How to convert from base10 (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 Base10 (decimal) to another base (base3, base5, base6, 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 AHHA! moment. Her teacher was not helping her down the path to her AHHA! moment with base conversion, so I decided that I would work with her over the phone until she reached the AHHA!
After showing her the method that I am going to call the “Base Chart, Breakdown Method“, she finally had her AHHA! 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 Base5.
First, a quick primer on what the 5 in Base5 means:
In Base10 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 onehundreds place)
 10^3 (the onethousands place)
 And so on…
For a different base, such as Base5, 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 twentyfives place)
 5^3 (The onehundredtwentyfives 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 (Base10) to Base5.
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:
5^35^2 5^15^0
    
Which can be also represented by solving the 5^0, 5^1, 5^2, and 5^3 like the chart below:
12525 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 Base10 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 (base10) to Base5. 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 8380=3 so we have 16 (fives) and 3 (ones)
 Write that down
Since we know that the only possible digits for Base5 are 0, 1, 2, 3, and 4 (5 digits, hence Base5), 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 1615=1 so we have 3 (twentyfives) and 1 (five)
 Write that down.
By now you should have the following written down:
 16 (fives) and 3 (ones)
 3 (twentyfives) 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 (twentyfives), 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.
12525 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 (Base10) converted to Base5 is 313.
 83 (Base10) = 313 (Base5)
You can convert from Base5 back to Base10 by using the Base Chart. If the problem had been reversed and you were given 313 (Base5) and you needed to convert it to Base10, you would just fill in the Base Chart, multiply the number on the bottom by the place number, and add the products.
12525 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 Base10 to Base5 conversion is correct.
I hope this post helped you achieve your AHHA! moment with Base Conversion problems. This method can be used to convert Base10 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 Base10 number you are trying to convert.
 Breakdown the Base10 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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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 base16 (not base 10) so you need to add them in base16.
It looks like what you did was take 28+28(base10), which equals 56. 56 does indeed convert to 38 in hexidecimal (base16).
What you need to do (since the 28’s are already base16 numbers) is convert the 28’s to base10 which would be:
28+28 (base16)
40+40 (base10)Now 40+40 = 80 (base10)
So just convert 80 (base10) to a base16 number. I’ll give you three guesses what the base16 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.
109=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 base10 to base4 … 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 1512? 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 base4, your way. ITS AWESOME THANKS!!
(I think I did it correct)>537/4 ~ 134
>134×4 = 536
>537536 = 1
(134 fours & 1 ones)
SO FAR
256641641
    1>134/4 ~ 33
>33×4 = 132
>134132 = 2
(33 sixteens & 2 fours)
SO FAR
256641641
   21>33/4 ~ 8
>8×4 = 32
>3332 = 1
(8 sixtyfours & 1 sixteen)
SO FAR
256641641
   1 21>8/4 = 2
>2×4 = 8
>88 = 0
(2 256s & 0 64s)
SO FAR
256641641
 2  0  1 21537(Dec) to Base4 = 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^(m1)+n3*k^(m2)+…+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^(m1)+…+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 writeup. 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 base7″. The answer is: 552426.
I’m now programming my calculator to convert from base10 to basen.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 
Josephine Liang said:
I can do conversion easily, but cannot seem to grasp using conversion theories and skills to algebraically solve problems such as the following:
When number X is converted to base 7, it becomes a 4 digit number. If the leftmost digit was removed, and written again as the rightmost number, the resulting number thus obtained is twice X. Find decimal representations of all such numbers X
July 9th, 2010 at 2:31 am 
sumaiya said:
can u plz explain the procedure to solve 134 to base 5 in hexa,octa and binary plzzzz
August 11th, 2010 at 10:33 am 
Seb said:
Merci pour m’avoir rémémoré mes cours de math
voici mon prog passe de base 10 to base n. il est en ABAP mais c’est un langage compréhensible pas très loin de l’algorithmique
REPORT ztestspas4.TYPES: coeff TYPE p LENGTH 2.
DATA: res(1024) TYPE c.
data: off type p LENGTH 4.
DATA: n TYPE p LENGTH 10.
DATA: l_valeur TYPE p LENGTH 10.
DATA: l_coeff TYPE p LENGTH 2.
DATA: i TYPE p LENGTH 10.
DATA: tab TYPE coeff OCCURS 0 WITH HEADER LINE.CONSTANTS: base(36) TYPE c VALUE ‘0123456789ABCDEFGHIJKLMNOPQRSTUVWXYZ’.
PARAMETERS p_valeur TYPE n LENGTH 10.
PARAMETERS p_base TYPE n LENGTH 2.INITIALIZATION.
STARTOFSELECTION.
CLEAR: tab[]
, l_valeur.n = LOG( p_valeur ) DIV LOG( p_base ).
l_valeur = p_valeur.
n = n + 1.
DO n TIMES.
l_coeff = l_valeur MOD p_base.
l_valeur = l_valeur DIV p_base.
APPEND l_coeff TO tab.
ENDDO.DESCRIBE TABLE tab LINES n.
i = n.
DO n TIMES.
off = n – i.
READ TABLE tab INDEX i.
res+off(1) = base+tab(1).
i = i – 1.
ENDDO.write res.
ENDOFSELECTION.
August 27th, 2010 at 7:59 am 
Seb said:
il faut peut être juste explicité un peu
p_valeur est une valeur saisie par l’utilisateur en base 10
p_base est la base dans laquelle il faut convertir la valeur saisie
dans ce prog max(p_base)=36 car représentation des nombres convertis avec les chiffres de 0 à 9 et les lettres de A à Z soit 10+26=36 possibilitésLOG est le logarithme népérien (ou normal) en math noté ln
Tab est le tableau des coefficients mutliplicateurs devant chaque puissance de la base de conversion en ordre inverse (ex:40 en base 8 =5*8^1 + 0*8^0=50 => tab={0,5})base+tab(1) signifie en abap : lit 1 caractère dans la chaine base en position tab.
idem res+off(1) ici se lit place dans la chaine de caractère “res” 1 caractère en position off.August 27th, 2010 at 8:16 am 
April Moore said:
I am having trouble with the whole “base” situation. I’m currently an Early Childhood Education Major and this is the stuff we’re going over in my Math2008 class. I can’t seem to grasp the concept. I understand counting in bases, but that’s about it. (For example… Base 5: 1,2,3,4,10,11,12,13,14,20,21,22,23,24,30…)
I am now being asked to do things like adding and subtracting with bases other than 10.
(Example… 24 BASE 5 + 33 BASE 5 = ???)I have spoken with 3 elementary school teachers that didn’t know how to help me, my professor doesn’t have time to help me after class and I’ve watched 3 hours worth of videos online and I STILL DON”T KNOW HOW TO DO IT. I can’t do the conversions either! I have a test in the morning and I’m freaking out.
This website is the only one that has given me a glimmer of hope. THANK YOU and GOD BLESSSeptember 8th, 2010 at 8:33 pm 
Ryan said:
I cant seem to figure out 46 from base 10 to the base sixteen. I found the answer on a site from a chart, its 2E. Please explain.
September 11th, 2010 at 9:15 am 
Steven said:
How do I write the digits 1016 in base 17? I am trying to write 567 in base 17, but I can’t figure out how to write the “16″ digit.
September 23rd, 2010 at 7:44 pm 
Jason said:
Mr. Stagner
Thanks for your wonderful example and explanations. you’re a gift to mankind!
October 25th, 2010 at 2:58 am 
Seasons said:
Maybe you should change the page subject How to convert from base10 (decimal) to any other base to something more better for your blog post you create. I loved the blog post still.
October 30th, 2010 at 4:58 am 
Kasey said:
Thank you sooo much, i can finally get this now after over a half a semester of being lost in my computer class.
November 7th, 2010 at 7:58 pm 
DC said:
thank you so much as a middle schooler this is referably easy to understand as pie because of the way you put it ! Now i understand bases and can be the smarticle of the class. You are the best.
November 13th, 2010 at 2:27 pm 
Christina said:
I am returning to school after many years to get a degree in Computers. Thanks so much for breaking down all the gobbledygook from my numbers and logic class into something easy to understand! Now I feel tons more confident about the Exam!
November 16th, 2010 at 10:25 am 
aj said:
thaaaaaaaaaaaaaaaaank u
may god make ur way easier like u made mine <3 <3December 12th, 2010 at 3:16 am 
Big Al said:
There is a way easier way to do this..
Convert 100 base 10 to base 5
Start by setting up a division symbol upside down with the 5 to left and the 100 in the division slot. Hard to represent here
5 ______ so it would look like this if I could get the underscores to
5 100
so 5 goes into 100 20 times no remainder the 20 goes under the 100 and becomes the next division, the remainder goes out to the right
5  100 0
20Now 5 goes into 20 , 4 times, remainder 0 now it looks like
5  100 0
December 21st, 2010 at 2:02 pm 
Big Al said:
There is a way easier way to do this, goes like this !
Suppose you want to convert 100 Base 10 into base 5.
start with a upside down division symbol _________
Imagine the 5 to left of the vertical bar and the 100 to the right
It would look like5  100 if I could get the underscores in there, they be under the 100, hard to do in text in computer, assume its there.
So now you have 5 goes into 100, 20 times with zero remainder, but the 20 under the 100 and the remainder out to the right,the 20 becomes your next divide
5100 0
20now 5 goes into 20, 4 times, zero remainder, looks like
5100 0
20
4 0now 4 is your next divide except
since 4 is less than five, your done, put the four out to the right as the last remainder , it looks like5100 0
20 0
4 —–> 4Now read your answer bottom to top of the remainders, 400
So 100 base 10 is 400 base 5 . You can check by expanding out
the 400 base 5 as normal 4×5**2 + 0×5**1 + 0×5**0 where
the ** represents raised to that power, WAH LAH !!You should always go to base 10 first before converting to other bases,its just easier.. hope this helps !!
December 21st, 2010 at 2:12 pm 
sofiah said:
Hello,
Having looked everywhere i can not seem to find how to convert a decimal number for example 0.5 into base 8. Please could you help, much appreciated
Sofiah
December 23rd, 2010 at 6:43 am 
Ryan Nuque said:
can someone answer this?
10 base10 convert to base12..
Thanks!January 1st, 2011 at 7:17 am 
alex said:
hello guys can anybody cud help me figure it out to make your formula in C programming.. / anybody here is a programmer.. this formula is easier to understnd ..
January 18th, 2011 at 1:33 am 
alex said:
guys
i mean can anybody here can help me making your formula for conversion of decimal to any base in C programming ..? Can u give me the code..? .. im really looking forward to it.. send me email if you have answers in my facebook acc..
xanderbaring@gmail.com here .. tnxx!
January 18th, 2011 at 1:55 am 
MINTA said:
I am trying to convert 742 base to to a base 5. I got 0422. This is not right. i used your system but I am caught up in the details. Help!!!
January 30th, 2011 at 7:28 pm 
MINTA said:
a student translated a base six quantity and writes 473 (base six)_ as his answer what was his error?
January 30th, 2011 at 7:34 pm 
Rahmat BoakyeAgyemang said:
Hi, kan any1 help me solve this question;
change 0.63479655 in base 16(hex) to base 10(denary). Tanks!February 7th, 2011 at 6:00 am 
Catwoman said:
I dunno but the easiest way to remember these stuff is to know that you just have to keep on dividing the number you need to convert by the base like for example take 65, you need to convert it to the base 9, so you do something like:
65/9=7 with remainder 3
7/9= 0 with remainder 7So you write the remainders you got right from the bottom so simply,
65(Base ten)=73(Base 9)So it goes on the same way what ever you do, take 65 as in base 11,
it’ll be something like,65/11=5 with a remainder of 10
5/11=0 with a remainder of 5So 65(base ten)= 5 10(Base 11)
Right???
March 8th, 2011 at 11:14 am 
apo said:
Your Logic is only correct if the numbers are devides exactly ?
I am trying to make 22 to base 4 but I can’t get 112 out from it (112 is the base 4 converstion)
Look what I do
22/4 = 5.5 (I drop the 0.5)
4/4 = 20
2220 = 2 so we have 2 at the moment the last digitnow 5/4 = 1.75 I dro the 0.75 and i am left with 1
1*4 = 4
54 =1 so for now I have 2 digits 1 and 2lets hit the end
1/4 = 0.25 drop the 0.25
I am left with 0×4
0X4 = 1 ? and am done ? or I go it wrong ?April 13th, 2011 at 8:17 am 
hstagner said:
When you get 0.25, just round it up to 1. Thank you for reading!
April 13th, 2011 at 8:56 am 
Aisha said:
OMG THANK YOU SOOOOOO MUCH!!!!!!!!! yesterday i had math team and we were learning this and i was like clueless!!! thanks now it make a lot of sense but could you email me the way to find what base something is in? thanks!
April 18th, 2011 at 9:43 am 
Carl said:
The article started out good, then strangely presented the breakdown in a confusing way.
Make the base chart into a times table (format with monospace font)
 5^3  5^2  5^1  5^0 
 125  25  5  1 
0  0  0  0  0 
1  125  25  5  1 
2  250  50  10  2 
3  375  75  15  3 
4  500  100  20  4 83 base 10 to base 5…
5^3 = 125. Too big, move over a column to 5^2.
5^2 = 25. How many fit into 83? 3*25=75, so this is our first digit.
8375=8. We now move to 5^1 column since we’re done with 5^2.
5^1=5. How many of these go into 8? Our next digit is 1.
85=3. Move to the last column, 5^0.
5^0=1. How many of these go into 3? Our last digit is 3.83 base 10 = 313 base 5.
Note: Base (anything) represents the number of digits. In this example, 5, the digits are 04 (5 digits). By nature, the base number becomes the first carry, 1 in the ^1 column.
Base 10 = 09, 10(10)=1*10^1 + 0*10^0 = 10
Base 5 = 04, 10(5)=1*5^1 + 0*5^0 = 5
Base 16 = 0F, 10(16) = 1*16^1 + 0*16^0 = 16Converting our example to from base 5 to base 10…
313 = 3*5^2 + 1*5^1 + 3*5^0
= 75 + 5 + 3
= 83Working with decimal points is no different.
 5^3  5^2  5^1  5^0  5^1  5^2 
 125  25  5  1  0.2  0.04 
0  0  0  0  0  0  0 
1  125  25  5  1  0.2  0.04 
2  250  50  10  2  0.4  0.08 
3  375  75  15  3  0.6  0.12 
4  500  100  20  4  0.8  0.16 23.6 (base 10) = 43.3 (base 5)
Converting back to base 10…
= 4*5^1 + 3*5^0 + 3*5^1
= 20 + 3 + .6
= 23.6Hope this makes things easier.
April 23rd, 2011 at 11:29 pm 
Carl said:
22 (base 10) = 112 (base 4)
 4^3  4^2  4^1  4^0  4^1  4^2 
 64  16  4  1  0.25  0.0625 
0  0  0  0  0  0  0 
1  125  16  4  1  0.25  0.0625 
2  250  32  8  2  0.50  0.1250 
3  375  48  12  3  0.75  0.1875 1*4^2 = 16
1*4^1 = 4
2*4^0 = 2
————
22April 23rd, 2011 at 11:45 pm 
Carl said:
742 (base 10) = 10432 (base 5)
 5^4  5^3  5^2  5^1  5^0  5^1  5^2 
 625  125  25  5  1  0.2  0.04 
0  0  0  0  0  0  0  0 
1  625  125  25  5  1  0.2  0.04 
2  1250  250  50  10  2  0.4  0.08 
3  1875  375  75  15  3  0.6  0.12 
4  2500  500  100  20  4  0.8  0.16 1*5^4 = 625
0*5^3 = 0
4*5^2 = 100
3*5^1 = 15
2*5^0 = 2
—————
742April 23rd, 2011 at 11:57 pm 
Carl said:
473 (base 6) is an obvious error.
Base 6 contains digits 05.
7 is not a digit within the base 6 system.April 24th, 2011 at 12:01 am 
Carl said:
10 (base 10) = A (base 12, or in any base system above 10)
Base systems above 10 typically use alpha to represent numbers above 9.
Base 11 = 09, plus A
Base 12 = 09, plus AB
Base 13 = 09, plus AC
…
Base 16 = 09, plus AF (A=10, B=11, …, F=15) 12^1  12^0 
 12  1 
0  0  0 
1  12  1 
2  24  2 
3  36  3 
4  48  4 
5  60  5 
6  72  6 
7  84  7 
8  96  8 
9  108  9 
A  120  10 
B  132  11 April 24th, 2011 at 12:16 am 
Carl said:
That would be
A*12^1 = 10
April 24th, 2011 at 12:18 am 
Carl said:
Base 8, digits 07
 8^1  8^0  8^1 
 8  1  0.125 
0  0  0  0 
1  8  1  0.125 
2  16  2  0.250 
3  24  3  0.375 
4  32  4  0.500 
5  40  5  0.625 
6  48  6  0.750 
7  56  7  0.875 .5 (base 10) = .4 (base
4*8^1 = .5April 24th, 2011 at 12:26 am 
Carl said:
567 (base 10) = 1G6 (base 17)
Base 17 = 09, plus AG
A = 10
B = 11
…
F = 15
G = 16 17^2  17^1  17^0 
 289  17  1 
0  0  0  0 
1  289  17  1 
2  578  34  2 
3  867  51  3 
4  xxx  68  4 
5  xxx  85  5 
6  xxx  102  6 
7  xxx  119  7 
8  xxx  136  8 
9  xxx  153  9 
A  xxx  170  10 
B  xxx  187  11 
C  xxx  204  12 
D  xxx  221  13 
E  xxx  238  14 
F  xxx  255  15 
G  xxx  272  16 1*17^2 = 289
G*17^1 = 272
6*17^0 = 6
—————–
567April 24th, 2011 at 1:19 am 
Carl said:
If you don’t want to make the times table, you can use the base chart with a bit of arithmatic.
In the first example, 83 to base 5…
 5^3  5^2  5^1  5^0 
 125  25  5  1 5^3 (125) is too big
83/(5^2) = 85/25 = 3.don’t care. 3 is our first digit.
833*5^2 = 8375 = 8
8/(5^1) = 8/5 = 1.don’t care. 1 is our second digit.
81*5^1 = 85 = 3
3/(5^0) = 3/1 = 3. 3 is our last digit.April 24th, 2011 at 1:41 am 
Alex said:
Hello Me Stagner,
Can you please tell me whether there is a quick method to convert numbers from one base (3 for example) to another base (2,5,6, etc.) without having to convert the number to base 10? It will really help me, I have a very important exam on Monday.
If it is too complicated to write up here, could you refer me to a website, as despite all my efforts I seem to be failing to find one on the internet…
Thank you! Have a nice day!
May 6th, 2011 at 4:48 am 
And said:
Hi,
I have a question, can a base be decimal?May 7th, 2011 at 6:22 pm 
And said:
Let me clarify, I meant can a base have a decimal in it?
May 7th, 2011 at 6:23 pm 
Sukhvir said:
What is the 16bit 2’complementary binary representation for the decimal number (1987)?
May 29th, 2011 at 10:47 pm 
Bike Storage said:
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?
June 8th, 2011 at 1:07 am 
Jeffry Manhulad said:
Thanks alot for this site!!! I have learned a lot!! from: Philippines
July 3rd, 2011 at 9:24 pm 
allie said:
Hi I need help with this question:
Explain why a number is even if and only if it has an even number of 1s when it is written in base 3?
Thanks!!August 7th, 2011 at 6:29 am 
Tani said:
Can u pls work on this. Convert 356 base 5 to base ten, then back to base 5. Can’t seem 2 get d same answer.
August 19th, 2011 at 2:17 am 
johnguackmbl said:
How would you do this problem?
5/8 in BASE10 –> ________ in BASE2September 1st, 2011 at 10:55 pm 
Ben said:
I will love to get more helps from u.
September 24th, 2011 at 11:49 am 
ASHWINI DAMODAR said:
HEY THANKS A LOT! I WAS TRYING TO LEARN BASE 10 CONVERSION FOR SO LONG, IN BOOKS THEY DIDN XPLAIN AS PROPERLY AS U DID HERE!! THANKS A LOT! YOUR ARTICAL IS VERY EASLIY UNDERSTANDABLE, U SHOLD WRITE A BOOK OR SOMTIN
September 26th, 2011 at 2:58 am 
Kamutungye said:
oohh… yes everything is right here,
October 9th, 2011 at 4:25 pm 
Amelia said:
why does base twelve use 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, T, E these symbols. Whats up with T and E ?
October 10th, 2011 at 10:27 am 
Durre said:
wat is the decimal form of 16/5
November 16th, 2011 at 7:27 pm 
Katie said:
Hey everyone, So I recently discovered a very easy/clean way to convert bases.
So lets just go ahead and do 22 to base 4
1.) Make a base 10 chart ( not neccesary but it helps visualize the numbers more accuratley) Let * mean to the power of
10*2–10*1–10*0
100 10 1
2 22.)Make a Base 4 Chart (Note that you only have to find base 4 factors up to 22…for instance finding the value for 4*6=4096 is a waste of time as you wont be using that number for anything)
4*3–4*2–4*1–4*0
64 16 4 1now you are going to divide.
3.) Start with 64…how many times does 64 go into 22…none so place a zero to hold the palce value.
4*3–4*2–4*1–4*0
64 16 4 1
04.)Now how many 16’s can go into 22? 1. so place a one
4*3–4*2–4*1–4*0
64 16 4 1
0 12216=6 now instead of finding factors that go into 22, you must find the factors that go into 6.
5.) How many 4’s go into 6? 1!! place a one
4*3–4*2–4*1–4*0
64 16 4 1
0 1 1Now subtract to get your last value. 64=2
6.) 4*3–4*2–4*1–4*0
64 16 4 1
0 1 1 27.) AWNSER!!!!! 112 in base 4
I hope this helped everyone as much as it has helped me!! Once you get the hang of it it takes only a few minutes to solve.
November 17th, 2011 at 2:49 am 
Quinn Del Val said:
Wow this was really helpful. My math teacher is pretty smart but not and I got in in class but looking over my notes I had no idea what was going on! I have a big test tomorrow and this really helped me understand the concept. Thanks!
November 17th, 2011 at 10:39 pm