# 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.
Old Base
Enter the input number base.
Allowed bases are from
-64 to 64, except -1 and 0.
New Base
Enter the output base.
Allowed bases are from
-64 to 64, except -1 and 0.
Output Format
Display the output numbers in the
new base as a power series sum.
Display the decimal value next to
each output number.
These two options don't work at
the same time. You must choose
one or the other.

## 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 = ∑anbn, where an is a digit from the alphabet and b is the given base. The sequence anan-1…a0 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 ∑an10n = a1*101 + a0*100 = 2*101 + 5*100. Here the indexes are a1 = 2 and a0 = 5 and a1a0 forms the number 25. In the base 2, the number 25 can be expressed as 1*24 + 1*23 + 0*22 + 0*21 + 1*20 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 2510 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, -2510 = 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
This example converts five numbers from base 10 to base 16. It decomposes each number using the formula x = ∑anbn = ∑an16n and then prints the coefficients an, which represent the number in base 16. For example, the number 30 = a1*161 + a0*161 = 1*161 + 14*161 and therefore the coefficients are a1 = 1 and a0 = 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 a1 = 1 and a0 = e, and we find that 3010 = 1e16.
30 260 -45 -100 2450
1e 104 -2d -64 992
Required options
These options will be used automatically if you select this example.
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 1102 = 1*22 + 1*21 + 0*20 = 4 + 2 + 0 = 610 and then to negabinary 610 = 1*(-2)4 + 1*(-2)3 + 0*(-2)2 + 1*(-2)1 + 0*(-2)0 = 11010-2. Similarly, -11102 (-1410) 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
These options will be used automatically if you select this example.
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.
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
These options will be used automatically if you select this example.
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 ∑an(-8)n, where an is the octal base alphabet made from the digits 0 to 7. From the coefficients before the sum terms, we find that the number 558 is 135-8, the number -128 is 26-8, and the number 448 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
These options will be used automatically if you select this example.
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
These options will be used automatically if you select this example.
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
These options will be used automatically if you select this example.
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://onlinetools.com/number/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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