# Why do we write numbers with decreasing place values?

This question came up while teaching ~16 year olds binary numbers. Why do place values increase to the left and not the other way round?

• Apropos as you are talking about binary numbers, computer designers have made both choices (and others as well) at various times: en.wikipedia.org/wiki/Endianness – Adam Nov 19 '20 at 23:27
• It’s even weirder when you consider that Arabs write everything right-to-left except numerals. – gen-ℤ ready to perish Nov 20 '20 at 5:11
• Are you aware of the existence of p-adic numbers theory en.wikipedia.org/wiki/P-adic_number for which calculations are done indeed from right to left ? But take care, it can look at first sight as counter-intuitive ! – Jean Marie Becker Nov 20 '20 at 10:29
• @user1027 I studied Arabic as a third language; I know what I’m talking about. Read here or try it out for yourself on an Arabic keyboard. – gen-ℤ ready to perish Nov 20 '20 at 17:56
• @Jean Marie Becker - APL ( A Programing Language ) implements all calculations Right to Left with no precedence. It's actually very effective but most people hate it. en.wikipedia.org/wiki/APL_(programming_language) – user10216038 Nov 27 '20 at 17:55

This is probably better suited to the History of Science and Mathematics SE, but I'll take a semi-informed stab at it.

Since we read from left to right, placing the most significant digits in that same order give us the best opportunity to quickly comprehend the magnitude of the number. For instance, the speed of light is 299,792,458 meters per second. I think your brain is much more equipped to jump to "Oh, around 300 million" by seeing the most significant digits and the number of commas than if the number were given as 854,297,992.

We obviously had a lot of history before the Hindu-Arabic numbers came into existence. But even things like Roman numerals would give you the ability to just think about the degree of precision you wanted by ignoring the "digits" like I and V and X and L that you felt like ignoring.

• What about it coming from how we say the numbers? one thousand two hundred thirty-four makes sense as 1000 200 30 4, which collapses to 1234. – IronEagle Nov 20 '20 at 2:00
• Your explanation about seeing the largest place values first (to whatever extent) doesn’t make sense considering that these numerals were adopted from speakers of a language that is written right-to-left. It’s serendipitous, but it’s not the actual reason. – gen-ℤ ready to perish Nov 20 '20 at 5:13
• @IronEagle: Once, we said "four-and-thirty" instead of "thirty-four". We still have thirteen, fourteen, etc. – J W Nov 20 '20 at 7:47
• I would think that it would be the number of digits, not the digits themselves, which would tell you most of the information about the magnitude. – Adam Nov 23 '20 at 16:09
• @gen-ℤreadytoperish: I think "Arabic" numerals with place value actually originated in the Brahmi script, which is written left to right. – Ben Crowell Nov 27 '20 at 1:49

Our numbering system comes from copying the Hindu-Arabic numbering system. They write from right to left, so in their system, they start from the least significant digit. However, taking over the ordering as-is was easier, otherwise operations would need to have been adapted. So we write numerals in the same order even if we generally write text in the opposite direction.

Note, that some operations are more "natural" in one order, others in the opposite. Comparison and division are naturally done from the most significant digit, addition and multiplication from the least significant digit.

Also note, that this is connected to endianness which is relevant for computer systems.

• The first paragraph is interesting, but needs reference. – Gerald Edgar Dec 13 '20 at 11:59
• endianness refers to byte addressing not bit order power interpretation. – user10216038 Dec 13 '20 at 16:35

It makes perfect sense if you consider the full range of real numbers instead of just integers. In any base (decimal, binary, hex, whatever) the radix point (decimal point in base 10) is in the middle with exponential powers increasing positively to the left and negatively to the right.

For any base ß :

ßⁿ + ßⁿ⁻¹ + ... + ß¹ + ß⁰ . ß⁻¹ + ß⁻² + ... + ß⁻ⁿ


This works in binary as well as any other number base. In fact it used to be common to operate on scaled binary like this in lieu of floating point.

• From this argument it would actually make more sense the other way, so that we have ...,-2,-1,0,1,2,... rather than the opposite. – BKE Dec 13 '20 at 9:22
• I don't see how this fact is an explanation for why we write numbers this way. – Jasper Dec 14 '20 at 13:13