How to Convert Hex to Binary and Binary to Hexadecimal

The base 16, also known as hexadecimal (abbreviated to hex) numbering system is regularly used in computer coding for conveniently representing a byte or word of data. This guide shows you how to convert from hex to binary and binary to hexadecimal.

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Before we learn how to convert hex to binary, let's try and understand how the base 10 system works.

The decimal, also known as the denary or base 10 numbering system that we use in everyday life makes use of ten symbols or numerals: 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9.

So to count you start with 0, then continue 1...2...3...4...5...6...7...8...9

What happens when you get to ten? There's no numeral for ten, so it's represented as

10

Which means 1 ten and no units

Similarly when you get to 99, there's no numeral for one hundred, so you write one hundred as 100.

So writing a number in the base 10 system involves using numerals in a "units", "tens", "hundreds", "thousands" place and so on

So 145 really means "one hundred, 4 tens and 5 units" although we just think of it as one hundred and forty five.

Hexadecimal or "hex" is a numbering system which uses 16 different numerals. We saw that decimal used ten numerals from 0 to 9. Hex expands on this by adding six more, the capital letters A, B, C, D, E and F.

So to count from 0 to 15 you go 0...1...2...3...4...5...6...7...8...9

But what happens next?

Simply continue with A...B...C...D...E...F which represents 10, 11, 12, 13, 14 and 15 decimal.

In the decimal system, we saw that when we got to nine, there was no numeral for ten so it was represented as 10 or "one ten and no units".

In the hex system when we get to F which is 15 decimal, we have to represent the next number sixteen as 10 or "one 16 and no units".

The binary system used by computers is based on 2 numerals; 0 and 1. So you count 0, 1, there is no numeral for 2, so 2 is represented by 10 or "one 2 and no units". In the same way that there is a units, tens, hundreds, thousands place in the decimal system, in the binary system there is a units, twos, fours, eights, sixteens place etc. in the binary system.

If a number isn't decimal (base 10), the base can be explicitly indicated by a subscript to avoid confusion. Sometimes the subscript is omitted to avoid excessive detail if the base has been specified earlier in a discussion or if numbers are listed in a table (e.g. numbers may be indicated as hex in the title of the table).

So for instance 1F hex (31 decimal) can be written 1F₁₆

Hex is very easy to convert to binary.

1. Write down the hex number and represent each hex digit by its binary equivalent number from the table above.
2. Use 4 digits and add insignificant leading zeros if the binary number has less than 4 digits. E.g. Write 10₂ (2 decimal) as 0010_2.
3. Then concatenate or string all the digits together.
4. Discard any leading zeros at the left of the binary number.

For a binary number, the most significant bit (MSB) is the digit furthermost to the left of the number and the least significant bit (LSB) is the rightmost digit.

Binary is also easy to convert to hex.

1. Start from the least significant bit (LSB) at the right of the binary number and divide it up into groups of 4 digits. (4 digital bits is called a "nibble").
2. Convert each group of 4 binary digits to its equivalent hex value (see table above).
3. Concatenate the results together, giving the total hex number.

Because of the ease of converting from hex to binary and vice versa, it's a convenient shorthand for representing byte values i.e. numbers from 0 to 255. Also it's compact requiring only 2 digits for a byte and 4 digits for a word.

• Hex dumps are used to display the values of bytes in files. Each byte value is displayed as hex
• Assembly language is written as a series of mnemonic (short, easy to remember word) instructions for a microprocessor. The operand (the data operated on by an opcode) is commonly specified as a hex value. It's also used to indicate the storage location of data

Example of assembly language instruction

In the short code segment below, MOV is the opcode (instruction) and 61 hex is the operand that the opcode acts on. AL is a register that stores a value temporarily so that arithmetic can be done on it before it's moved to memory. A program called an assembler converts the human understandable assembly language to machine code.

MOV AL, 61H ; Load AL register with 61 hex (97 decimal )

To convert decimal to binary and binary to decimal, see my other guide:

How to Convert Decimal to Binary and Binary to Decimal

For more details on how binary is used in computer systems and digital electronics, see my other article:

Why is Binary Used In Computers and Electronics?

You can convert hex to decimal by simply multiplying each hex numeral by the placeholder's value as a power of 16 and adding the result. (F₁₆ = 15 decimal and A₁₆ = 10 decimal)

Example: What is the decimal equivalent of 52FA₁₆ ?

52FA₁₆ = 5 x 16³ + 2 x 16² + 15 x 16¹ + 10 x 16⁰

= 5 x 4096 + 2 x 256 + 5 x 16 + 10 x 1

= 21,242