 3 — the bonus columns. Group into fours from the right: 0010 = 2, 1101 = D → hex 2D. Group into threes: 101 = 5, 101 = 5 → octal 55.

Line 4 — the reverse. Decimal 42: the largest power of two that fits is 32, remainder 10; then 8, remainder 2; then 2 → 101010. Check: 32 + 8 + 2 = 42.

Branch — 11111111 (all bits set): every position kept: 128 + 64 + 32 + 16 + 8 + 4 + 2 + 1 = 255 — the famous 8-bit maximum (2⁸ − 1). Hex FF, octal 377.

Branch — 100000000: a 1 followed by eight 0s = 2⁸ = 256. The pattern holds forever: a 1 with n zeros after it is exactly 2ⁿ. Hex 100, octal 400.

Branch — leading zeros: 00101101 is still 45 — leading zeros never change the value, they just fill out the field width (this is why you see 8-bit values written as 00101101).

Branch — the error path: entering 1021 gets “‘2’ is not a binary digit. Only 0 and 1 are valid.” — the calculator names the exact offending character instead of a generic complaint.

Powers-of-two reference table (2⁰ – 2¹⁶)

The skeleton key for mental binary math: every binary digit is one of these. 2¹⁰ = 1,024 (about a thousand) and 2²⁰ = 1,048,576 (about a million) are the two to memorize first.

Power	Value	Power	Value
2⁰	1	2⁹	512
2¹	2	2¹⁰	1,024
2²	4	2¹¹	2,048
2³	8	2¹²	4,096
2⁴	16	2¹³	8,192
2⁵	32	2¹⁴	16,384
2⁶	64	2¹⁵	32,768
2⁷	128	2¹⁶	65,536
2⁸	256

Bits, bytes, and why it works this way

A bit is one binary digit — the smallest unit of digital information, a single on/off. Eight bits make a byte, the standard unit of storage: one byte holds any value from 0 to 255. With N bits you can represent 2^N different values, numbered 0 through 2^N − 1 — that’s why 8-bit color gives 256 shades per channel and 16-bit audio gives 65,536 levels. Hexadecimal is just shorthand: 4 bits = 1 hex digit, so FF is easier to read than 11111111. Octal groups 3 bits per digit.

Output · formula · meaning

Output	Formula	What it means
Decimal	Σ bit_i × 2ⁱ	45 for 101101 — add up the powers of two wherever the digit is 1. The everyday number system.
Binary	repeated division by 2, remainders reversed	101010 for decimal 42 — the same value in base 2. Leading zeros are cosmetic.
Hexadecimal	group bits in fours, 0–9 then A–F	2D for 101101 — compact shorthand: every 4 bits become one hex digit. The programmer’s favorite base.
Octal	group bits in threes	55 for 101101 — base 8, the classic Unix-permission notation (chmod 755 is octal).
Bit length	⌈log₂(n+1)⌉	6 bits for 101101 — how many binary digits the value needs; a 1-byte field holds it, topping out at 63.

Who uses this

The CS student: converts 101101 by hand for the homework, then checks the positional breakdown — 32 + 8 + 4 + 1 = 45 — against her own work.
The embedded hobbyist: reads a register dump full of hex like 0x2D and turns it back into binary to see which individual bits (flags) are set.
The networking learner: converts subnet masks between dotted decimal and binary — 255.255.255.0 is 24 one-bits followed by 8 zero-bits — and the octet table makes it click.
The curious Googler: wondered what 42 looks like in binary (101010, nicely symmetric) and stayed for the powers-of-two table.

Accuracy & limitations

Conversions are exact for binary inputs up to 53 bits and decimal inputs up to 9,007,199,254,740,991 (2⁵³ − 1) — beyond that, integers can’t be represented exactly in this calculator and are rejected rather than rounded. This converter is unsigned only: negative decimals are refused, because a negative binary value needs a stated convention (sign-magnitude, one’s complement, or two’s complement). Leading zeros are accepted and don’t change the value; spaces and underscores between digits are ignored. The positional breakdown is shown for inputs up to 16 bits.

Cite this calculatorBright Side Kit. “Binary to Decimal Converter: Hex, Octal and Bit Table | Bright Side Kit.” brightoolbox.com, 2026.

Bright Side Kit. "Binary to Decimal Converter: Hex, Octal and Bit Table | Bright Side Kit." https://brightoolbox.com/binary-to-decimal/ Accessed October 24, 2026.

Data sources

Positional notation — value = Σ bit_i × 2ⁱ; hex and octal are regroupings of the same bits (4 and 3 per digit).
IEEE 754 double-precision range — integers are exact up to 2⁵³ − 1 = 9,007,199,254,740,991, the bound enforced by this calculator.

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Frequently asked questions

Why do computers use binary instead of decimal?

Hardware only needs to tell two states apart. A transistor is on or off, a voltage is high or low, a pit on a disc reflects or it doesn’t — two states are far more reliable to detect than ten. Binary trades compactness for reliability: 45 becomes the longer 101101, and everything above it (hex, bytes, protocols) is built to keep the long strings manageable.

What exactly is a bit?

One binary digit — the smallest unit of digital information. A bit is a single 0 or 1. Eight bits make a byte; a byte holds one character of text in classic ASCII, or any value from 0 to 255. Everything you see on a screen is, at the bottom, a very long string of bits like the ones this converter reads.

What is the maximum value for N bits?

2^N − 1. With N bits you can write 2^N different values, numbered from 0: 8 bits → 255, 16 bits → 65,535, 32 bits → 4,294,967,295. This converter reports the max for your exact bit count with every result.

Can this converter handle negative numbers?

No — it’s unsigned only, by design. A negative binary value needs a stated sign convention: two’s complement (the modern standard, where the top bit means negative), one’s complement, or sign-magnitude. Without agreeing on the convention first, “101101 as a negative” has three different answers — so the calculator refuses negatives rather than guessing.

Why is a kilobyte 1,024 bytes instead of 1,000?

Because 2¹⁰ = 1,024. Memory is addressed in binary, so sizes naturally land on powers of two: 1,024 bytes ≈ a thousand, close enough that “kilo” stuck. Strictly, 1,024 bytes = 1 kibibyte (KiB) and 1,000 bytes = 1 kilobyte (KB) — the distinction matters in storage marketing, where drives are sold in decimal GB and read smaller in binary GiB.

Do leading zeros change a binary number’s value?

Never. 00101101 is 45, exactly like 101101 — the zeros just fill out the field. You’ll see them everywhere in computing: 8-bit values are written as full octets (00101101), and this converter accepts them and notes that they’re cosmetic.

How do I convert binary to decimal by hand?

Memorize the powers of two, then add the ones under the 1s. Write the powers above the digits (32, 16, 8, 4, 2, 1 for 101101), add the ones sitting under a 1: 32 + 8 + 4 + 1 = 45. The powers-of-two table on this page covers 2⁰ through 2¹⁶; with it and this trick you can convert any 16-bit number in your head.