# Number systems and binary codes

1. Jun 15, 2015

### ranju

1. The problem statement, all variables and given/known data
Given (135)x + (144)x = (323)x. What is the value of base x?
Now we can find out the value of x by simply converting to decimal both the sides , form a quadratic equation and solve but once in a random lecture our prof. taught us a kind of short method to solve this in less time followed by " here with base is unknown 5+4 is giving 3 , now 5+4 is 9 so we'll write digits from 0-8 and then further repeat it by adding the same with carry until we get 3.. See I am not sure that what I stated is correct.. I maybe skipping somet data.. So , I just wanted to know about this short method
what actually this method is like??

2. Relevant equations

3. The attempt at a solution
In one method we simply convert in quadratic as : (1*x2 +3*x+5*x0) + (1*x2+4*x+4*x0) = 3*x2+2*x+3*x0
and x will come out to be 3

2. Jun 15, 2015

### RyanH42

First thing you should consider that the base cannot be smaller then "inside"numbers.Example $(1204)_x$ here x must be higher then 4, $x>4$.You cannot write $(143)_3$
Here the rule If you add 5+4 you get 9 but 9 is here 3 so.You need to think this way.First consider the rule which I wrote above Second theres a number ( which here x) , If I do this (9/x) remaining will be 3.

3. Jun 15, 2015

### RyanH42

Here x will not come out 3.The way of thinking is true.I mean equation is true but result is false

4. Jun 15, 2015

### ranju

I didn't get this , what is ths logic??

5. Jun 15, 2015

### Staff: Mentor

When the least significant digit of each is summed we know the result is nine and if we were working to base ten
we would write the resut as digit 9. But we are working in a base where this result is written as a 3.

So what do you conclude is the base?

6. Jun 15, 2015

### ranju

but how to conclude about the base in this method??

7. Jun 15, 2015

### phinds

Do you understand how positional notation and radix number systems work? If you don't have a solid grasp on that, you should study it before attempting this technique.

8. Jun 15, 2015

### Staff: Mentor

When you can't see a clear path, be adventurous: try various possibilities in turn and something will eventually leap out at you.

In the case in point, try ALL possible bases, one at a time.

9. Jun 15, 2015

### ranju

Yes I am quite familiar with it.. and I have solved it accordingly but about the short method I am confused..

10. Jun 15, 2015

### phinds

Your statement in post #6 suggests that this is not true.

11. Jun 15, 2015

### ranju

see about that radix number system , I have done with the conventional method as I stated in my question..but in the short method the picture is different ..if 5+5 =4 and we are getting 3 for base x in the result , how to take the possibilities that's the problem..!!

12. Jun 15, 2015

### ranju

How to try the possibilities ..?? what I am seeing your answer is to compare both sides ..is it so??

13. Jun 15, 2015

### Staff: Mentor

Here's a three digit number: 1 3 5
I haven't told you the base it is written in. ( I'll give you a hint: it is not binary!)
What possible bases might it have? List some of them.

14. Jun 15, 2015

### phinds

Huh? Where did that set of numbers come from?

15. Jun 15, 2015

### ranju

the base will be any number greater than 5..!!

16. Jun 15, 2015

### ranju

sorry its 5+4 .. these are the LSBs of the given problem..!!

17. Jun 15, 2015

### Staff: Mentor

Right!

So work your way up through the first dozen of these possibilities, applying it to the arithmetic expression in your first post.

18. Jun 15, 2015

### phinds

Right. So now we're back to NascentOxygen's question in post #5, which you never answered, which led me to believe that you don't have a good understanding of positional notation and radix number systems. If you did have that understanding, you would not have even asked the question, so I again suggest that you go back to the basics and learn them before trying to apply them.

Look, I'm not trying to insult you or give you a hard time. We see this all the time here on the forum. Someone feels that they have a good grasp of particular material but their questions make it clear that this is not the case. In such cases, I always feel that it's a good idea for them to go back to the basics.

If you don't want to do that then NascentOxygen has given you enough hints to lead you to the answer for this particular problem and ones similar to it (but possibly not enough to generalize to all such radix number system problems, which is why I suggest going back to the basics).

Last edited: Jun 15, 2015
19. Jun 15, 2015

### ranju

that's the problem actually I tried this thing..if we see as a whole all and all base 6 suits it..b
it will be better if you just have a look..I answered his question...
And by the way ..the doubt is clear by now to me...

20. Jun 15, 2015

### Staff: Mentor

So you are all sorted now? Make up some exercises for yourself as practice, so you don't forget again.

Try this quick one: $\mathrm {\ \left(523\right)_x\ +\ \left(117\right)_x\ =\ (641)_x}$

Determine x.

Be ambitious, try some with a hexadecimal base.