How to convert from base-10 (decimal) to any other base
- 101 Comment
A friend of mine just recently decided to go back to college after being out of school for a while. Tonight, I helped her with a math operation that she had been struggling with. She needs to convert from Base-10 (decimal) to another base (base-3, base-5, base-6, etc.). She had all but given up on the subject after several attempts to get her teacher to help her. She is an intelligent person, but math is one of those subjects that you need to practice and find that AH-HA! moment. Her teacher was not helping her down the path to her AH-HA! moment with base conversion, so I decided that I would work with her over the phone until she reached the AH-HA!
After showing her the method that I am going to call the “Base Chart, Breakdown Method“, she finally had her AH-HA! moment and can now do base conversions in her sleep. I figured I would share the method so that, one day, if your life depended on it, you could convert 83 to Base-5.
First, a quick primer on what the 5 in Base-5 means:
In Base-10 the counting system uses 1’s, 10’s, 100’s, 1000’s and so on. This is represented by:
- 10^0 (The ones place)
- 10^1 (The tens place)
- 10^2 (The one-hundreds place)
- 10^3 (the one-thousands place)
- And so on…
For a different base, such as Base-5, this is no different. You have the 1’s, 5’s, 25’s, 125’s, and so on. This is represented by:
- 5^0 (The ones place)
- 5^1 (The fives place)
- 5^2 (The twenty-fives place)
- 5^3 (The one-hundred-twenty-fives place)
- And so on…
This is true for any different base system.
So, how do we solve a base conversion problem? The best way to learn is to do. So let’s do one.
-Convert 83 (Base-10) to Base-5.
-Start by writing a Base Chart for the base you are trying to convert (this will be used later in the problem). The chart will look like the one below:
| | | | |
Which can be also represented by solving the 5^0, 5^1, 5^2, and 5^3 like the chart below:
|125|25 |5 |1 |
| | | | |
Save this “Base Chart” for later use. Notice, that I filled in the chart until I had a number that was greater than the Base-10 number that I am trying to convert. In this case 125 is greater than 83.
Back to the problem. We said we wanted to convert 83 (base-10) to Base-5. Now comes the Breakdown part.
-How many times does 5 go into 83?
- 83/5=16.6 (drop the .6)
- 5 goes into 83, 16 times, or 16 sets of 5’s
- 16×5=80 and 83-80=3 so we have 16 (fives) and 3 (ones)
- Write that down
-Since we know that the only possible digits for Base-5 are 0, 1, 2, 3, and 4 (5 digits, hence Base-5), we also know that we cannot have 16 sets of 5’s. We have to keep Breakin’ It Down. Each time you Breakdown, you move up a place in the Base Chart.
- 5 goes into 16, 3 times, or 3 sets of 25’s (since we are now into the 25’s place in the Base Chart).
- 3×5=15 and 16-15=1 so we have 3 (twenty-fives) and 1 (five)
- Write that down.
By now you should have the following written down:
- 16 (fives) and 3 (ones)
- 3 (twenty-fives) and 1 (five)
Take all the numbers below the Base digits (below 5 in this case) and write down what you have in order from highest to lowest place. This would mean you cannot write down 16 (fives). You should end up with this:
- 3 (twenty-fives), 1 (five), and 3 (ones)
Now, remember that Base Chart we set aside for later use? Fill in the numbers that you have come up with in the appropriate slots in the base chart. It would look like the chart below.
|125|25 |5 | 1 |
| | 3 | 1| 3 |
It turns out that we didn’t need the 125’s place after all. Now, reading from left to right, the answer to our original question is:
83 (Base-10) converted to Base-5 is 313.
- 83 (Base-10) = 313 (Base-5)
You can convert from Base-5 back to Base-10 by using the Base Chart. If the problem had been reversed and you were given 313 (Base-5) and you needed to convert it to Base-10, you would just fill in the Base Chart, multiply the number on the bottom by the place number, and add the products.
|125|25 |5 |1 |
| | 3 | 1| 3|
This is also a good way to check and see if your Base-10 to Base-5 conversion is correct.
I hope this post helped you achieve your AH-HA! moment with Base Conversion problems. This method can be used to convert Base-10 to any Base. It’s just a matter of following these steps:
- Create the Base Chart so that it goes out to a place that is greater than the Base-10 number you are trying to convert.
- Breakdown the Base-10 number into legal groups of your target Base number places.
- Write the number of groups into the appropriate places in the Base Chart.
If you are having trouble with this method, please let me know in the comments section below. A special thanks to Denise for letting me teach her the “Base Chart, Breakdown Method.” I’m sure you will ace that exam!
No related posts.
Related posts brought to you by Yet Another Related Posts Plugin.