2
$\begingroup$

I'm trying to convert from base $x$ to base $y$, but am having trouble understanding why the following method only works when converting to base $10$.

Take for instance the number $2132$ (base $4$). I can convert it to base $10$ the following way:

$2*4^3 + 1*4^2 + 3*4^1 + 2*4^0 = 158$

So that means $2132$ (base $4$) = $158$ (base $10$).

Now what if I want to convert the same number, $2132$ (base $4$) to base $6$? Why can't I do the same method? Example: $2*4^3 + 1*4^2 + 3*4^1 + 2*4^0 = 158$

Why is this method specific to base $10$ only?

I know that I can use a different method to convert from base $4$ to base $6$, but I'm not sure why base $10$ has this method that no other base can use?

$\endgroup$
  • 3
    $\begingroup$ The problem relies in how you define the sum $+$ and the product (exponentiation is also a product). Of course the usual $+,\times $ work only for base $10$ and not for other bases. $\endgroup$ – b00n heT Jul 12 '16 at 21:11
  • 1
    $\begingroup$ You calculate the sum in the normal way, this is in base $10$. You can convert this number ($158$) into base $6$ and you are done. $\endgroup$ – Peter Jul 12 '16 at 21:15
  • $\begingroup$ Thanks for the help. I should have realized I was multiplying in base 10. This helped me realize that: mathforum.org/dr.math/faq/basetables.html $\endgroup$ – Esand Jul 12 '16 at 21:28
2
$\begingroup$

Take for instance the number $2132$ (base $4$). I can convert it to base $10$ the following way:

$2*4^3 + 1*4^2 + 3*4^1 + 2*4^0 = 158$

So that means $2132$ (base $4$) = $158$ (base $10$).

Now what if I want to convert the same number, $2132$ (base $4$) to base $6$? Why can't I do the same method? Example: $2*4^3 + 1*4^2 + 3*4^1 + 2*4^0 = 158$

Why is this method specific to base $10$ only?

Because the basic mathematics for arithmetic you have learned in school (addition table, multiplication table etc.) is the maths for base $10$.

E.g. $3 \cdot 4^1 = 3 \cdot 4 = (12)_{10} = (C)_{16} = (1100)_2$ etc.

Your method is not general purpose, it is only valid to convert to a base $10$ (decimal) representation.

$\endgroup$
1
$\begingroup$

You got the number to decimal, which is good.

Now just take it into base $6$.

$6^3 = 216$ is greater than $158$ so we need just three digits.

$6^2 \times 4$ is $144$, leaving $14$. $6 \times 2$ is $12$, leaving $2$, and we're done:

$$2132_4 = 158_{10} = 422_6.$$

Now, if you wanted to work in base $6$, you could, but then you're just converting each number along the way (which, by the way, is what you do when you convert to base $10$ first!):

$$1000_4 = 144_6, 100_4 = 24_6, 10_4 = 4_6, 1_4 = 1_6$$

Then, have at it!

$$2 \times 144_6 = 332_6, 1 \times 24_6 = 24_6, 3 \times 4_6 = 20_6, 2 \times 1_6 = 2_6$$ $$332_6 + 24_6 + 20_6 + 2_6 = 422_6.$$

There's really no difference. We're all just so familiar with decimal that it seems different.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.