What is Octagonal Addition?

[chinmayrshah]

I ran through some previous years GATE papers and found one question like this:

 

If 137+276=435, then how much is 731+672?

(a.) 534 (b.) 1403 (c.) 1623 (d.) 1531

 

The answer is c.) 1623 and their explanation is that it is an octagonal addition. I surfed over the net and received unsatisfactory answers that used calculators. Please help here!

 

Thanks in advance!

 

Chinmay

 

This is what I tried. I don't know where I have mistaken, but there is some gross of it, I believe.

S.O.S

 

Thanks in advance!

 

Chinmay!

[Schrödinger's hat]

To other readers:

Explained base conversion and the process of addition and carry in another base in chat. Posting to save someone covering the same ground.

 

For your entertainment, here's Tom Lehrer:

[John Cuthber]

Just curious.

Did they call it octal or octagonal?

[Fuzzwood]

Simple. 7 + 6 doesnt make 13 but 7 + 1 = 10 (because of base 8), 10 + (6-1) = 15. You don't count to 10 going from 0 1 2 3 4 5 6 7 8 9 10, but 0 1 2 3 4 5 6 7 10

Edited January 28, 2012 by Fuzzwood

[chinmayrshah]

John Cuthber: I got that problem solved by Schrodinger's hat. And yes, it is octal and not octahedral or octagonal, unlike what I saw in the book. Apologies! My only question is now how to find out whether the given arithmetic is based on octal calculations?

 

Thanks in advance!

 

Chinmay!

[Schrödinger's hat]

@John: I get the impression from chinmayrshah that someone was referring to it as octagonal. I don't know who though.

 

Answering this here in case the answer is useful to someone else

I have fully understood the octal system but the last thing which I ask is the way to identify them. In the problem which I have asked, there was a very convincing discussion that the numbers were reversed and the option given was fitting there well too. How to judge whether the number is octal then?

That's a good question, I hadn't considered different order of digits.

Let's have a look again at the problem:

If 137+276=435, then how much is 731+672?

(a.) 534 (b.) 1403 (c.) 1623 (d.) 1531

 

Looking at the first sum, we could try the digits being reversed.

Start with 1+2,=3

Do we see a 3 in the left-most place in the answer?

No.

Perhaps the left hand side is reversed and the right hand side isn't? This would be extremely odd, but let's consider it for completeness.

No, that doesn't work either.

So we can reject the idea of it being reversed digits, it didn't work.

(Aside: We could try it with the second sum as well, just for fun. 7+6=13, we have a 3. That fits 1403 and 1623 if the question is reversed and the answer isn't, but we have a carry 7+3+1(carry) gives 11, that doesn't fit either.)

 

We can instead explore the idea of a different (but unknown) radix (ie. binary or octal or something).

My thought process went:

It's not decimal.

No letters, try octal first (also you'd mentioned octagonal, so I thought octal quite likely).

Octal works for the first sum, try the second.

Correct answer.

 

If I'd seen something like 1A34 (or even if I hadn't had the octagonal clue), I'd try hexadecimal first (base 16, you run out of numbers at 9 so the ones after go A B C D E F, F corresponds to the decimal 15). Hex is probably the most common base humans use after decimal (it's good for talking to computers as it's easily converted to/from binary).

 

I'd try these because I already knew that hex and octal are very common in programming, and without further context I thought they'd be the most probable.

 

A more general strategy would go something like this (assuming knowledge of modulo arithmetic, feel free to ask if you don't understand):

The word radix means the base of your arithmetic, decimal is radix 10, binary radix 2 etc.

Look at:

137+276=435

Assume left is the most significant digit.

Assume the digits have the same meaning as they do in decimal (ie. 5 is this many ..... )

Assume + is addition.

 

6+7 = 5 (mod radix)

I also know that 6+7 is less than twice the radix, because 6 and 7 individually must be less than the radix.

So either radix + 5 = 6 + 7 or 5 = 6+7

I know 5 is not 6+7, so take the first one, then rearrange to get

radix = 6 - 5 + 7 = 8

 

I know there is a carry, so try the next digit

3 + 7 + 1 (mod 8) = 3

Carry again

1 + 2 + 1 (mod 8) = 4

435

 

 

So either one of my assumptions is false, or it is octal.

[doG]

John Cuthber: I got that problem solved by Schrodinger's hat. And yes, it is octal and not octahedral or octagonal, unlike what I saw in the book. Apologies! My only question is now how to find out whether the given arithmetic is based on octal calculations?

 

Thanks in advance!

 

Chinmay!

Google 'octal prefixes'. There have been a number of them used to indicate octal notation. In reality you won't encounter it much any more. Its use is from the 8 bit computing era. Hex is now more practical with longer binary words.

[DrRocket]

Just curious.

Did they call it octal or octagonal?

 

It is usually just called base 8, though in parallel with the hexadecimal arithmetic used in computersit is sometimes called octadecimal.

 

In fact, outside of applications in computers, there is no good reason for working with arithmetic in unusual bases and it is usually not called anything at all.

 

The question given in the OP is rather quirky and of zero mathematical interest -- parlor tricks at best.

Edited January 29, 2012 by DrRocket

[chinmayrshah]

The problem is that here in India, we have an entrance named GATE for pursuing masters. The previous years papers had this sum. It has nothing to do with Computer/IT/Electronic Engineering. I myself am pursuing Civil, but this octal question was nevertheless asked. That's it. The importance of this was a seat in a good institute! I know that hex is now prevalent but I was keen to know how to figure out such mind numbing questions when you have less that 2 minutes to solve such questions! Thanks to all of my ScienceForum folks! You all are amazing and very helpful!

 

Chinmay!

[Schrödinger's hat]

In fact, outside of applications in computers, there is no good reason for working with arithmetic in unusual bases and it is usually not called anything at all.

 

The question given in the OP is rather quirky and of zero mathematical interest -- parlor tricks at best.

 

'applications in computers' is an awfully wide field to dismiss out of hand, it includes a lot of computer science (including mathematics) as well as the simple grunt-work of sitting down to program something in C or assembly.

It has a lot of practical interest if you're trying to operate a debugger.

It also has some pedagogical interest. By understanding that radix is an arbitrary choice, you come to better understand decimal arithmetic.

It can also help to drive home the distinction between repeating decimals and irrational numbers. Many students lump these together as 'number that keeps going'.

If you're not thinking in terms of using different bases you'll also have a devil of a time programming anything using large numbers that works efficiently (or even understanding the explanations of said things).

 

There are tricks from number theory using exotic bases, such as a primorial or other variable radix, or base e.

Sometimes they can result in tighter data packing or faster algorithms.

 

Granted, looking at a sum and inferring that it is in another base is an exercise which has little mathematical value, but then so is finding the derivatives of a bunch of polynomials along with almost everything else encountered in a highschool or college exam.

Sadly 'get the students to do a bunch of related, but worthless calculations and hope they think about the concepts involved' seems to be the main way of teaching maths these days.

Edited January 30, 2012 by Schrödinger's hat