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 $10$?

for some reason it won't let me edit the question. I made a mistake and I meant to type 12 and not 1 for the addition answer. — Jessica M., Oct 11 '13 at 04:41
You have a critical misunderstanding of what converting bases is. See answer below for more information. Feel free to ask any questions you have. — MT_, Oct 11 '13 at 04:48

score 2 · Answer 1 · answered Oct 11 '13 at 04:45

2

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.

answered Oct 11 '13 at 04:45

MT_

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9
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Ok thanks. I think I understand why I was confused. But how do you calculate the different bases? So you have the numbers 12, 14, and 18 in base 10. What is the next step to calculate base 8? Do you divide the number ex. 12 by the base? I read some online tutorials but I don't really understand how to actually calculate the base 8 number if you start with a base 10 number. – Jessica M. Oct 11 '13 at 20:49
1

There is no easy formula (none that I know of, at least) to converting bases. What you have to go through is this:
For a number $n$ if you want to convert it to base $k$ you first think, what is the greatest power of $k$ that is less than $n$? Then, subtract that power of $k$ and then ask yourself that question again with the new number, repeating this process.

For example, let's say you want to convert $162$ to base $8$. The greatest power of $8$ would be $8^2 = 64$, and you can subtract that $2$ times to get $162-64-64=44$. Now you can subtract $8$ 5 times to get $4$, so it equals $254$.
– MT_ Oct 11 '13 at 21:19
Ok I'm with you until 162 - 64- 64 = 44. After that I get lost. I don't get the last sentence. I'm a little confused with this point. Is that 8 base 5 in the last sentence? Are you staying with base 8 throughout the conversion? – Jessica M. Oct 12 '13 at 08:55
I'm trying to convert 735 base 10 to base 8. Greatest exponent of 8 that goes into 735 is 8 to 3rd power = 512. So 735 - 512 = 223. (Only able to subtract 512 1 time from 735). 223 - 64 - 64 - 64 = 31 (able to subtract 64 3 times). 31 - 8 - 8 - 8 = 7 (able to subtract 8 3 times) and 7 is left. I think I'm beginning to grasp the concept but is that the right rationale? So you take 1337 is the conversion. Does this look correct. I goggled the conversion and it is 1337 but is that how 1337 is calculated? – Jessica M. Oct 12 '13 at 09:35

score 2 · Answer 2 · answered Oct 11 '13 at 04:46

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.

score 0 · Answer 3 · answered Oct 11 '13 at 04:45

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 $12_{10}$.

score 0 · Answer 4 · answered Oct 11 '13 at 04:46

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.

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

4 Answers4