base-26 (alphabet)

Question

I need help converting the number 3,209,386 to a base-26 number, writing my answer using letters as our digits. Also I would like an explanation of how to do this. Thank You!

Answer 1

You can convert any number from base 10 to another base by repeatedly dividing it by the new base and keeping track of the remainders.  To start with an easy example, I will convert 10 into binary (base 2).

 

10/2 = 5 R0

5/2  =  2 R1

2/2  =  1 R0

1/2  =  0 R1

 

Once you get to zero, you simply read the column of remainders from the bottom to the top to get your final answer, which in this case is 1010.

 

Moving on to the big problem, we can start by dividing 3209386 by 26.  The answer is 123437 with remainder 24.  Since we are expressing the digits of the solution by letters, we can use the 25th letter of the alphabet, y.  (We use the 25th letter rather than the 24th because we are using a=0, b=1, ..., z=25).  Then we can continue as above.

 

3209386/26 = 123437 R24 y

123437/26  =  4747    R15 p

4747/26     =  182      R15 p

182/26      =   7         R0   a

7/26         =   0         R7   h

 

Thus, the answer is happy.

Answer 2

You can use letters to signify numbers as a single character where the number is >9, so 10 would be A and 15 would be F, because F is the 6th letter of the alphabet, and 9 + 6 = 15.

 

So we convert from powers of 10 (10⁰,10¹,10² etc.) to powers of 26:

26⁰, 26¹, 26², 26³, 26⁴  = 1, 26, 676, 17576, 456976, etc.

 

Dividing 3209386 successively by 26⁴, giving 70 in the ten-thousands place, then the remainder of that by 26³ , giving 0 in the thousands place, and 15 in the hundreds place, and then also 15 in the tens place, and 24 in the ones place.

 

So this looks like 70  0  15  15  24

 

So this becomes  70FF0   (since 24 is <26, then the ones column = 0) 

 

Answer 3

base 26 has the following values:26^0, 26^1, 26^2, 26^3, 26^4, 26^5, 26^6, ...

 

26^0=1, 26^1=26, 26^2=676, 26^3=17,576, 26^4=456,976, 26^5=11,881,376

you see that we only need 5 places

 

456,976     17,576,     676     26     1

divide 3,209,386 by 456,976 to get 7 remainder 10,554

10,554 is smaller than 17,576 so a 0 goes in that place

10,554 divided by 676=15 remainder 414

414 divided by 26=15 remainder 24

the 24 goes in the ones place

the answer is:

7   0   15   15   24 base (26)