So, I'm studying base number addition (AoPS book) and I'm kind of stumped. Any help would be thankful. So the question was:
Perform the following base number addition: $ 1_6 + 5_6 = ? $
So initially, I thought it was 6 but turns out that I'm wrong. Here's my work. So I did $1_6 = 1*6^0$ and $5_6 = 5*6^0$ So I did $(5+1) * 6^0$ and my answer was 6 but it's actually 10. Can someone explain why?
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It's only a question of how one represents numbers. The underlying value, say $8$, can be represented in many ways, whether it's $8_{10}$, or $1000_2$, or $4_{10} + 4_{10}$: the underlying quantity is the same concept of "the number eight".
The "base", or "radix", is the number of distinct symbols you can use, when representing numbers.
For example, decimal (base 10) has 10 symbols: $\{0, 1, 2, 3, 4, 5, 6, 7, 8, 9\}$. Binary (base 2) has 2 symbols $\{0, 1\}$.
You'll notice that no matter the base, the "carry" always happens for powers of the radix, which are always written as $1, 10, 100, 1000$, etc. Each power of the radix looks like what a power of $10$ would look like in base $10$. In particular, the radix is always written as $10_n$ in base $n$.
One way to think about this is with each digit existing on a clock of radix hours. When you hit the "0" hour again, you have a carry, and need to start counting on a new clock (representing a new digit).
Also, you can think of "base 1" (a single symbol), as the way you would count numbers on a dice, or on cards. Every new element needs a new digit.
So what happens in your case ? Well with $5_6 + 1_6$, you run out of symbols on your $6$-hour clock. So you add (carry) a new digit, and get $10_6$, representing a power of the radix, here, $6_{10}^1 = 6_{10}$.
Notice $1+5=6$. But $6_6$ is not the proper way to represent the number! $12346_6$ is not a proper way, just like you can't put a $10$ in one digit in base $10$. If you recall how bases work, if you wanted to represent $6$ in base $6$, this is $\boxed{10_6}$, since $6=1*6^1+0*6^0$.