What is a Number Base?

Date: 04/19/2003 at 13:37:55
From: Cathy
Subject: Base Numbers

The base eight number 7n3 is equal to the base 7 number 1n35.
What digit does n stand for?

I don't know what a base number is.

Date: 04/19/2003 at 22:59:16
From: Doctor Samus
Subject: Re: Base Numbers

Hi Cathy,

In order to understand this question, it is necessary to understand 
what it means for a number to be represented in a particular "base."

Usually, we do math in the "base 10" system. What this means is that 
the right-most digit of a number represents the number of 1's (i.e. 
the number of 10^0's), the next digit over to the left represents the 
number of 10's (i.e. the number of 10^1's), and the nth digit over to 
the left from the right-most digit represents the number of 10^n's. 
For example, for the number 7068 in base 10, we know that the number 
represented is:

    7    1000's (10^3's) +
    0     100's (10^2's) +
    6      10's (10^1's) +
    8       1's (10^0's) = 7068 (in base 10).

In general, a number represented in the "base x" system will have its 
right-most digit represent the number of 1's (since x^0 is always 1), 
and the nth digit over to the left from the right-most digit will 
represent the number of x^n's. For example, if x=4 (as in "base 4"), 
the number 3201 represents:

    3    64's (4^3's) +
    2    16's (4^2's) +
    0     4's (4^1's) +
    1     1's (4^0's) = 225 (in base 10).

For your problem, we know you have 7n3 in base 8, so it represents 
the number:

    7    64's (8^2's) +
    n     8's (8^1's) +
    3     1's (8^0's) = 448 + 8n + 3 = 8n + 451 (in base 10).

You also have the number 1n35 in base 7, so it represents the number:

    1    343's (7^3's) +
    n     49's (7^2's) +
    3      7's (7^1's) +
    5      1's (7^0's) = 343 + 49n + 21 + 5 = 49n + 369 (in base 10).

What would you do now to figure out the value of n?

I hope this helps, but feel free to write back if you need further 
assistance. Thanks for writing in.

- Doctor Samus, The Math Forum
  http://mathforum.org/dr.math/