Number System Conversion Example

We will convert to binary using the "division method." We will start by dividing 200 by the base (in this case 2) and writing down the remainder (use long division!) in the rightmost numerical position (in this case, the one's spot). We will continue this cycle as in the table below:

Calculation Step	Explanation	Current Result
200/2 = 100r0	As described above.	0
100/2 = 50r0	We continue by dividing 100 (the result from above) by 2, and writing the remainder to the left of the one found in the above step.	00
50/2 = 25r0	Continue as above.	000
25/2 = 12r1		1000
12/2 = 6r0		01000
6/2 = 3r0		001000
3/2 = 1r1		1001000
1/2 = 0r1	Now we have reached 0, so we know we are done.	11001000

Now we'll convert 11001000 back to decimal (base 10). We do this by multiplying each digit by the appropriate power of 2 (the base):

1*2⁷ + 1*2⁶ + 0*2⁵ + 0*2⁴ + 1*2³ + 0*2² + 0*2¹ + 0*2⁰

Remember these powers directly correspond to the location in the binary number... for example, the first '1' on the left is in the 128's spot, which is 2⁷.

We now multiply this out and add it up:

128 + 64 + 0 + 0 + 8 + 0 + 0 + 0 = 200, and we're done.

We will once again use the division method to covert 200 to octal. The only difference from the above is that we use 8 instead of 2 to divide by!

Calculation Step	Explanation	Current Result
200/8 = 25r0	As described above.	0
25/8 = 3r1	We continue by dividing 25 (the result from above) by 8, and writing the remainder to the left of the one found in the above step.	10
3/8 = 0r3	Continue as above. Now that we are at 0, we are done.	310

Now we'll convert 310 back to decimal (base 10). We do this by multiplying each digit by the appropriate power of 8 (the base):

3*8² + 1*8¹ + 0*8⁰

Remember these powers directly correspond to the location in the octal number... for example, the spot the '3' occupies corresponds to 8² which is the 64s spot.

We now multiply this out and add it up:

192 + 8 + 0 = 200, and we're done.

By now you should have seen the patterns of which numbers we change to convert a number between any base and decimal and back again. We will once again use the division method to covert 200 to hex. The only difference from the above is that we use 16 instead of 2 or 8 to divide by!

Calculation Step	Explanation	Current Result
200/16 = 12r8	As described above.	8
12/16 = 0r12	We continue by dividing 12 (the result from above) by 16, and writing the remainder to the left of the one found in the above step. Remember 12 corresponds to C in hex. Sincer we are at 0, we are done.	C8

Now we'll convert C8 back to decimal (base 10). We do this by multiplying each digit by the appropriate power of 16 (the base):

12*16¹ + 8*16⁰

Remember these powers directly correspond to the location in the hex number... for example, the spot the 'C' occupies corresponds to 16¹ which is the 16s spot.

We now multiply this out and add it up:

192 + 8 = 200, and we're done.