Question

Right now I’m preparing for my AP Computer Science exam, and I need some help understanding how to convert between decimal, hexadecimal, and binary values by hand. The book that I’m using (Barron’s) includes an example but does not explain it very well.

What are the formulas that one should use for conversion between these number types?

Answer 1

Are you happy that you understand number bases? If not, then you will need to read up on this or you’ll just be blindly following some rules.

Plenty of books would spend a whole chapter or more on this…

Binary is base 2, Decimal is base 10, Hexadecimal is base 16.

So Binary uses digits 0 and 1, Decimal uses 0-9, Hexadecimal uses 0-9 and then we run out so we use A-F as well.

So the position of a decimal digit indicates units, tens, hundreds, thousands… these are the “powers of 10”

The position of a binary digit indicates units, 2s, 4s, 8s, 16s, 32s…the powers of 2

The position of hex digits indicates units, 16s, 256s…the powers of 16

For binary to decimal, add up each 1 multiplied by its ‘power’, so working from right to left:

1001 binary = 1*1 + 0*2 + 0*4 + 1*8 = 9 decimal

For binary to hex, you can either work it out the total number in decimal and then convert to hex, or you can convert each 4-bit sequence into a single hex digit:

1101 binary = 13 decimal = D hex

1111 0001 binary = F1 hex

For hex to binary, reverse the previous example – it’s not too bad to do in your head because you just need to work out which of 8,4,2,1 you need to add up to get the desired value.

For decimal to binary, it’s more of a long division problem – find the biggest power of 2 smaller than your input, set the corresponding binary bit to 1, and subtract that power of 2 from the original decimal number. Repeat until you have zero left.

E.g. for 87:

• the highest power of two there is 1,2,4,8,16,32,64!
• 64 is 2^6 so we set the relevant bit to 1 in our result: 1000000
• 87 – 64 = 23
• the next highest power of 2 smaller than 23 is 16, so set the bit: 1010000
• repeat for 4,2,1
• final result 1010111 binary
• i.e. 64+16+4+2+1 = 87 in decimal

For hex to decimal, it’s like binary to decimal, only you multiply by 1,16,256… instead of 1,2,4,8…

For decimal to hex, it’s like decimal to binary, only you are looking for powers of 16, not 2. This is the hardest one to do manually.