How to convert from base-10 (decimal) to any other base
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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:
|5^3|5^2 |5^1|5^0|
| | | | |
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.
- 16/5=3
- 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|
- 3x25=75
- 1x5=5
- 3x1=3
- 75+5+3=83
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!
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50 Comments on this post
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Cherith Welter said:
I am trying to convert base 10 to base 16 and I can’t seem to wrap myself around how to do this. I understand how you did the base 5, but the answers that I am coming up with aren’t matching the book so I am confused!
January 11th, 2008 at 11:33 am -
hstagner said:
Hello Cherith,
Thank you for reading Searchmarked.com. I am sorry that you are struggling with base 16 conversions.
Perhaps you could send me a specific example that I can work out with this method. I will post it and it might be able to help everyone.
Regards,
Harley Stagner
January 15th, 2008 at 2:04 pm -
Cherith said:
Dear Harley Stagner,
The specific problem is 28+28=50. I get it to come out as 38, but I can’t seem to figure out how to make it 50. What am I doing wrong?January 15th, 2008 at 2:37 pm -
hstagner said:
Hello Cherith,
It looks like the 28’s in this problem are already in base-16 (not base 10) so you need to add them in base-16.
It looks like what you did was take 28+28(base-10), which equals 56. 56 does indeed convert to 38 in hexidecimal (base-16).
What you need to do (since the 28’s are already base-16 numbers) is convert the 28’s to base-10 which would be:
28+28 (base-16)
40+40 (base-10)Now 40+40 = 80 (base-10)
So just convert 80 (base-10) to a base-16 number. I’ll give you three guesses what the base-16 number will be
.January 15th, 2008 at 4:50 pm -
Cherith said:
Thank you for explaining that to me. It makes a lot of sense now!
CherithJanuary 16th, 2008 at 9:51 am -
omar said:
2. Convert the following number to from decimal to hexadecimal, stop at 4 decimal points
a. 0.6640625
b. 0.3333
c. 69/256February 15th, 2008 at 11:42 am -
omar said:
1. Using division method convert the following decimal numbers:-
a. 13750 to base 12
b. 6026 to hexadecimals
c. 3175 to base 5February 15th, 2008 at 11:45 am -
omar said:
3. Convert the following from the given base to decimal
a. 0.1001001(2)
b. 0.3A2 (16)February 15th, 2008 at 11:45 am -
omar said:
4. Convert the following decimal to binary and hexadecimal
a. 27.625
b. 4192.37761February 15th, 2008 at 11:46 am -
omar said:
5. What is the decimal value of the binary number
a. 1100101.1
b. 1110010.11
c. 11100101.1February 15th, 2008 at 11:46 am -
hstagner said:
Omar,
Thank you for reading. I will help you, but I will not do your whole test or homework for you. I will give you a representative example for each of these problems by answering part a. in each of these problems.
You should be able to follow along by example after this. Stay tuned for the post.
Regards,
Harley Stagner
February 15th, 2008 at 2:01 pm -
Parag said:
thank u really!!
i urgently needed to know this..
thanks againJune 19th, 2008 at 12:04 pm -
Ahmad said:
It is very nice way to teach base conversion
September 10th, 2008 at 4:46 am -
hannah said:
OMG thank u soooo much
i m a junior high student and we are learning this
my teacher didn’t explain so that i understood but this page reallllllly helped me with my hw
thx angainSeptember 14th, 2008 at 2:13 pm -
hstagner said:
You are very welcome Hannah! Thank you for reading…
I am glad that I could help you.
September 14th, 2008 at 3:13 pm -
Danielle said:
I am trying to wrap my head around base conversions and I understand your examples but am struggling with converting numbers into base three. I was trying to determine what 10 and 102 would be in base three and am stuck…
Thank you so much for your help!
September 15th, 2008 at 8:42 pm -
jenn said:
hello,
can u teach me using division method how to convert 13750 to base 12??
6026 to hexadecimal and 3175 to base 5?September 20th, 2008 at 2:03 pm -
hstagner said:
Hello Danielle,
You use the exact same method no matter which base.
So for base 3 let’s do 10.
Draw your base 3 chart:
| 27 | 9 | 3 | 1 |
| | | | |-3 goes into 10 how many times?
-10/3 = 3.33333333 (drop the trailing 3’s)
-3 goes evenly into 10, three times or 3 sets of 3’s and 3×3=9.
-10-9=1 so we have three sets of 3’s and one 1.
-However, we can’t have a 3 because the only digits in base 3 are 0, 1, and 2. So, we have to break it down.
-Every time we break it down we move up (to the left) in our base chart.
-3/3=1, so we have one 9, zero 3’s and one 1. Let’s write that in the chart.
| 27 | 9 | 3 | 1 |
| | 1 | 0 | 1 |-Tada! Our answer is 101.
Thank you for reading! I hope that I could help you.
September 24th, 2008 at 10:05 pm -
Gordon Ma said:
I’ve been looking at your examples, and they’re very helpful. Can you teach me how to convert something in base 10 to base 1?
December 8th, 2008 at 7:12 pm -
indu said:
explain
why 72 base 10 is both 187 base5 &242 base 5?
December 18th, 2008 at 8:23 am -
Jeffrey said:
I was trying to figure these out. I can get it to work for every problem except this one: Convert 61 from base-10 to base-4 … I do 61/4 = 15. 4*15 = 60 . So 15 fours and 1 one. Then I do 15/4 and get 3. 3*4 = 12 so It’d be 3 sixteens and two fours. I plug them into my chart. and I get it to look like this:
| 64 | 16 | 4 | 1 |
| 0 | 3 | 2 | 1 |I try to convert it back to check: 16 * 3 = 48
4 * 2 = 8
1 * 1 = 1
Giving me: 57 as an answer. 4 short of what I was looking for. Where did I go wrong?January 7th, 2009 at 5:08 pm -
hstagner said:
Hello Jeffrey,
Thanks for reading! I’ll give you a hint (okay well maybe more than a hint).
The answer is 331
For the 4’s place, what is 15-12? Just a small math error, that’s all.
January 11th, 2009 at 10:12 am -
Rimpy said:
@jeffery: It should be like below:
64 | 16 | 4 | 1
0 | 3 | 3 | 1March 22nd, 2009 at 12:51 pm -
Alex said:
Thanks a lot Harley! I finally get how to convert bases!
July 13th, 2009 at 6:48 pm -
hstagner said:
You are quite welcome Alex. I’m glad that I could help! Thank you for reading SearchMarked.com.
July 13th, 2009 at 7:46 pm -
Nate said:
Yes, i’m sure we can all do this easly thankyou. However, one thing they do not show just about anywhere in textbooks or on the web is how to work with DECIMALS!
please…. how abouse something like 256.233(base 8)–>base 10
September 1st, 2009 at 4:27 pm -
dog kennel said:
hey,
thanks for sharing this method.it is not much simple but effective.keep it up.October 12th, 2009 at 4:12 pm -
sanket said:
brilliant!!!
October 28th, 2009 at 1:00 pm -
dog kennel said:
I know my calculator can do base 2, base 18 and base 16 conversion, but that doesn’t help me when I need to convert to others bases (3 etc)…
November 10th, 2009 at 8:19 am -
Sukhraj said:
what about negabinary aka base – 2 (negative 2) does it work for negative bases?
November 15th, 2009 at 10:50 am -
celseq said:
it needs 2 b more clear
November 21st, 2009 at 3:15 pm -
hstagner said:
Hello Celseq,
Thank you for reading Searchmarked. Judging from the other comments, it is perfectly clear. I don’t know how else to explain it. Perhaps you can give me a suggestion after you brush up on your grammar? Try this:
“I am having a hard time understanding the explanation in this blog post. Can someone please help me?”
That’s better.
November 21st, 2009 at 7:02 pm -
Adam Murray said:
WOW. thank you so much you have helped me with my degree
November 22nd, 2009 at 12:51 pm -
Stephen said:
DUDE !!!
LIFESAVER
November 23rd, 2009 at 6:21 am -
Saurabh said:
Thank You!!! After almost 6 years working in the industry I decided to write my MBA Entrance exam and your site was extremely helpful.
November 24th, 2009 at 1:32 pm -
JacobF said:
I did 537 (decimal) to base-4, your way. ITS AWESOME THANKS!!
(I think I did it correct)->537/4 ~ 134
->134×4 = 536
->537-536 = 1
(134 fours & 1 ones)
SO FAR
|256|64|16|4|1|
| | | | |1|->134/4 ~ 33
->33×4 = 132
->134-132 = 2
(33 sixteens & 2 fours)
SO FAR
|256|64|16|4|1|
| | | |2|1|->33/4 ~ 8
->8×4 = 32
->33-32 = 1
(8 sixty-fours & 1 sixteen)
SO FAR
|256|64|16|4|1|
| | | 1 |2|1|->8/4 = 2
->2×4 = 8
->8-8 = 0
(2 256s & 0 64s)
SO FAR
|256|64|16|4|1|
| 2 | 0 | 1 |2|1|537(Dec) to Base-4 = 20121
December 12th, 2009 at 3:16 pm -
Elin said:
Good time Harley!
I’m a bit confused about how to convert 5 in decimal to hexadecimal. Thank you for response!December 13th, 2009 at 2:25 pm -
Elin said:
oooo! sorry for question! actually 5 in decimal = 5 in hexadecimal! i had to got the idea earlier! again sorry! thank you for your help and your explanations!
December 13th, 2009 at 2:29 pm -
will said:
I think there is a better and also a easier way to understand this.
If we have a number N based on k, we know we can represent it as:
N= n1*k^m+n2*k^(m-1)+n3*k^(m-2)+…+ni*k^1+nj*k^0,
where n1,n2….ni,nj is in [0,k).
Now we get out one k , we get
N=k(n1*k^(m-1)+…+ni*k^0)+nj
And this nj is the last bit of N. Therefore, we can extract k recursively to get each bit : nj, ni, ..n2, n1December 20th, 2009 at 1:09 am -
Elizabeth said:
I’m trying to figure out how to find the base 10 value of 45 ^ 8. Sorry this is easier than what you are talking about now, but I don’t quite get it. Thanks.
December 24th, 2009 at 12:03 pm -
dharmender said:
10*8/7+20*4/7+2*3/14+6*11/21
pls/. solve this questionJanuary 23rd, 2010 at 5:18 am -
oyun said:
Perhaps you could send me a specific example that I can work out with this method. I will post it and it might be able to help everyone.
January 25th, 2010 at 2:36 pm -
Maga said:
Hi, i know it has been asked before, but do you know how I can convert a fractional decimal, like 1.78 to another base, 60? I can’t find it anywhere! thanx
January 27th, 2010 at 7:37 pm -
elaine said:
Thank you for such a fine explanation of how to do this problem. I have been working on it for 3 days and finally have a clear understanding and feel ready for the test.
February 12th, 2010 at 2:02 am -
maggie said:
Ugh! Thank you so much! What a huge help. I’m studying for the FE and I COULD NOT get it until now.
April 14th, 2010 at 12:19 pm -
Dave said:
Nice write-up. The one thing I never really “got”, and that wasn’t covered very well in my text (like your friend, I’m an adult back in school) was how to convert directly from one base to another without using an “easy” base like decimal or binary as an intermediary. Is there a general method to convert from any base to any other base, directly?
May 14th, 2010 at 9:26 am -
libby said:
hi
could you please teach me how to convert a ternary number to base five without converting back to a decimal number?
thanksMay 15th, 2010 at 6:26 pm -
Samol said:
thanks
June 8th, 2010 at 9:01 am -
Daniel said:
Excellent explanation! It helped me to solve a problem from a recreational math exam. The question is “write 96942 in base-7″. The answer is: 552426.
I’m now programming my calculator to convert from base-10 to base-n.June 21st, 2010 at 2:16 am -
Deeps said:
WoW! It’s of so much help.Well explained and is very fast to do.Thanx a Lot buddy!
July 5th, 2010 at 1:55 am
