Why do we order our numbers from most significant digit and not the least significant?

Question

Not sure if this belongs here, but it occurred to me that when I add two numbers, I start adding them from the left to the right. Probably because of the simple fact that I read from left to right, or maybe because it feels natural to repeat myself while adding (in the example below): "5+1.. six hundred...4+3..six hundred seventy... 8+3.. six hundred eighty one".

If I were to add 148 and 533, in my head, I'd go:

• 1+5, makes 6
• 4+3, makes 7
• 8+3, makes 11, let's modify that last one and make that 8 instead

Consider flipping it around. Assume were to write it like this: 841 + 335:

• 8+3, makes 11, in other words, 1 and I'm going to add 1 to the next sum
• 4+3 makes 7, but we have 1 from the last sum, so 8
• 1+5 makes 6

Is there any obvious good (historic?) reason to why we write it the way we do? Are the numbers easier to read from left to right because of that's the way it has always been done, or because of some other reason? Do we ever speak of endianness in other situations than for base 2? For languages where you read from right to left, how does one usually mentally add numbers?

Answer 1

Not a specifically historical reason because I don't know the history, but two plausible practical reasons that come to mind:

• with infinite decimals, there's no last digit to start from
• in everyday situations, the most significant digit is literally the one most significant to us: for example if something costs £$1249.99$ then the $1$ at the beginning is far more important to my ability to afford it than the $9$ at the end. The notation gives us the important information first—it's over £$1000$—then fills in the details.

This contrasts, though, with older phrases like five-and-twenty (and their equivalents still in use in some languages). But maybe that phrase is designed to emphasise twenty. I can't really take that further without making this a linguistics or psychology question.

Answer 2

When working with pencil and paper the advantage of starting at the ones place is that you don't have to change anything you have already written down because of carries. That is the way I was taught in school. When working in my head I tend to go down from the top as you do. I do the same when multiplying, I multiply the two most significant figures, then each most significant by the other next most, and so on, adding carries as I go. One big advantage is if you just want an approximate answer you can stop at the appropriate time.

Answer 3

There is a difference between ordering numbers and operating on them.

When ordering them, the basic operation is (usually) comparing them. Especially when comparing reals, there may be no lowest order digit, so the comparison is best done from the high order digits.

When operating (e.g., add or multiply), we are usually given the complete number and want to obtain the result, so it does not matter if the operation is done from the high digits or the low digits.