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Adding in Base 9 and Base 5


Date: 10/21/97 at 10:13:13
From: Costa, Jose
Subject: Rule for adding two numbers in base 9 

I'm a Computer Programmer and am aware of the binary system as well as 
hexadecimal. My 8-year-old son is in 3rd grade and is learning how to 
add other bases other than 10. I have already found useful 
information, but I need to know more about some methods.

Observation:

  - the number 243 in base 10 decomposed is:
      2x10^2 + 4x10^1 + 3x10^0

  - the same number in base 9 decomposed is:
      2x9^2 + 4x9^1 + 3x9^0
    ( I think this is correct! )

Givin the following addition:

    base 10        base 9

     4687           4687
   + 8456         + 8456
   ------         ------
    13143          ?????   What is the rule for the addition?

Question 1: If we apply the decomposition rule for the number 13143
            to base 9, would the result be the same as if we sum the
            operators 4687 and 8456 using base 9?

Question 2: Is there a general rule for adding numbers in any base?

I need to help him to understand how this works.
Thank you very much in advance.
Congratulations for this site.

Best regards.
Costa, Jose - Portugal


Date: 10/21/97 at 16:10:15
From: Doctor Rob
Subject: Re: Rule for adding two numbers in base 9 

You have the first part correct. Good thinking! Now to answer your
questions.

1. No, applying the decomposition rule for 13143 to base 9 would not 
   give the same result as adding 4687 and 8456 using base 9.

2. Yes, this is how it goes.
  a. Start with the rightmost column of digits (be sure the numbers 
     are properly aligned with units digits under each other).
  b. Begin with carry zero.
  c. Add the digits in the current column plus the carry.
  d. If the sum is less than the base, put it down at the bottom and 
     set the carry to zero.  If it is not less than the base, subtract 
     the base, put down the result, and set the carry to one.
  e. If you are not out of columns, move to the next one to the left, 
     and go back to step c above.
  f. If the carry is not zero, write it down as the leftmost digit of 
     the sum.
  g. Stop.

In your example, the work would look like this:

   4687
 + 8456
 ------
  ?????

7 + 6 = 13 > 9, 13 - 9 = 4, write this digit down, with carry of 1.

     1
   4687
 + 8456
 ------
      4

1 + 8 + 5 = 14 > 9, 14 - 9 = 5, write down 5, carry 1.

    11
   4687
 + 8456
 ------
     54

1 + 6 + 4 = 11 > 9, 11 - 9 = 2, write down 2, carry 1.

   111
   4687
 + 8456
 ------
    254

1 + 4 + 8 = 13 > 9, 13 - 9 = 4, write down 4, carry 1.

  1111
   4687
 + 8456
 ------
   4254

Write down the carry, 1.

  1111
   4687
 + 8456
 ------
  14254

Stop, you are done.

The answer is 14254 (base 9), or 1*9^4 + 4*9^3 + 2*9^2 + 5*9 + 4, 
which equals 9688 (base 10). Notice that this is different from 13143 
(base 9), which equals 8868 (base 10).

-Doctor Rob,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/   


Date: 10/29/97 at 10:47:58
From: Costa, Jose
Subject: Adding in base 5

Dear Dr.Math,

I recently asked you a question about base 9. The question was clearly 
answered by Dr. Rob. Now I have another question. Consider the 
following:

  4687  base 10
+ 8456
------
 13143   adding the units we get 7+6 = 13 and the result
         is 13 - 10(base) = 3 and we borrow 1 to the ten's.
         If we're in base 9 it would be 13 - 9(base), right?

The same addition in base 5 should be:

  4687  base 5
+ 8456
------
     ?  adding the units we get 7+6 = 13 and the result
        is 13 - 5(base) = 8 (8 is greater than the own base)

Question 1: What should we carry in this case? Should we borrow
            more than 1?

Question 2: Is it logical to say that the result should never 
            exceed the figures within the base range?

Looking forward to hearing from you.
Best regards.


Date: 11/04/97 at 13:48:52
From: Doctor Pipe
Subject: Re: Adding in base 5

Jose,

The numbers 4687 and 8456 are not base 5 numbers. A base 5 number will 
not contain any digits that are 5 or greater. 4687 and 8456 could be 
base 9 or base 10 numbers but not base 8 or base 7 or base 6 and so 
on. Your question concerns base 5. so let's use that as an example.

When I write a number like this: b^n, I mean the base b to the nth 
power.  For example, 5^2 is 5 to the second power, or 5 squared, which 
is 5 x 5 or 25.

In the decimal, or base 10, system, a three-digit number contains the 
hundreds, tens and units positions - or 10^2, 10^1, and 10^0.  The 
decimal number 527 is interpreted as:

     (5 x 10^2) + (2 x 10^1) + (7 x 10^0)
   = (5 x 100) + (2 x 10) + (7 x 1)
   = 500 + 20 + 7
   = 527

A three-digit number in base 5 contains the 5^2, 5^1, and 5^0 
positions. The base 5 number 243 is not two hundred forty-three; 
instead, it is two four three base 5. But we can convert it to a 
decimal number:

     (2 x 5^2) + (4 x 5^1) + (3 x 5^0)
   = (2 x 25) + (4 x 5) + (3 x 1)
   = 50 + 20 + 3
   = 73

So 243 base 5 is equal to 73 base 10.

Why is the digit 5 invalid in a base 5 number? Look at the following 
two columns. The left column is base 5 and the right column is 
base 10:

      1      1
      2      2
      3      3
      4      4
     10      5     - the base 5 number is: (1 x 5^1) + (0 x 5^0)
     11      6
     12      7
     13      8
     14      9
     20     10     - the base 5 number is: (2 x 5^1) + (0 x 5^0)
     21     11
     22     12
     23     13
     24     14
     30     15     - the base 5 number is: (3 x 5^1) + (0 x 5^0)
     31     16
     32     17
     33     18
     34     19
     40     20     - the base 5 number is: (4 x 5^1) + (0 x 5^0)
     41     21
     42     22
     43     23
     44     24
    100     25     - the base 5 number is: (1 x 5^2) + (0 x 5^1) 
                     + (0 x 5^0)
 
In a base 10 number, there is no single digit that means ten of 
something; in a base 5 number, there is no single digit that means 
five of something. When you have five of something, you write 0 and 
carry a one to the left.

So, when you add two base 5 numbers, you carry "fives" to the left in 
the same way that you carry "tens" to the left in the decimal system.

  1243  base 5
+ 4243  base 5
------
 11041  adding the units we get 3+3 = 11 base 5; write down 1 
        and carry 1

        to the 5^1 column.  1+4+4=14 base 5; write down 4 and carry 1
        to the 5^2 column.  1+2+2=10 base 5; write down 0 and carry 1
        to the 5^3 column.  1+1+4=11 base 5; write down 11.

I hope that this helps!

-Doctor Pipe,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/   
    
Associated Topics:
Elementary Number Sense/About Numbers
High School Number Theory
Middle School Number Sense/About Numbers

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