Binary to Decimal Calculator

Q: What is binary 10 in decimal?

Binary 10 equals 2 in decimal, not ten. The left digit sits in the 2 1 place (worth 2) and the right digit sits in the 2 0 place (worth 1). So 2 + 0 = 2.

Q: Why do I get an error message when I type?

The box only accepts 0 , 1 , one dot, one minus sign at the front, plus spaces and commas for grouping. Common causes of an error are: A digit from 2 to 9, or a letter Two or more dots A minus sign in the middle More than 63 binary digits

Q: Why did my signed answer change when I picked a different bit width?

The sign depends on the leftmost bit of the padded pattern. For example, 1011 in 4-bit is -5 , because the first bit is 1. In 8-bit it becomes 00001011 , so the first bit is 0 and the answer is 11 . Same bits, different width, different meaning.

Q: Do leading zeros change the answer?

No. 0001010 and 1010 both equal 10 . Leading zeros only matter for the signed result, because they push the sign bit to 0 and keep the value positive.

Q: Is the decimal answer for a binary fraction exact or rounded?

It is exact . Every binary fraction ends after a set number of decimal places, because 2 divides evenly into 10. For example, 0.0001 in binary is exactly 0.0625 . Nothing is rounded off.

Q: Is there a quick way to convert binary in my head?

Yes, use the doubling method . Start at 0. Read the bits left to right. For each bit, double your total and add the bit. For 1101 : 1, then 3, then 6, then 13 . No powers of 2 needed.

Q: What does 11111111 equal?

Eight 1s equal 255 unsigned, which is the largest value one byte can hold. Read as an 8-bit signed number, the same bits mean -1 . Set the bit width to 8-bit to see both answers side by side.

Binary to Decimal Calculator

Introduction

Computers count with just two digits: 0 and 1. That is called binary, or base 2. People count with ten digits, 0 through 9. That is decimal, or base 10. This binary to decimal calculator changes a binary number into the decimal number it stands for.

Type in bits like 101011 and press Convert. You get the answer right away, plus the math that leads to it. You can use spaces or commas to group bits, add a radix point for binary fractions like 1010.11, or put a minus sign in front.

The tool also shows:

The place value of every bit, so you can see which powers of 2 you are adding.
A step-by-step solution you can copy for homework or notes.
The signed (two's complement) value for 4, 8, 16, 32, or 64 bits.
The same number written in hexadecimal (base 16).
A chart of how much each bit adds to the total.

It is a fast way to check your work and to learn how number systems really work.

How to use our Binary to Decimal Calculator

Type a binary number (base 2), pick a bit width, and the calculator shows the decimal value, the signed value, the hex value, a place value diagram, and step-by-step math.

Binary number (base 2): Enter your bits in the box. You can use 0 and 1, one dot (.) for a fraction, and one minus sign (-) at the front. Spaces or commas help you group bits, like 1010 1100. You can enter up to 63 binary digits.

Signed interpretation width: Choose how many bits to use for the two's complement (signed) answer. Pick "Auto" to use the bits you typed, or pick 4, 8, 16, 32, or 64-bit. If the first bit is 1, the answer is negative.

Display options: Turn on the switch to add commas to the decimal answer, like 1,024. This only changes how it looks. The Copy button still gives you the plain number.

Convert: Click this to run the conversion. You can also press Enter in the input box.

Clear: Click this to erase the input and all results so you can start over.

Random: Click this to fill the box with a random binary number and convert it right away. This is a fast way to practice.

Binary to Decimal Conversion Explained

Binary is a number system that uses only two digits: 0 and 1. It is called base 2. Decimal is the number system we use every day. It uses ten digits, 0 through 9, so it is called base 10. Converting binary to decimal means finding the everyday number that a string of 0s and 1s stands for.

How Place Value Works in Binary

In decimal, each place is worth ten times more than the place to its right (1, 10, 100, 1000). In binary, each place is worth two times more (1, 2, 4, 8, 16, 32, and so on). The place values are powers of 2. The rightmost digit of a whole number sits in the 2⁰ place, which equals 1.

The Conversion Steps

Write the binary number and give each digit its power of 2, starting at 2⁰ on the right.
Multiply each digit by its place value.
Ignore every place with a 0, since 0 times anything is 0.
Add up the place values where the digit is 1. That sum is the decimal answer.

Example: 101011 in binary. The place values are 32, 16, 8, 4, 2, and 1. The digits 1 sit in the 32, 8, 2, and 1 places. So 32 + 8 + 2 + 1 = 43 in decimal.

Binary Numbers With a Point

A binary number can have a radix point, like 101.11. Digits to the right of the point use negative powers of 2: 2^-1 = 0.5, 2^-2 = 0.25, 2^-3 = 0.125. So 101.11 equals 4 + 1 + 0.5 + 0.25 = 5.75.

Unsigned vs. Two's Complement (Signed)

An unsigned binary number is always zero or positive. Every bit just adds value. Computers also need negative numbers, so they use two's complement. In that method, the leftmost bit is the sign bit. If it is 0, the number is positive and reads the same as unsigned. If it is 1, the number is negative, and you find its value by subtracting 2ⁿ, where n is the total number of bits. For example, 8-bit 11111111 is 255 unsigned, but 255 − 256 = −1 as a signed value. This is why the bit width (4, 8, 16, 32, or 64 bits) matters. Wrap-around behavior like this is closely related to remainder math, which you can explore with the Modulo Calculator.

Binary and Hexadecimal

Hexadecimal is base 16 and uses the digits 0–9 plus the letters A–F. Each hex digit stands for exactly 4 binary digits, so long bit strings become short and easy to read. For example, 1010 1100 in binary is AC in hex. Programmers often use hex as a shortcut for binary, and the Hex Calculator handles arithmetic directly in base 16.

Why Binary Matters

Computers store everything with switches that are either off (0) or on (1). Text, pictures, music, and video are all saved as binary. Knowing how to convert binary to decimal helps with computer science classes, digital electronics, network work like IP addresses and subnet masks, and reading memory or color codes.

Handy Powers of 2

2⁰ = 1, 2¹ = 2, 2² = 4, 2³ = 8, 2⁴ = 16, 2⁵ = 32, 2⁶ = 64, 2⁷ = 128, 2⁸ = 256, 2⁹ = 512, 2¹⁰ = 1024. Learning these makes short binary numbers quick to read in your head.

Formulas used

Binary to decimal positional expansion

(b_{n-1}\dots b_1 b_0 . b_{-1} \dots b_{-m})_2 = \sum_{i=-m}^{n-1} b_i \times 2^{i}

Integer part value (Horner accumulation)

V_{int} = \sum_{i=0}^{n-1} b_i \times 2^{i}

Fractional part value

V_{frac} = \frac{\sum_{j=1}^{m} f_j \times 2^{m-j}}{2^{m}} = \sum_{j=1}^{m} f_j \times 2^{-j}

Signed value (two's complement, width w)

V_{signed} = \begin{cases} V_{unsigned}, & b_{w-1}=0 \\ V_{unsigned} - 2^{w}, & b_{w-1}=1 \end{cases}

Binary to hexadecimal grouping (4 bits per hex digit)

H_k = \sum_{t=0}^{3} b_{4k+t} \times 2^{3-t}

Negative input (sign-magnitude)

(-B)_2 = -\left(\sum_{i=-m}^{n-1} b_i \times 2^{i}\right)_{10}

Frequently asked questions

What is binary 10 in decimal?

Binary 10 equals 2 in decimal, not ten. The left digit sits in the 2¹ place (worth 2) and the right digit sits in the 2⁰ place (worth 1). So 2 + 0 = 2.

Why do I get an error message when I type?

The box only accepts 0, 1, one dot, one minus sign at the front, plus spaces and commas for grouping. Common causes of an error are:

A digit from 2 to 9, or a letter
Two or more dots
A minus sign in the middle
More than 63 binary digits

Why does the signed result say Not applicable?

Two's complement only works for whole bit patterns with no written sign. If your input has a radix point or a minus sign, the calculator skips two's complement and just shows the plain signed decimal value instead.

Why did my signed answer change when I picked a different bit width?

The sign depends on the leftmost bit of the padded pattern. For example, 1011 in 4-bit is -5, because the first bit is 1. In 8-bit it becomes 00001011, so the first bit is 0 and the answer is 11. Same bits, different width, different meaning.

Do leading zeros change the answer?

No. 0001010 and 1010 both equal 10. Leading zeros only matter for the signed result, because they push the sign bit to 0 and keep the value positive.

Does the Copy button include the commas?

No. The display switch adds commas only for easier reading. Copy always gives the raw number, like 1024, so you can paste it into a spreadsheet, a code editor, or another calculator.

Is the decimal answer for a binary fraction exact or rounded?

It is exact. Every binary fraction ends after a set number of decimal places, because 2 divides evenly into 10. For example, 0.0001 in binary is exactly 0.0625. Nothing is rounded off.

Is there a quick way to convert binary in my head?

Yes, use the doubling method. Start at 0. Read the bits left to right. For each bit, double your total and add the bit. For 1101: 1, then 3, then 6, then 13. No powers of 2 needed.

What does 11111111 equal?

Eight 1s equal 255 unsigned, which is the largest value one byte can hold. Read as an 8-bit signed number, the same bits mean -1. Set the bit width to 8-bit to see both answers side by side.

Why does my hex answer have a dot in it?

Because your binary number had a radix point. Fraction bits are grouped in fours from left to right, padded with zeros if needed. So 1010.11 becomes A.C in hex.

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