We’re writing numbers wrong. We write “365” starting with the most significant digit of “3” (hundred). The “biggest number on the left” rule is both algorithmically bad and clashes with how humans intuitively represent numbers in their minds. I propose an innocent and totally practical fix: flip the written order of all numbers, writing “↗563” instead of “365.” I analyze the implications of this change as they propagate through our language and thought.
Read this article in a prettier form on my website.
If I’m writing “three hundred and sixty-five”, “365” becomes “↗563”, with the “↗” read as “flip.” Likewise, “21,514” becomes “↗415,12.” As you move right (→), the each digit’s magnitude goes up (↑). If you’re writing an expression with multiple numbers, just include it at the beginning (not before every number): “50 + 2” becomes “↗05 + 2”.
If somehow this system were ever adopted, we would need to preface every relevant expression with the up-right arrow. That sucks, but otherwise we couldn’t tell if the author was using the old system or the new one.
I have no illusions: this system will not be adopted anytime soon, and for good reason. The switching cost would be large and the benefits minor. If you were going to swap systems, start with getting the US off of the Imperial system and onto metric. Setting aside practicality, the fact remains: English writes its numbers backwards.
Prior work. I am apparently not the first person to have this thought. @lsusr wrote a short note titled “We write numbers backward”. However, they promote the misconception that Arabic numerals are backwards in English because Arabic reads right-to-left. I thought that at first too, but it’s not true; I explain why. I also found a tiny HackerNews thread whose original poster promotes the same misconception.
For people who are adults today, switching from the current system would be cursed. I will generally imagine that people had grown up with this system.
Some of the benefits apply under a serial computation model (reading one digit
at a time), which is not usually how people read. Other benefits apply under realistic human reading conditions.
Simplifies long addition and multiplication
Imagine you have a number in mind (like “1,000”) and you want to add another number to it. You’re serially processing the other number one digit at a time. If you’ve received the prefix “I have $320”, you can’t start adding the number to 1,000 because you don’t know what place the “3”, “2”, and “0” correspond to. If I instead write “I have ↗023 dollars”, you can perform long addition as you process each digit.
Since the pronunciation of “↗023” should be “twenty and three hundred”, the primary real-life speed-up to long addition would be in spoken English: adding a medium-length number as you hear its increasingly large components. You would propagate carries in real time. In the current system, you cannot finalize any high-order digit in the result until all lower-order digits have been processed and their carries accounted for.
Likewise, flipped-number (“little endian”) algorithms are slightly more efficient at e.g. long addition.
Endianness in computer science
I’m dancing around a classic “holy war” in computer science: should 32-bit integers start with their least significant or their most significant byte? In some ways, I have replayed parts of this debate, but in the arena of human communication instead of synthetic computation.
Big-endian: The integer’s most significant byte (the “big end”) is stored at the lowest memory address. Since we write integers with the “biggest” digit first (the “3” in “365”), English’s current system could be called “big-endian.”
Little-endian: The integer’s least significant byte (the “little end”) is stored at the lowest memory address. Little-endian implementations are often slightly more efficient. My proposal argues for little-endian notation in written English.
To read more about these ideas, see Understanding Big and Little Endian Byte Order.
Aligns with the direction of reading
In real life, people look at the printed page and see the entire number all at once. Imagine it… you’re reading left-to-right, you come across a long number (e.g. 521,300,421,503), and then your eyes flick to the right end of the number and begin scanning left. Err… why are we doing that?
| Kinda like having a single paragraph which is aligned to the right. That paragraph isn’t impossible to read, but it’s out-of-place. |
The first digits are informative on their own
Here’s another serial processing benefit. Suppose you’re processing a string of text one character at a time. As you receive each character, your knowledge of the sentence looks like:
- “I have $”
- “I have $3”
- “I have $32”
- “I have $320”
What comes next? Maybe the full sentence is “I have $320.”, or maybe it’s “I have $320,000”.
More importantly, by this point in time, what do we actually know about the number? We know it “begins” with the digits “320”. That… doesn’t actually tell us much. It has a “3”? Three of what, exactly? Is the number big or is it small? Is the number even? Who even knows!
Spoken English partially solves this ambiguity. A speaker doesn’t say “three two zero”, they say “three hundred and twenty.” You are quickly given the information that there are three hundreds – not just three somethings of unknown magnitude.
Better aligns with human intuition
Mental number line: small on the left, big on the right
In our current writing system, the “biggest” digit is on the left. That’s bad, because we associate left with smaller.
Subjects compared two-digit target numbers to a fixed standard number, 65. For one group of subjects, the larger response was assigned to the right-hand key and the smaller to the left-hand key… the reverse assignment was used for the larger left group… The larger right group responded faster on average than the larger left group.
— The Mental Representation of Parity and Number Magnitude
Our current number system fights our mental number line. The most significant digit is on the left (“3” hundred in “365”) and the numbers get “smaller” as you read to the right – but that’s intuitively the “bigness” direction! We’re so used to this mismatch that we don’t notice it anymore.
In contrast, flipped numbers are internally congruent with the mental number line. In “↗563”, the value of the components increases from left to right: “5” -> “60” -> “300”. Thus I align the direction of reading, the significance of digits, the spoken order of components, and the mental number line. Children would learn a single unified rule: bigness is to the right.
We probably associate “right” with “big” because we read from left to right
Dehaene et al. (1993) found that Iranian subjects (who write right-to-left) displayed no or reversed effects.
The organization of Western writing system has pervasive consequences on the everyday use of numbers. Whenever a series of numbers is written down, small numbers appear first in the sequence; hence, they are located to the left of larger numbers. In this manner a left-to-right organization is imposed on numbers on rulers, calendars, mathematical diagrams, library bookshelves, floor signals above elevator doors, typewriter or computer keyboards, and so on.
How does immersion in this left-to-right-oriented environment shape spatial conceptualization of numbers? American children tend to explore sets of objects from left to right, whereas the converse is true of Israeli children… This is likely to become the order in which they normally count a set.
— The Mental Representation of Parity and Number Magnitude
People would learn to estimate magnitude at a glance
Humans don’t process numbers a single numeral at a time. There are two main ways to read a number.
The exact value
Consider 3,124,203,346 (or ↗643,302,421,3). Suppose we don’t just care about the rough magnitude, but about its exact value. In our current system, you have to count the number of digits in a large number – reading to the right – and then jump back to the beginning of the number in order to read off its exact value. For example, you only know to say “3 billion” because you count the number of digits (or perhaps the number of comma-separated groups). You read to the end and then jump back to read it again – an extra eye movement.
In contrast, the flipped number system works perfectly: you start reading on the left and process each digit one at a time. You gain information immediately. The flipped number system leads to smoother reading.
The rough magnitude
When reading “320,000”, your visual system perceives the entire word at once and you quickly grasp the magnitude of the number. The most significant digit is on the left (e.g. the “3” in “320,000”). These two facts establish the important information: the rough magnitude (“three hundred thousand” in 320,000).
In contrast, when reading “↗000,023”, you first land upon the “↗” and then the “0”. As before, you immediately grasp that this number is in the hundreds of thousands. However, you have to move your eye again over to the right-most digit (“3”) in order to know how many hundred thousand. The flipped number system apparently complicates magnitude estimation.
However, on further thought, the situation looks less problematic. Yes, the flipped system complicates magnitude estimation for folks who grew up with the current system. But if you had grown up reading flipped numbers, might you not read seamlessly? In search of magnitude information, would your eyes not be trained to jump from the previous word directly to the “3” at the end of “↗000,023”?
For example, when reading “↗000,000,05”, a native reader wouldn’t count the zeroes. Their eyes would jump to the right, see the “5”, and notice it’s in the third group past the decimal. They would quickly grasp “fifty million.” The commas do the heavy lifting, just as they do right now.
My best argument that magnitude estimation will be harder: In English, you always read words starting from the left. Therefore, it would be unnatural to follow the rule, “when reading to discover the magnitude of a number, saccade your eyes to the right end of e.g. ‘↗000,023′”. This new rule adds a small but frequent tension.
Why I think the above argument isn’t that good: The argument claims that a person would not learn to flawlessly switch between the two well-practiced rules: “focus on the left side of normal words” versus “focus on the right side of numbers whose magnitude you want to learn”. In other words, that there is an inherent friction in switching between rules.
Psychology studies language switching costs for bilingual folks. While fluent in each language in isolation and somewhat used to switching, some studies support the idea of inevitable friction. But Adamou & Shen (2017) show that for people who practice switching frequently and naturally, this cognitive cost can disappear entirely. The key is to actually measure their speed at switching languages in realistic ways. Their work suggests that only unpredictable switches impose costs.
In contrast, reading is far less unpredictable than reacting to spoken language. In a book, the “future” is frozen and you can see it with your peripheral vision. You will see that a number is coming later in the sentence, and you will probably know if you want the exact value. These incidents will be predictable and – under this theory – free from switching costs.
Lastly, the mental number line gets bigger to the right. Therefore, we would be quite comfortable learning the rule “look right to determine how big the number is.”
Right-to-left scripts already swap directions for reading numbers
While Arabic scripts read right-to-left, surprisingly, they both write numbers in the same order we do, and also read those numbers in the same order. So they might write “I have 1,300 dollars” as “السعر 1,300 دولار”. They start on the right side of the sentence and read left until they hit the number. At that point, they saccade their eyes to the left side of the number and read to the right, and then saccade back to the word to the left of the number and continue reading left.
This rule is strictly more complicated than what flipped numbers require. Flipped numbers only require you skip past a few digits to quickly determine magnitude, but still allow you to smoothly move your eyes in the usual direction to read the exact value of a number. In contrast, the Arabic rule always requires that you skip past the digits, switch directions (to read the number), and then switch directions again to continue reading the text. Even so, hundreds of millions right-to-left readers execute this rule every day. I don’t yet see evidence that the Arabic rule makes it harder to read numbers even after the rule is learned.
Scientific notation should flip
If you’re writing a number where most readers will only care about the magnitude, then write the number in scientific notation.
The point of scientific notation is to quickly communicate approximate magnitude, only including the digits which are relevant. Consider the standard notation of . You read the first part: ““. 5 what? You don’t know. Instead, we might write in order to communicate the most important information ASAP. (Or technically, ↗.)
However, for with , this would run counter to the “left is smaller” mental number line by putting the big magnitude to the left of a number in the one’s place. We can’t win – no matter which way we order the scientific notation, the mental number line will be violated for either or . On the other hand, given that readers would be used to looking for the most significant digit on the right, writing would be congruent with the more usual way of writing ↗000,000,05. On balance, I think that “” is the way to go.
Flip the decimal part
Let’s consider “5.37”. I propose we write that as “ 73.5″, with the “-2” indicating “the first digit has the place of .”
Decimal long addition and multiplication are easier, as you never revise digits you’ve already computed. The first digits are informative. By flipping the decimal part, we preserve the symmetry of powers of ten around the decimal point. Having the fractional part on the left accords with the mental number line – smaller components on the left, bigger on the right.
As explained earlier, if the reader wants the exact number, they start reading from the left. If the reader wants the rough magnitude, they saccade to the right end of the number and estimate how many digits (or comma-separated triplets) come after the decimal point. This is what readers currently do, except now the eye lands on the right end of the number instead of the left.
Flipped pronunciation
If you read “↗563”, you should not read it aloud as “three hundred and sixty five” – that would require scanning to the end of the flipped number and then reading backwards. Instead, read aloud “↗563” as “five, sixty, and three hundred” and “↗023” as “twenty and three-hundred.”
💡Flipping the local ordering of pronunciation: If we’re truly optimizing, we might as well say “twenty and hundred-three” while we’re at it. The first words “and three-” don’t tell you much until you know “three of what“? Whereas “and hundred-three” tells you the order of magnitude as soon as possible.
At first glance, there’s a tempting and obvious culprit. We call the numbers “Arabic numerals”, and Arabic is written right-to-left. I can imagine an ancient Arab merchant writing “٣٦٥” (“365”), which reads right-to-left as “five, sixty, and three-hundred.” A European, unfamiliar with the reading direction, copied the digits in the same sequence “365” but read the number according to their left-to-right convention. Thus spawned our current system.
It’s a neat theory. It’s a theory I came up with. It’s also wrong.
Reality is not so neat. Although we call them “Arabic numerals”, they are more accurately known as “Hindu–Arabic numerals.” While mathematicians like the Persian Al-Khwārizmī (after whom we coined “algorithm”) introduced the system to Europe, the numerals still originated in India. The relevant ancient Indian scripts (like Brahmi) were written left-to-right. Thus dies the “merchant miscommunication” hypothesis. Writing the most significant digit on the left was not a translation error.
Why did the right-to-left Arabic keep the left-to-right numbers?
Arabic forces its readers to change reading directions entirely to read numbers. That initially seemed like strong evidence that the Arabs had a strong reason to retain the orientation of the numbers. So what happened? I can only speculate, but let’s put ourselves in Al-Khwārizmī’s shoes.
It’s the early 9th century in Baghdad, the heart of the Islamic Golden Age. Al-Khwārizmī is a brilliant scholar sponsored by the caliph’s court.
Al-Khwārizmī encounters a revolutionary system of calculation from India. Before, arithmetic was a chore. You might use an abacus. Multiplying or dividing large numbers was complex and error-prone. In contrast, the Indian system was mind-blowing.
- In positional notation, the value of a digit depends on its position. The “5” in “50” is different from the “5” in “500”.
- A symbol for zero allowed for clear distinctions between “5”, “50”, and “501”.
Al-Khwārizmī recognized the system’s genius and wrote On the Calculation with Hindu Numerals. This book introduced the system to the Arab world and, later, to Europe.
So, why didn’t he flip the order to match their right-to-left script? The direction was part of the technology. The numerals were not just a new set of fancy symbols to replace familiar ideas. The Hindu numerals were the front-end of a brand-new computational engine. The positional logic was baked into its left-to-right structure: as you move one way, the value of the digit changes by a power of ten.
Al-Khwārizmī likely prioritized quickly integrating a system that made commerce, astronomy, and engineering calculations vastly easier. Flipping the numbers wasn’t simply a matter of flipping the written order of the Hindu numerals – he would’ve needed to re-invent the algorithms which came with those numerals and translate the Indians’ existing mathematical work. Left-to-right numbers came in a package deal.
Even before the Arabs, these ancient Indian mathematicians were already “doing it wrong” by the logic of this proposal.
Why? I don’t know. Probably they had spoken numbers first. To write their numbers, they retained the order in which they spoke numbers. That order happened to be our current rule of “biggest part first” – e.g. “three hundred” in “365”. But once established, the switching costs became too high – even when it creates obvious inefficiencies, like Arabic readers changing direction mid-sentence.
Our number system fights our mental number line and complicates mental arithmetic. Why did we end up here? I’d guess that Al-Khwārizmī couldn’t just flip the Hindu numerals around because the notation was part of the technology. Now we’re all trapped by a coordination problem too big to solve. Who’s going to convince all English speakers to flip their numbers? No matter how you quantify the switching costs, or how you write the number representing that cost, that cost is big.
Still, understanding might bring value. For example, maybe this essay helps explain why kids find positional notation to be difficult (Fuson, 1990). We know that learning two contradictory patterns makes both harder to learn (McNeil and Alibali, 2005). Children simultaneously learn “biggest on the left” from the notation but “biggest on the right” from their teacher writing ascending sequences (“1, 2, 3…”) on the blackboard. Maybe someone should take a look at that?
The next time you encounter a long number and have to read to the end to figure out what the first digits even mean, remember that that silly design choice was made thousands of years ago. Our entire civilization agreed to write numbers backwards, and now it’s too late to fix it.
You can’t unsee it now. You’re welcome.