Number System Conversion Handout

There are four bases to learn:

Decimal (base 10) Hexidecimal (base 16) Binary (base 2) Octal (base 8)

0 0 0 0

1 1 1 1

2 2 10 2

3 3 11 3

4 4 100 4

5 5 101 5

6 6 110 6

7 7 111 7

8 8 1000 10

9 9 1001 11

10 A 1010 12

11 B 1011 13

12 C 1100 14

13 D 1101 15

14 E 1110 16

15 F 1111 17

16 10 10000 20

Conversions:

Recall that each of the binary digits has a value of twice the previous digit (instead of 10 times the previous digit, as in the decimal system). Consider the binary digit values of the binary number 1111, which is 15 in the decimal system.

Binary digit values:

| 128 | 64 | 32 | 16 | 8 | 4 | 2 | 1 |

------------------------------------------------------------------------

| 0 |0 | 0 | 0 | 1 | 1 | 1 | 1 |

To convert from binary to decimal, we add up the value of each of the digits. For example:

The binary number: 1 1 1 1 = 8+4+2+1 = 15

To convert from decimal to binary is only slightly more involved. Let's try the number 35, for example. First, we determine the highest binary digit value that is less than 35. The highest is 32. So, we put down a 1 in the 6th digit place and subtract 32 from 35 yielding 3.

1_ _ _ _ _

35 - 32 = 3

Then we check to see if the next lowest binary digit value will fit into the remaining value of 3. 16 is greater than 3, so we put down a zero and continue.

1 0 _ _ _ _

Eight and four are also too large, so we put down two more zeros and continue.

1 0 0 0 _ _

The next binary digit value, 2, is smaller than 3, so we put down a 1 and subtract 2 from 3.

1 0 0 0 1 _

3 - 2 = 1

The last binary digit value is 1, and since we have a 1, we put down a 1 and stop.

1 0 0 0 1 1

So, the binary number 100011 is 35 decimal.

Octal:

Each digit in octal is worth three binary digits. This is because the values 0-7 can be held in three bits. (zero in octal = zero in binary, 7 octal = 111 binary). So to convert, we simply break up a binary number into three bit segments and write in an octal digit for each segment.

1 1 0 1 1 0 0 1 = 1 1 0 1 1 0 0 1

3 3 1

So, 11011001 = 331 octal. To convert from octal to binary, we reverse the process.

Hex:

Converting from hex to binary works the same way, but each digit is worth four binary digits, intead of just 3. Convert in the same fashion as described above for octal.

Converting other bases:

To perform any other conversion not covered on this sheet, convert first to binary, then from binary to the new base.