# Number's base changer

## World's simplest number tool

Quickly change a number's base in your browser. To get your number in the new base, just enter it in the input field, set the input number's base and the output number's base, and this utility will convert the number to the new base. Created by developers from team Browserling.

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Number's base changer tool
What is a number's base changer?

This is an online browser-based utility for changing a number's base. Numbers are an abstract idea and to write them down, you need to select a base. The base can be anything and the most common base is 10 (decimal) and it's the default base in this tool. When you write a number 25 or 1044, it's assumed that it's in base 10 but it could be in any other base as there are infinitely many numeral systems. Other well-known numeral systems are binary (base 2), octal (base 8), and hexadecimal (base 16). Slightly less common are negative bases, such as negabinary (base -2), negaoctal (base -8), negadecimal (base -10), and negahexadecimal (base -16). Even less common and virtually unknown bases are rational (for example, base 2/3), irrational (for example, base e), fractional (for example, base π/2), imaginary and complex (for example, base i), and many more. At the moment we support only positive and negative integer bases. All bases use the same principle to represent a number with the only difference being the alphabet set of the digits and whether the sign symbol is used for negative numbers. For example, the binary base expresses each number using two characters "0" and "1", the decimal base uses ten digits "0123456789", and the hexadecimal base uses six more letters "0123456789abcdef". Given the alphabet and the base, we can express any number by the formula x = ∑a

_{n}b^{n}, where a_{n}is a digit from the alphabet and b is the given base. The sequence a_{n}a_{n-1}…a_{0}of indexes in this formula is the number x in the given base b. For example, in base 10, the number 25 can be written as a sum ∑a_{n}10^{n}= a_{1}*10^{1}+ a_{0}*10^{0}= 2*10^{1}+ 5*10^{0}. Here the indexes are a_{1}= 2 and a_{0}= 5 and a_{1}a_{0}forms the number 25. In the base 2, the number 25 can be expressed as 1*2^{4}+ 1*2^{3}+ 0*2^{2}+ 0*2^{1}+ 1*2^{0}and from this sum, we find that 11001 is the number 25 in base 2. With negative bases, it works the same way. For example, in base -8, the number 25_{10}is 1*(-8)^{2}+ 5*(-8)^{1}+ 1*(-8)^{0}, which is 151_{-8}. If the base is negative, then the minus sign is not used. For example, -25_{10}= 4*(-8)^{1}+ 7*(-8)^{0}= 47_{-8}. In positive bases, the minus sign needs to be used to express negative numbers. This tool accepts any number with the base from -64 to 64 as input (except bases 0 and -1, which don't exist) and outputs it in any other base from this range. It can convert multiple numbers at the same time if they are entered one per line. As it's not always easy to know what each number means, you can also print the decimal number after each converted number. Also, as sometimes you may need the full sum formula, you can select the "Use Sum Formula" option to print the sum terms. That's numberwang!
Number's base changer examples
Click to use

Decimal to Hexadecimal

This example converts five numbers from base 10 to base 16. It decomposes each number using the formula x = ∑a

_{n}b^{n}= ∑a_{n}16^{n}and then prints the coefficients a_{n}, which represent the number in base 16. For example, the number 30 = a_{1}*16^{1}+ a_{0}*16^{1}= 1*16^{1}+ 14*16^{1}and therefore the coefficients are a_{1}= 1 and a_{0}= 14. The hex base uses letters a, b, c, d, e, and f for coefficients 10, 11, 12, 13, 14, 15, so we get that the coefficients actually are a_{1}= 1 and a_{0}= e, and we find that 30_{10}= 1e_{16}.
30
260
-45
-100
2450

1e
104
-2d
-64
992

**Required options**

Enter the input number base.

Enter the output base.

Display the output numbers in the
new base as a power series sum.

Display the decimal value next to
each output number.

Create a Negabinary

In this example, we change a binary number's representation to negabinary, which is the base minus two (base -2). It's often used to encode negative binary numbers as it doesn't use the "-" sign. We have also activated the use decimal notation option so that we knew what the input numbers were without much thinking. The input number is first converted to decimal 110

_{2}= 1*2^{2}+ 1*2^{1}+ 0*2^{0}= 4 + 2 + 0 = 6_{10}and then to negabinary 6_{10}= 1*(-2)^{4}+ 1*(-2)^{3}+ 0*(-2)^{2}+ 1*(-2)^{1}+ 0*(-2)^{0}= 11010_{-2}. Similarly, -1110_{2}(-14_{10}) is 110110_{-2}and as you can see, the minus sign disappears in this base.
110
-1110
10101

11010 (6)
110110 (-14)
10101 (21)

**Required options**

Enter the input number base.

Enter the output base.

Display the output numbers in the
new base as a power series sum.

Display the decimal value next to
each output number.

Decode a Negadecimal

This example decodes negadecimal numbers. "Nega" means negative and "decimal" means ten, so it converts numbers from base -10 numbers to base 10. As the option "Print Sum" is enabled, it displays the new numbers as a decomposition sum. It seems that there are no signed numbers in the input as all numbers look positive but in fact, the third and fifth numbers are signed. After the conversion, you can see that these numbers are -45 and -505 as they use minus signs in the sum formula.

184
176
55
722
1515

2×(10)¹ + 4×(10)⁰
3×(10)¹ + 6×(10)⁰
-4×(10)¹ - 5×(10)⁰
6×(10)² + 8×(10)¹ + 2×(10)⁰
-5×(10)² - 0×(10)¹ - 5×(10)⁰

**Required options**

Enter the input number base.

Enter the output base.

Display the output numbers in the
new base as a power series sum.

Display the decimal value next to
each output number.

Encode to Negaoctal

In this example, we convert three octal (radix 8) numbers to negaoctal (radix -8) numbers. We activate the "Use Sum Formula" option to get the numbers' power series from the formula ∑a

_{n}(-8)^{n}, where a_{n}is the octal base alphabet made from the digits 0 to 7. From the coefficients before the sum terms, we find that the number 55_{8}is 135_{-8}, the number -12_{8}is 26_{-8}, and the number 44_{8}is 144_{-8}.
55
-12
44

1×(-8)² + 3×(-8)¹ + 5×(-8)⁰
2×(-8)¹ + 6×(-8)⁰
1×(-8)² + 4×(-8)¹ + 4×(-8)⁰

**Required options**

Enter the input number base.

Enter the output base.

Display the output numbers in the
new base as a power series sum.

Display the decimal value next to
each output number.

Unary Numeral System

This example loads two negahexadecimal numbers and expands them as repeated 1's in the unary numeral system. The unary number system, also known as base 1, uses only one symbol to represent each number. This symbol can be anything and we have hard-coded it to be "1". If a number in the decimal base is N, then in the unary base it has N ones. In our case, the number 1f2

_{-16}is 18 in base 10, thus in base 1 it's equal to eighteen ones. The number 1c_{-16}is -4 in base 10 and in base 1 it's equal to minus four ones.
1f2
1c

111111111111111111 (18)
-1111 (-4)

**Required options**

Enter the input number base.

Enter the output base.

Display the output numbers in the
new base as a power series sum.

Display the decimal value next to
each output number.

Base -32 to Base -64

In this example, we convert from one negative base (-32) to another negative base (-64). To do it, this utility first finds decimal base 10 values of the input base -32 values and then converts them to base -64 values. As both bases are negative, the numbers don't use the "-" sign. These bases use an alphabet that not only contains numbers but also letters and even special characters ("$" and "_" in base -64). We also display the intermediate decimal value in brackets next to each base -64 output number.

g
1uk
1sm
1cr
4jh

g (16)
1_k (84)
1$m (150)
1Sr (667)
1aN (3505)

**Required options**

Enter the input number base.

Enter the output base.

Display the output numbers in the
new base as a power series sum.

Display the decimal value next to
each output number.

Pro tips
Master online number tools

You can pass input to this tool via

__?input__query argument and it will automatically compute output. Here's how to type it in your browser's address bar. Click to try!
https://onlinenumbertools.com/change-number-base

__?input__=30%0A260%0A-45%0A-100%0A2450&input-base=10&output-base=16&show-representation-sum=false&show-decimal=false
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