why $10$ in any base number system written as $10?$

Question

I am a student trying to write an article in number system

can same one give me an idea

why $10$ in any base number system written as $10$ $?$

Answer 1

I'm not sure what you mean. In 'why 10 in any base number system write as 10' what is the supposed base of each '10'...?

For example in base 2, 'two' is written as 10, but it is not decimal 'ten'; it's two. We use subscript to denote a base (with a common convention that the base in subscript is in decimal), so:

• two in binary is $2 = 10_2$,
• ten in decimal is $10 = 10_{10}$,
• and sixteen in hexadecimal is $16_{10}=10_{16}$.

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EDIT - another answer for the question refinement given in a comment

As for comment 'by the way i didnt mean ten i mean 10':
if you mean literary $10$, that is 'two digits: one and zero' in both cases of '10', then the question means 'why we use 10 to write 10'.

And then the answer is: '10' is written as '10', because what we write with digits one-and-zero is written with two digits, one and zero.

Possibly you meant 'a system base' instead of '10' in the first place? That would make the question:

• why in any number system a system base is written as 10?

Then the answer is: because the number represented with digits $$d_nd_{n-1}\ldots d_1d_0$$ in a system with base $b$ is $$d_n\cdot b^n + d_{n-1}\cdot b^{n-1} + \ldots + d_1\cdot b^1 + d_0\cdot b^0$$ so the number $b$ in system with base $b$ is written as $$b = 1\cdot b^1 + 0\cdot b^0 = 10_b$$

Answer 2

In the number system to base $n$, the number represented by the digits $$a_ra_{r-1}\dots a_1$$ is $$a_rn^{r-1}+a_{r-1}n^{r-2}+\dots a_2n+a_1$$

If we set $a_k=0$ except for $a_2=1$ this sum is equal to $n$ so that $10=n$ in base $n$.

Answer 3

It's just the way the positional system works. For example consider the number $101$ (no base mentioned).

The positional system works so that each digit in the number have different "weight". If it's base ten the first one has weight hundred, the zero has weight ten and the last one has weight one. The number is then the sum of the digits multiplied with it's weight. So $101$ fx in base ten means one hundred plus zero tens plus one.

The weights of the digits is got by multiplying the weight of the next by the base. So for example for base ten the last digit has weight one, the one preceding it would have the weight ten, the one preceding that ten times that which is hundred and so on.

This means that since the last always has the weight one and the one preceding that has the weight of the base (since it should be base times the last one which is one). This means that since that has that weight the number $10$ will be interpreted as one times the base plus zero (ie the base).