Are arithmetic operations are only defined to base $10$?

Question

I want to know if the simple math operations that we use, per example, $5 + 5$, are only defined on base $10$. If I want to change $5$ to base $3$ and then sum it, or double it, can I do directly $12_{\textrm{base } 3} + 12_{\textrm{base } 3}$?

Answer 1

Based on my understanding of the question, I do not think there are too much magic here. So for naturally defined summation, itself does not rely on the base-representation of numbers, but it just relies on the numbers it tries to sum.

e.g. 2 + 8 = 10 in base 10, it means, starting from number 2, going forward by 8 steps and find 10 there.

If you represent the numbers in base $3$: $2=2_{(3)}$, $8=22_{(3)}$, and $10=101_{(3)}$.

So you get $2_{(3)}+22_{(3)}=101_{(3)}$, so you start from number $2_{(3)}$, go forward by $22_{(3)}$ steps, and arrive at $101_{(3)}$

Thus the definition of summation $a+b$ over natural numbers (in the natural sense) should be - starting from $a$, and you find the $b^{th}$ successor of $a$, regardless of how to represent the numbers.

But I think here you might be talking about the calculation rules instead of the definition for summation. So the set of calculation rules for base 10 is of course different for base 3.

To see this: under base 10, we count to 9, and if we want to count up one more, to increment to the higher position; but by contrast, under base 3, we could only count to 2, and if we want to count up one more, we need to increment to the higher position.

Answer 2

You can use the arithmetic operators with any base of numeration whatsoever, and you can even use it with archaic or nonstandard numeral systems. For example: $$\text V + \text V = \text X$$

The important thing is that you and I, reader and writer or vice-versa, understand what base we're working on.

To that end, you can write something like "in the following section, we'll be using the ternary numeral system," and just go ahead and write 12 + 12 = 101. Look for example at this page from Princeton University on binary addition:

Now that we know binary numbers, we will learn how to add them. Binary addition is much like your normal everyday addition (decimal addition), except that it carries on a value of 2 instead of a value of 10.

For example: in decimal addition, if you add 8 + 2 you get ten, which you write as 10; in the sum this gives a digit 0 and a carry of 1. Something similar happens in binary addition when you add 1 and 1; the result is two (as always), but since two is written as 10 in binary, we get, after summing 1 + 1 in binary, a digit 0 and a carry of 1.

Therefore in binary:

0 + 0 = 0

0 + 1 = 1

1 + 0 = 1

1 + 1 = 10 (which is 0 carry 1)

It would get tiresome if they had to put in a little subscript 2 every time they wanted to put a number in binary in there.

Of course if you want to be extra clear, you can always specify the base in a subscript: $$12_3 + 12_3 = 101_3$$

Apparently there is such a thing as a ternary computer, but most of our computers operate in binary, and they generally convert the results to decimal to display them to us. So there is a difference between doing a calculation in a particular base and displaying it in different base.

The basic principles are the same: you add two positive numbers together, the result is also a positive number, you subtract a number from itself the result is 0, etc.

However, if you depend on tables of addition and multiplication to help you in base 10, then you will need different tables of multiplication to help you in a different base. But the basic principle is the same.

Beware of division, however: $$\frac{1}{3} = 0.3333333333333\ldots$$ But if we're working in duodecimal (base 12), we'd have $$\frac{1}{3} = 0.4.$$ And there are still much subtler issues that can come up.