Number Converting:

# 5. Hexadecimal Number Conversion

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## Hexadecimal to Binary

Converting from hexadecimal to binary is as easy as converting from binary to hexadecimal. Simply look up each hexadecimal digit to obtain the equivalent group of four binary digits.

 Hexadecimal: 0 1 2 3 4 5 6 7 Binary: 0000 0001 0010 0011 0100 0101 0110 0111
 Hexadecimal: 8 9 A B C D E F Binary: 1000 1001 1010 1011 1100 1101 1110 1111

 Hexadecimal = A 2 D E Binary = 1010 0010 1101 1110 = 1010001011011110 binary

## Hexadecimal to Octal

When converting from hexadecimal to octal, it is often easier to first convert the hexadecimal number into binary and then from binary into octal. For example, to convert A2DE hex into octal:

(from the previous example)

 Hexadecimal = A 2 D E Binary = 1010 0010 1101 1110 = 1010001011011110 binary

Add leading zeros or remove leading zeros to group into sets of three binary digits.

Binary: 1010001011011110 = 001 010 001 011 011 110

Then, look up each group in a table:

 Binary: 000 001 010 011 100 101 110 111 Octal: 0 1 2 3 4 5 6 7

 Binary = 001 010 001 011 011 110 Octal = 1 2 1 3 3 6 = 121336 octal

Therefore, through a two-step conversion process, hexadecimal A2DE equals binary 1010001011011110 equals octal 121336.

## Hexadecimal to Decimal

Converting hexadecimal to decimal can be performed in the conventional mathematical way, by showing each digit place as an increasing power of 16. Of course, hexadecimal letter values need to be converted to decimal values before performing the math.

 Hexadecimal: 0 1 2 3 4 5 6 7 Decimal: 0 1 2 3 4 5 6 7 Hexadecimal: 8 9 A B C D E F Decimal: 8 9 10 11 12 13 14 15

A2DE hexadecimal:
= ((A) * 163) + (2 * 162) + ((D) * 161) + ((E) * 160)
= (10 * 163) + (2 * 162) + (13 * 161) + (14 * 160)
= (10 * 4096) + (2 * 256) + (13 * 16) + (14 * 1)
= 40960 + 512 + 208 + 14
= 41694 decimal

Finally, why you might choose one number system over another...