freeCodeCamp/guide/english/mathematics/binary-decimal-hex-conversion/index.md

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Binary Decimal Hexadecimal Conversion

Conversion:

You can easily convert numbers from one base to another applying the definition of n-based number which requires you to know how our positional system works: Let's start from a two digits number like 12 for example. In order to obtain it's base-10 value we need to multiply its single digit by 10^n where n is the digit position starting from right and counting from 0.Then we simply sum all the values. For example the base-10 value of 12 will be obtained in this way:

1*(10^1) + 2*(10^0) = 10 + 2 = 12

This was obvious but what if you had a base-2 number and wanted to know its base-10 value? First of all mind that a base n number only has n total symbols to represent its values. In the binary base we have then just 2 (base-2) symbols: 1 and 0. Applying the procedure you have seen before you will be able to obtain a decimal number starting from a binary one like 101:

101 = 1*(2^2) + 0*(2^1) + 1*(2^0) = 4+0+1 = 5

In the same way a hexadecimal (base-16) number has 16 symbols to represent its values: 0, 1, 2, 3, 4, 5, 6 ,7, 8, 9, A, B, C, D, E, F. Converting a base-16 number like 7AF to a decimal will be easy then:

7AF = 7*(16^2) + A*(16^1) + F*(16^0) = 7*256 + 10*16 + 15*1 = 1967 

What if you wished to convert a decimal number into a n-based number? A common way to accomplish this is dividing the decimal number by the n base repeatedly. Take note of all remainders, and when your quotient reaches 0 stop. Now simply write all your remainders setting the last one as the most significant digit (your newly converted n-based number should have as last digit your first remainder). EG: Let's convert the base-10 12 to its base-2 value

12/2 = 6 with remainder 0
6/2  = 3 with remainder 0
3/2  = 1 with remainder 1
1/2  = 0 with remainder 1

base-10 12 = base-2 1100

Now using the first method written above you can check if everything worked fine:

1100 = 1*(2^3) + 1*(2^2) + 0*(2^1) + 0*(2^0) = 8+4+0+0 = 12

Binary Decimal Hexadecimal Converter

A Binary, decimal and hexadecimal converter it's a tool that allows you to convert one number in the corresponding one expressed in a different numeral system. The numeral systems allowed are base-2 (binary), base-10 (decimal) which is the one we commonly use and base-16 (hexadecimal). The are plenty of this tools available online:

• [Binary Hex Converter] (www.binaryhexconverter.com/)
• [Calculator website] (http://www.calculator.net/) Usually scientific calculators too include base conversion tools and in MacOSX default calculator you can use this function using its programmer view pressing Cmd+3 or under menu View->Programmer.

Your own converter:

A good idea to practice programming and fully understand numbers conversion would be to code your own online conversion tool. If you want to know more about this topic, please check this wikipedia entry.