Number Base Converter (Binary, Octal, Decimal and Hexadecimal)

Choose the base of your number and enter the number written in that base. The page converts it to binary, octal, decimal and hexadecimal at once and shows the steps.

Input base

Number to convert

Whole numbers from 0 up to 64 binary digits (18,446,744,073,709,551,615 in decimal) are supported. Hex digits A to F can be entered in upper or lower case.

Result

Choose the base of your number on the left, enter the number and press "Calculate". The number will appear here in all four bases.

[ CLOSE ]

What you can do on this page
What is this calculation used for?
How to Use
Formula
Symbols and terms
- Symbols
- Terms
Good to know before you start
How to calculate it in Excel
How to calculate it in Google Sheets
How to calculate it in Python
How to write it in LaTeX and other math languages (copy and paste)
How to have ChatGPT do the calculation
DataChef Features
Related Features
NumberChef Calculators List

What you can do on this page

Choose the base of your number (binary, octal, decimal or hexadecimal) and enter it, and you see the same number in all four bases at once
Answers questions like "What is hex 2AA in decimal?" or "What is 170 in binary?" in one step
It shows the place value expansion (each digit × its place value, all added up) and the repeated division by the base with the remainders, so you learn how the conversion works
Useful for studying base conversion and checking answers for computer science classes and IT certification exams
A plain-language explanation of the formulas and copy-and-paste formulas for Excel, Google Sheets and Python are all on this page

You can convert whole numbers from 0 up to 64 binary digits (about 18.4 quintillion in decimal). Fractions and negative numbers are not supported.

What is this calculation used for?

A standard topic in computer science classes and IT exams

Base conversion is a standard topic in high school and college computer science courses, such as AP Computer Science Principles, and in IT certification exams such as CompTIA A+ and Network+.
Questions like "What is binary 1010 in decimal?" or "Write decimal 45 in binary" use exactly the two methods on this page: the place value expansion and the repeated division with remainders. Practicing with the steps shown helps you work through them without getting stuck on the exam.

Color codes in web design and programming (hexadecimal)

Colors on web pages are set with hexadecimal color codes such as "#FF5733". The code lists the strength of red, green and blue as two hex digits each (00 to FF = 0 to 255 in decimal), so #FF5733 is red FF (255), green 57 (87) and blue 33 (51).
Once you can move between hexadecimal and decimal, you can picture color changes such as "a little less red" directly from the numbers in the code.

Linux file permissions are octal

The 755 in "chmod 755", a command often used on servers, is an octal number. Each octal digit matches 3 binary digits, and 755 in binary is 111 101 101 - on/off switches for "read, write, execute", 3 bits each for the owner, the group and everyone else.
Once you understand number bases, this number is no longer a secret code but a table of bit switches.

Understanding IP addresses and subnet masks (binary)

An IP address, such as 192.168.1.1, is really a binary number split into four 8-bit parts: 192 is 11000000 and 255 is 11111111.
Networking basics such as "the subnet mask 255.255.255.0 marks the first 24 bits are the network part" only make sense once you can convert between decimal and binary. This knowledge is the foundation for network engineering work and for certifications such as CCNA.

Learning how computers work (bits, bytes and memory)

"One byte = 8 bits, which can hold 0 to 255 (00 to FF in hex)" and "memory addresses and error codes are shown in hex, such as 0x7FFF" - binary and hexadecimal are the common language for learning what happens inside a computer.
Topics that often trip up programming students, such as bitwise operations, character codes and color values, become much clearer once you can convert between bases freely.

Formula

Base \(n\) → decimal (positional notation)

Standard notation (the usual math form)

\(N\) \(=\) \(\displaystyle\sum_{i=0}^{k}\) \(d_i\) \(\times\) \(b^{i}\)

In words (symbols replaced with words)

④ decimal value \(N\) \(=\) ③ add them all \(\displaystyle\sum\) ① each digit \(d_i\) \(\times\) ② place value \(b^i\)

The formula in words

① Multiply each digit \(d_i\) by

② its place value \(b^i\) (the base \(b\) multiplied by itself \(i\) times, where \(i\) is the digit position) ,

③ add them all up \(\displaystyle\sum\) ,

④ and you get the decimal value \(N\)

Quick example

Converting hexadecimal 2AA to decimal (A is 10 in decimal, and the base is \(b=16\))

decimal value \(N\) \(=\) \(2\) \(\times\) \(16^2\) \(+\) \(10\) \(\times\) \(16^1\) \(+\) \(10\) \(\times\) \(16^0\)

\(2 \times 256 + 10 \times 16 + 10 \times 1 = 512 + 160 + 10 = 682\)

Key idea

Positional notation is a way of writing numbers where each digit position has its own place value. It works exactly like the everyday decimal system, \(682 = 6 \times 100 + 8 \times 10 + 2 \times 1\); the only difference is whether the place values are powers of 10 or powers of 2, 8 or 16. The rightmost digit is position \(i=0\) (place value \(b^0=1\)), and each step to the left multiplies the place value by the base \(b\). In hexadecimal, the letters A to F stand for 10 to 15.

Decimal → base \(n\) (divide by the base and read the remainders in reverse)

Standard notation (the usual math form)

\(N\) \(=\) \(b\) \(\times\) \(q\) \(+\) \(r\)

In words (symbols replaced with words)

① decimal value \(N\) \(=\) ② base \(b\) \(\times\) ③ quotient \(q\) \(+\) ④ remainder \(r\)

The formula in words

① Divide the decimal value \(N\) by

② the base \(b\) to get

③ a quotient \(q\) and

④ a remainder \(r\) . Repeat this with each new quotient until the quotient is 0. The remainders, read from the last one to the first (so the last remainder becomes the leftmost digit), are the digits of the base \(n\) number

Quick example

Converting decimal 682 to hexadecimal (the first division; the remainder 10 is the hex digit A)

decimal value (682) \(=\) base (16) \(\times\) quotient (42) \(+\) remainder (10 = A)

\(682 = 16 \times 42 + 10\)

\(42 = 16 \times 2 + 10\)

\(2 = 16 \times 0 + 2\)

Key idea

The remainders come out starting from the rightmost (lowest) digit. For 682, the remainders appear in the order 10 (A) → 10 (A) → 2, so reading them in reverse gives \(2\mathrm{AA}_{16}\). Remember: "the last remainder is the leftmost digit". To convert to binary, do the same with \(b=2\); for octal, use \(b=8\).

Binary ⇔ hexadecimal (groups of 4 bits)

Standard notation (the usual math form)

\(h\) \(=\) \(b_3 \times 2^3 + b_2 \times 2^2 + b_1 \times 2^1 + b_0 \times 2^0\)

In words (symbols replaced with words)

② one hex digit \(h\) \(=\) ① value of 4 bits

The formula in words

① Split the binary number into groups of 4 digits from the right, and find the value of each group of 4 (\(b_3 \times 8 + b_2 \times 4 + b_1 \times 2 + b_0 \times 1\)) . That value is

② one hex digit \(h\) (for octal, use groups of 3 digits in the same way)

Quick example

Converting binary 001010101010 to hexadecimal by splitting it into groups of 4 from the right

0010 / 1010 / 1010 \(\rightarrow\) 2 / A / A

\(0010_{2} = 2,\ \ 1010_{2} = 10 = \mathrm{A}\)

\(001010101010_{2} = 2\mathrm{AA}_{16}\)

Key idea

One hex digit matches exactly 4 binary digits (a nibble). Without going through decimal, you can use a table of 4-digit groups (0000=0, 0001=1, …, 1111=F) to turn even a long binary number into hexadecimal mechanically. This is why programmers use hexadecimal so much: it is a compact way to write binary. Octal matches 3 binary digits, so the same method works with groups of 3.

Base conversion comes down to two directions. From base \(n\) to decimal, multiply each digit by its place value and add them all up. From decimal to base \(n\), divide by the base repeatedly and read the remainders from bottom to top. Between binary and hexadecimal, groups of 4 bits (3 bits for octal) let you convert directly without going through decimal.

Symbols and terms

Symbols

\(b\)	bee	The base (radix), the number that tells which number system is used. On this page it is 2, 8, 10 or 16. (From "base".)
\(d_i\)	dee sub i	The digit in position \(i\), counting from the right (the rightmost digit is position 0). In hexadecimal it can also be A to F (10 to 15 in decimal). (From "digit".)
\(i\)	i	The digit position. The rightmost digit is \(i=0\), and it goes up by 1 for each digit to the left.
\(N\)	N	The value in decimal. The conversion steps on this page always go through this decimal value.
\(\sum\)	sigma	The symbol for "add them all up" (summation). \(\sum_{i=0}^{k}\) says "let \(i\) go from 0 to \(k\) and add up all the terms".
\(q\)	q	The quotient of a division. To convert decimal to base \(n\), keep dividing until the quotient is 0. (From "quotient".)
\(r\)	r	The remainder of a division. The remainders become the digits of the base \(n\) number, starting from the rightmost digit. (From "remainder".)
\(2\mathrm{AA}_{16}\)	two A A, base sixteen	The small 16 written at the lower right of a number tells you "this number is written in base 16 (hexadecimal)". In the same way, \(1010_{2}\) is a binary number. This notation is used when the base might be unclear. Programmers also write hexadecimal with the prefix 0x, as in 0x2AA.

Terms

base (radix)	The rule for how many of something make one group (a carry to the next digit) when you count. Everyday numbers group by 10 (decimal), and computers work internally in groups of 2 (binary). The group size (2, 8, 10, 16 and so on) is called the base or radix.
positional notation	A way of writing numbers where each digit position has its own place value. It works just like decimal 682 standing for \(6 \times 100 + 8 \times 10 + 2 \times 1\); in binary, octal and hexadecimal, the place values are powers of 2, 8 and 16.
binary	A way of writing numbers with only two digits, 0 and 1 (base 2). Computers handle information as electrical on and off states, so all their internal calculations are done in binary.
octal	A way of writing numbers with the eight digits 0 to 7 (base 8). One octal digit matches 3 binary digits. It is used for file permissions in Linux and Unix, such as "755".
hexadecimal	A way of writing numbers with sixteen symbols, 0 to 9 and A to F (A is 10, B is 11, …, F is 15), also called hex (base 16). One hex digit matches 4 binary digits, so programmers use it widely to write long binary numbers in a short form.
bit	One binary digit. It is the smallest unit of information a computer handles, and its value is either 0 or 1.
nibble	A group of 4 bits (one hex digit). Two nibbles make one byte.
byte	A group of 8 bits (two hex digits). One byte holds 256 different values, from 00 to FF in hex, which is why bytes and hexadecimal go so well together.

Good to know before you start

Here is what helps you use the calculation on this page with real understanding, not just by pressing the button.

Place value in the decimal system (elementary school)	Being able to break 682 into "6 hundreds, 8 tens and 2 ones" = \(6 \times 100 + 8 \times 10 + 2 \times 1\) Knowing that each place to the left is worth 10 times as much
Division with remainders (Grades 3–4)	Being able to find the quotient and remainder of a division such as 682 ÷ 16 Understanding that dividend = divisor × quotient + remainder
Powers and exponents (Grades 6–8)	Knowing that "to the \(n\)th power" is multiplying the same number \(n\) times, as in \(2^3 = 2 \times 2 \times 2 = 8\) Knowing that any number to the power 0 is 1, as in \(16^0 = 1\)
Number bases (high school math and computer science)	Knowing that numbers can be written not only in decimal but in many systems with different bases (you can learn this on this page)

How to calculate it in Excel

Copy the whole table below and paste it into cell A1 in Excel. It works as is.

Table to convert decimal to binary, octal and hexadecimal

Decimal value	170
Binary	=DEC2BIN(B1)
Octal	=DEC2OCT(B1)
Hexadecimal	=DEC2HEX(B1)

Table to convert binary to decimal

Binary	10101010
Decimal	=BIN2DEC(B1)

Table to convert hexadecimal to decimal

Hexadecimal	2AA
Decimal	=HEX2DEC(B1)

Table to convert large numbers (BASE and DECIMAL functions)

Decimal value	1500
Hexadecimal (BASE)	=BASE(B1,16)
Back from hexadecimal to decimal (DECIMAL)	=DECIMAL(B2,16)

In the first table, enter a decimal number in B1, and B2 to B4 show it in binary, octal and hexadecimal (170 gives 10101010, 252 and AA).
Note that these functions have size limits. DEC2BIN works only from −512 to 511 (results of up to 10 binary digits), DEC2OCT from −536,870,912 to 536,870,911, and DEC2HEX from −549,755,813,888 to 549,755,813,887. The reverse functions BIN2DEC, OCT2DEC and HEX2DEC also accept at most 10 digits; more than 10 digits gives an error, and with exactly 10 digits and a leading 1 bit (a leading 8 to F in hex), the value is read as a negative number (two's complement).
For larger numbers, use the BASE function (decimal → base n) and the DECIMAL function (base n → decimal) in the fourth table (Excel 2013 or later; they handle numbers up to about 9 quadrillion, 2 to the 53rd power minus 1).

How to calculate it in Google Sheets

Copy the whole table below and paste it into cell A1 in Google Sheets. It works as is.

Table to convert decimal to binary, octal and hexadecimal

Decimal value	170
Binary	=DEC2BIN(B1)
Octal	=DEC2OCT(B1)
Hexadecimal	=DEC2HEX(B1)

Table to convert binary to decimal

Binary	10101010
Decimal	=BIN2DEC(B1)

Table to convert hexadecimal to decimal

Hexadecimal	2AA
Decimal	=HEX2DEC(B1)

Table to convert large numbers (BASE and DECIMAL functions)

Decimal value	1500
Hexadecimal (BASE)	=BASE(B1,16)
Back from hexadecimal to decimal (DECIMAL)	=DECIMAL(B2,16)

The same functions as in Excel work as is. Copy the whole table, paste it into cell A1, and replace B1 with your own number.
The size limits, such as DEC2BIN working only from −512 to 511, are the same as in Excel. For larger numbers, use the BASE and DECIMAL functions.

How to calculate it in Python

number = "2AA"   # the number to convert (written in its own base)
from_base = 16   # the base of that number (2, 8, 10 or 16)

value = int(number, from_base)    # first turn it into a decimal integer

binary = format(value, "b")       # binary
octal = format(value, "o")        # octal
hexadecimal = format(value, "X")  # hexadecimal (upper case)

print(f"Decimal: {value}")
print(f"Binary: {binary}")
print(f"Octal: {octal}")
print(f"Hexadecimal: {hexadecimal}")

Runs with the standard library only. int(text, base) converts base n → decimal, and format(value, "b"/"o"/"X") converts decimal → binary/octal/hexadecimal. Python integers have no size limit, so any number, however large, can be converted. Change the first two lines and run it.

How to write it in LaTeX and other math languages (copy and paste)

Base \(n\) → decimal (positional notation)

N = Σ dᵢ × bⁱ

N = \sum_{i=0}^{k} d_i \times b^{i}

<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
  <mrow>
    <mi>N</mi>
    <mo>=</mo>
    <munderover>
      <mo>&#x2211;</mo>
      <mrow><mi>i</mi><mo>=</mo><mn>0</mn></mrow>
      <mi>k</mi>
    </munderover>
    <msub><mi>d</mi><mi>i</mi></msub>
    <mo>&#xD7;</mo>
    <msup><mi>b</mi><mi>i</mi></msup>
  </mrow>
</math>

N = sum_(i=0)^k d_i * b^i

FromDigits[{2, 10, 10}, 16]

N := add(d[i]*b^i, i = 0 .. k);

N = polyval(d, b);

N = ∑_(i=0)^k d_i × b^i

Decimal → base \(n\) (divide by the base and read the remainders in reverse)

N = b × q + r (0 ≤ r < b)

N = b \times q + r \quad (0 \le r < b)

<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
  <mrow>
    <mi>N</mi>
    <mo>=</mo>
    <mi>b</mi>
    <mo>&#xD7;</mo>
    <mi>q</mi>
    <mo>+</mo>
    <mi>r</mi>
  </mrow>
</math>

N = b q + r, \ 0 <= r < b

QuotientRemainder[682, 16]

q := iquo(n, b); r := irem(n, b);

q = floor(n/b); r = mod(n, b);

N = b × q + r (0 ≤ r < b)

Binary ⇔ hexadecimal (groups of 4 bits)

h = b₃×2³ + b₂×2² + b₁×2¹ + b₀×2⁰

h = b_3 \times 2^3 + b_2 \times 2^2 + b_1 \times 2^1 + b_0 \times 2^0

<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
  <mrow>
    <mi>h</mi>
    <mo>=</mo>
    <msub><mi>b</mi><mn>3</mn></msub><mo>&#xD7;</mo><msup><mn>2</mn><mn>3</mn></msup>
    <mo>+</mo>
    <msub><mi>b</mi><mn>2</mn></msub><mo>&#xD7;</mo><msup><mn>2</mn><mn>2</mn></msup>
    <mo>+</mo>
    <msub><mi>b</mi><mn>1</mn></msub><mo>&#xD7;</mo><msup><mn>2</mn><mn>1</mn></msup>
    <mo>+</mo>
    <msub><mi>b</mi><mn>0</mn></msub><mo>&#xD7;</mo><msup><mn>2</mn><mn>0</mn></msup>
  </mrow>
</math>

h = b_3*2^3 + b_2*2^2 + b_1*2 + b_0

FromDigits[{b3, b2, b1, b0}, 2]

h := 8*b3 + 4*b2 + 2*b1 + b0;

h = 8*b3 + 4*b2 + 2*b1 + b0;

h = b_3 × 2^3 + b_2 × 2^2 + b_1 × 2^1 + b_0 × 2^0

How to have ChatGPT do the calculation

You are a number base conversion assistant. Do the following calculation by actually running Python code, and base your answer only on the numbers from the execution result (do not answer by mental math or guessing).

Convert hexadecimal 2AA to each of these bases:
1. Decimal
2. Binary
3. Octal

For the hexadecimal-to-decimal conversion, also show the expansion of each digit × its place value (a power of 16).
Show the code you used and the numbers from the execution result.

How to Use

1

Enter your numbers

Type the numbers you want to calculate with into the input fields
2

Calculate

Press the "Calculate" button
3

Check the result

The result appears on the spot. The same page also explains the idea behind the calculation and the formula