converting a number from base $10$ to base $8$

Question

I'm really confused and I can't get the concept. How is $14$ in base $8 = 16$? And how does $8$ in base $8 = 10$?

In case of $14$ isn't $1\times8 + 4\times1 = 12$?

Shouldn't any number in base $8$ be lower than a number in base $10$?

Answer 1

In base $n$, the number $123456$ represents $1(n^5) + 2(n^4) + 3(n^3) + 4(n^2) + 5(n^1) + 6(n^0)$

The problem that you're having is you're converting the other way around.

$14$ in base $10$ is, well, $14$. If you were to convert that to base $8$, you have to find a number $ab$ such that $8a + b = 14$ with $a, b < 7$. What you did was take $14$ as if it were a number in base $8$ and convert it to base $10$.

And to answer your question, no, a number in a lower base should always be "higher" (if interpreted as a higher base). You are showing a huge misunderstanding of what converting bases implies, and if you have any further questions, feel free to ask.

Two equivalent numbers written in different bases have to be exactly that: EQUIVALENT. Converting a number in a base does NOT mean making that number smaller, it means expressing the same numerical value using a different numerical system.

Answer 2

Changing bases is a matter of "repeated subtraction" or "repeated modulus" for showing how a number changes between bases. For $14_{10}=16_{8}$, note that this looks like:

$$1\cdot 10 + 4\cdot 1=14=1\cdot 8 + 6\cdot 1$$

If you convert $14_8$ to base $10$, then your addition is correct for arriving at the value $12_{10}$.

Answer 3

It appears that you’ve misinterpreted your source. What it’s saying is that the number that is written $14$ in base ten is written $16$ in base eight, and that the number that is written $8$ in base ten is written $10$ in base eight. In more compact form: $14_{\text{ten}}=16_{\text{eight}}$ and $8_{\text{ten}}=10_{\text{eight}}$. You’re interchanging the two representations and reading it as if $14$ were the base eight and $16$ the base ten notation.

You're quite right that if $14$ is read as a base eight representation, it’s the representation of $1\cdot8+4\cdot1=12_{\text{ten}}$, the number twelve, but in your source it’s intended to be read as a base ten number.

Answer 4

There are lots of decent online resources for learning how to deal with bases. It's a little tricky to wrap your head around. I found this resource, maybe you'll find it useful.

Given that we're talking about pretty small numbers here, maybe it's a good idea if I just make a table so you can see how it works:

Number in Base 8 | Number in Base 10

1 = one
2 = two
3 = three
4 = four
5 = five
6 = six
7 = seven
10 = eight
11 = nine
12 = ten
13 = eleven
14 = twelve
15 = thirteen
16 = fourteen
17 = fifteen
20 = sixteen

I hope that can help you figure it out.

For the sake of proper attribution, I lifted this table from this site.