# How could I justify switching phone numbers from decimal to hexadecimal?

A very common media technique is to replace area or exchange codes of phone numbers with 555 to prevent people from dialing real numbers and harassing people by accident. However, I find this a bit limiting for my world, since I think a lot can be conveyed through phone numbers. I still wanted to avoid this issue though, and I’ve landed on the use of the hexadecimal system, which provides the numbers 0-9 and letters A-F for me to work with.

My issue now comes in justifying this change for a world that otherwise is Earth, for all intents and purposes. I see two options here. The first is to simply handwave the problem away by saying this is how it has always been, but I feel this creates more problems than it solves. The second is the one I want to address here.

What is the best way, making the least amount of changes to Earth, to justify the use of a hexadecimal dialing system such that everyone would understand this change by the 1990s and accept it as completely normal?

• In ye olde dayes, telephone exchanges were named - you would dial them by converting the first letter(s) of the name to a number, the same way you used to write text messages. Jun 24, 2023 at 21:33
• @SurpriseDog isn't wrong. We've already graduated from base-10 dialing. There are so many IP addr in the world today that we're planning to move from IPV4 to IPV6. The real problem you're going to face is practicality. Putting a grid of 3x4 buttons on the phone (after rotary dial) was convenient and easy to use. A grid of 4x4 is less practical and harder to use. As we once used text to dial (why buttons still have associated letters), we could shift to text-style typing to use any base. But it would be more likely to be base-36 (all the letters & numbers). Innovation tends to be organic.
– JBH
Jun 24, 2023 at 23:18
• The DTMF system which replaced rotary-dial phones originally had 16 buttons -- see en.wikipedia.org/wiki/Dual-tone_multi-frequency_signaling . Perhaps it wouldn't take much for an alternate history to keep all 16, rather switching to a cut-down 12 key layout. Jun 25, 2023 at 5:19
• Gonna be honest, this probably doesn't matter as much as you think it does. Just imagine some quirky mathematician was in charge of the system who was really into base-16. Jun 25, 2023 at 17:12
• this probably doesn't matter - In 1996 everyone was pissed off that they had a new prefix and had to start using them, but no one complained because there was no one to complain to, except yourself for forgetting to dial one first, and it was probably the wrong area code anyway. - "a lot can be conveyed through phone numbers" ? People thought a lot should be conveyed through spelling out web sites too: double you, double you, double you, dot... we stopped doing that, finally, and just starting googling the keyword when we got home. You need operators or Ima sue for bloody finger syndrome Jun 25, 2023 at 19:23

The civilization behind those telephones simply uses a base 16 numeral system.

Using base 10 is not grounded in the fundamental laws of physics, it's just a convention. Since we are accustomed to it, for us it is totally normal that our phone numbers are decimal and not binary or hex.

So it will be for them: they are used to count in base 16, so having number in base 16 is equally logic and straightforward.

• In fact, L. Ron Hubbard's Battlefield Earth used a base-11 system to rationalize how hard it was to crack Psychlo math - and that was rationalized by giving the Psychlos 11 fingers. Humanity itself has used a variety of bases before settling on base-10.
– JBH
Jun 24, 2023 at 23:11
• Base 10 is grounded in the very fundamental number of fingers that we have. It looks like most civilisations in the word have independently come up with counting systems which used the number 10 prominently. Arabic numerals, Chinese numerals, Roman numerals, Babylonian numerals, Egyptian numerals, they all give a special significance to the number 10. Maya numerals give special significance to 5 and 20.
– Stef
Jun 25, 2023 at 17:02
• There are several systems that also give significance to 12 (or 60, or other multiples thereof), though this is usually due to mathematical convenience (many integer factors in base 10), so this is definitely in addition to rather than instead of. Jun 26, 2023 at 5:43
• @Miral if you count the bones of your fingers on one hand with the thumb on that hand, you get 12 (3 per finger, 4 times), do that 5 times for the fingers and thumb on the other hand and you get 60. It's just as natural as counting to 10 with a bit of practice. Jun 26, 2023 at 8:31
• @Separatrix Interesting way of counting, however I count in binary on my fingers so can count to 1023 using both hands. I start with my thumb, so I just have to be careful when counting to 4. :) Jun 26, 2023 at 15:25

Telephone equipment communicates using a technology called Dual-tone multi-frequency signaling - you've probably heard this if you use a landline - the sequence of tones when you dial a number (or just hit a button on the keypad).

Under the covers, this technology uses four high frequencies and four low frequencies. For instance to convey the number 5 you use a combination of a 770 Hz and a 1336 Hz tone.

Four high frequencies and four low frequencies give us 16 possible combinations. Ten of these are used for the regular digits - 0-9, two are used for the star and hash (pound) sign, and the final four are normally labelled "A", "B", "C" and "D". They're mostly unused in modern telephone networks, but still have wide support (I believe).

So, there's the technology - perhaps in your world when the first DTMF phones were being introduced all 16 keys were included rather than the twelve we are more familiar with. Then, when number exhaustion started to be a problem, rather than choosing to extend the length of phone numbers, phone companies assigned numbers including *#ABCD.

This actually is quite a bad idea though - if you have ten possible digits and 7 digit phone numbers that's (at most) 10^7 or 10 million combinations. Adding in ABCD would give you about 105 million combination, or 270 million if you also allow * and #. In contrast, adding a three digit area code gets you to 10^10, or 10 billion possibilities.

• Mayhaps, but if you use all sixteen digits, you can get away with a two digit area code and still have more than a ten digit decimal number Jun 25, 2023 at 18:51
• 10+2 digit keypad with 4 modifiers. Esthetic changes necessary to a standard 80s office phone: zero. +1 Jun 25, 2023 at 19:36
• Note that base-12 phone numbers (using two of the six additional DTMF codes) are a thing: en.wikipedia.org/wiki/Mobile_dial_code and have been spotted in the wild: i.stack.imgur.com/ULbpp.jpg Jun 25, 2023 at 19:52
• @RadovanGarabík I use those today, for example *111# is used to check my mobile balance. I didn't realise they could be used for calling too.
– stan
Jun 26, 2023 at 7:28
• TIL that's a USSD code: en.wikipedia.org/wiki/Unstructured_Supplementary_Service_Data
– stan
Jun 26, 2023 at 7:29

Here's my answer - Telephone adoption was much more widespread globally and this meant that (like IPv4) that all the decimal numbers were used up, switching to Hex added significantly more numbers, whilst still being reasonably human readable.

• It's a misconception that IPv6 has more addresses because it's hex. That's just the user interface. IPv4 and IPv6 addresses are both binary numbers, v6 is 128 bits long instead of 32. You could write IPv4 addresses in hex (which would make subnetting actually easier to understand) but it wouldn't give you more addresses. It would make the addresses shorter to input though. (If you wrote IPv4 addresses as a single decimal number instead of one number for each byte, that would also make them less digits). Jun 26, 2023 at 11:44
• That said, if your constraint is the actual number of digits required to be entered by the user then yes absolutely this does give you more combinations (just pointing out that wasn't what happened with IPv4) Jun 26, 2023 at 11:48
• IPv6 has 128-bit addresses. IPv4 uses 32-bit addresses. A 32-bit IPv4 address can be written as four pairs of hex digits perfectly well, like ff.ff.00.01 = 255.255.0.1 (four decimal numbers from 0 to 255 is exactly the same thing as four hex numbers from 0 to FF). What opened up more addresses is having more bits in the address, regardless of how you write them. The last sentence, about making them more manageable for humans, is accurate. The length of IPv6 addresses is why we typically use a more compact number system for them. Jun 28, 2023 at 1:13
• For phone numbers, primarily aimed at manual human entry, we might use base 36 (all the arabic numerals and all the letters of the alphabet.) But then words as phone numbers become possible, like 1-800-f***-you on your incoming call display... Jun 28, 2023 at 1:14
1. Area code 555 appears in American movies and only in American movies. I have never seen a European movie with anything similar; and I have never seen a written story, American or non-American, with this silly convention.

2. Once upon a time, telephone numbers looked like RIV-1489, with the three letters abbreviating the name of the exchange.

3. Nowadays, when we can type letters and symbols on our mobile phones, there is no technical reason why we cannot use them; for example, //USA/SYR/xxxxxxx instead of +1315xxxxxxx.

4. Even in America, the land of 555, there are combinations which are reserved and are never used in real life:

• Area codes in the North American Numbering Plan cannot begin with 1.

Technically, they cannot begin with 0 or 1; but an area code of the form 0xy would create a lot of complications.

• The second digit of an area code in the North American Numbering Plan cannot be 9.

• Area codes in the North American Numbering Plan which have identical second and third digit are reserved for special use.

Which means that you can safely use area codes 1xy, x9y and most of xyy (except, for example, 911) as area codes in your story with no risk of annoying any real-life telephone user.

• Many countries reserve blocks for fictitious phone numbers and unless you have a very good handle on how those rules work I'd be cautious claiming no non-US media you've encountered has used such a number. If you've ever watched Doctor Who there's half decent odds you've seen an 07700-900xxx or 020-79460xxx number on-screen at some point, for example Jun 25, 2023 at 10:10
• I think you are overreacting to the "555" area convention ... Jun 25, 2023 at 17:09
• If you're restricting yourself to North American dialing, it was 2L-5N, not 3L-4N. A 555 was "KLondike 5" Jun 25, 2023 at 18:42
• @SaraJ - Reserved blocks are far less silly that just having 555 though... It's far less obvious that an 07700900xxx number is fake compared to it blatantly starting with 555. I find 555 to be quite immersion breaking Jun 26, 2023 at 9:04
• @ScottishTapWater: 555xxxx is a reserved block. (Though now only 55501xx is reserved.) en.wikipedia.org/wiki/555_(telephone_number) Jun 26, 2023 at 16:41

First one should realize that we use decimal digits for phone addresses is arbitrary. When phone addressing was introduced, if we had used letters, or coloured shapes (to reach me, dial red square, blue circle, green triangle), everyone would have though that was normal, and using digits would be seen as strange. So, it doesn't require a huge leap of faith to address phones with hex digits instead of decimal digits.

As for a reason, phone numbers tend to run out. My parents have had the same phone line for more than 60 years. And when they first got their phone, it used a four digit local number. But then they ran out of numbers and their area went from four digit local numbers to five digit local numbers in the 1970s. Later, in the 1990s, that wasn't enough and since then they use seven digit local numbers. (This is outside of the North American plan). Now while adding digits is a way to increase the pool of addresses, it comes with a disadvantage. It means everyone phone number changes. Another way to increase the pool of possible numbers is to start using hexadecimal digits. And while this means big changes in infrastructure, it does have one advantage: everyone can keep their existing phone number, as any decimal digit is also a hexadecimal digit.

• There's also a few major disadvantages - 1. the older phones cannot call the newer numbers, unless a special prefix system is introduced, and 2. requiring overhaul of the entire country's electromechanical switchers Jun 25, 2023 at 22:32
• @catalogue_number Well, yes, that's why I wrote "this means big changes in infrastructure". But it's a choice a society could make. Jun 26, 2023 at 5:29
• @Abigail also: OP wasn't asking about the possibility of switching the system, just about the likelyhood of it coming about differently FROM THE START. And that has a really really low hurdle, it would've been enough if some higher-up at one of the bigger telephone providers (when the switch from named exchanges to numbers happened) was a bit of a nerd/ liked/preferred base 16. Jun 26, 2023 at 8:43
• @Hobbamok That's not true though - named exchanges still had numbers for the users served by that exchange. "Pennsylvania 6-5000", for example. Considering we have a world where base-10 counting is common to nearly every civilisation ever, the overwhelming majority of phone systems would always have used base-10 numbers. So even if a vanishingly-small minority used base-16 at some point, it is inconceivable that when exchanges were automated, the overwhelming majority would be forced to adopt that scheme. Jun 26, 2023 at 11:27
• @Hobbamok ... And the starting point when telephone systems started was not users' numbers as a lookup table, it was literally a count of users. So this would have needed someone at that local phone company to require all their staff to count in base-16 at all times in all situations. In a world where almost all human civilisations count in base-10, that is an almost impassable hurdle. Base-16 is only popular today in some niches of electronics and software because of genuine technical reasons, not because of one person's whim. Jun 26, 2023 at 11:58

## Binary keyboards

At some point in your history, phone numbers had to be typed in binary, because dials had only two keys, or some other reason related to the technology.

Writing down long binary numbers, and memorising long binary numbers, is quite a pain. Therefore there is a huge incentive for people to convert between binary phone numbers and another base.

Converting back-and-forth between binary numbers and decimal numbers is quite a pain. It's not hard per say, but there is no convenient way to do it, you need a pen and paper to write all the intermediary calculations and it is easy to make mistakes.

Converting between binary numbers and hexadecimal numbers on the other hand, is extremely natural. Just replace each group of 4 bits by the corresponding hexit:

0000 0001 0010 0011 0100 0101 0110 0111 1000 1001 1010 1011 1100 1101 1110 1111
0    1    2    3    4    5    6    7    8    9    A    B    C    D    E    F


For instance, the hexadecimal phone number F90122 corresponds to the binary phone number 1111.1001.0000.0001.0010.0010, and it is easy to convert back and forth on the fly.

The simplest solution is to let AT&T maintain their monopoly for a little while longer. Then they can lobby to have the digit base increased rather than increasing the string length, so they can lease (yes, "lease" not "sell"; AT&T was a strong company) their "new and improved" "full dialing power" telephones.

There is precedent for this too. The * and # keys were not on the telephone dial until AT&T put them there, and there are specialty models that include the ABCD keys that Neil Tarrant explains in his answer. They even sell 16-digit telephone pads you can use to upgrade you own set up! (not an endorsement, they're just the first one I found that claims to still have them in stock)

### Because Western Union Already Did It

Nature abhors a vacuum, and people loathe change.

Drawing inspiration from the Font of All Knowledge, it appears that rotary dials are not exactly new ideas in the 20th century. As early the 1830s, dial mechanisms were considered for telegraphy. Essentially, a certain telegrapher would turn the dial first to U, then X then Q, then space-A-K-H. The receiving telegrapher would check his code book and know that the shipment had been placed on the Empress of Canada.

In your world, telegraphy has been done with alphanumeric dials for decades. When telephony came on the scene, the inventor simply combined his voice over wire device with an off the shelf dialing device. Dialing of Exchange plus number, e.g., PEnnsylvania-65000 becomes a trivial matter of dialing the two letters, then the five numbers.

The exchange system is cool because it gives you an easy to remember alphanumeric address that has expandability built into the system. It's easy enough to make longer alphanumeric exchanges, like SUSquehanna7-4800.

In your world, the end user won't be inconvenienced or really even notice anything unusual. With 3,656,158,440,062,976 available telephone addresses using 10 characters, I doubt you'd run out of addresses! Even so, the switch from 212-PE6-5000 to a number like 796f-6e65-2e is trivial. Your end user won't know the difference, except that the familiar exchanges are all gone. Everything would happen at the underlying technological level. The switch could easily be done in the 1950s, to much fanfare as it was introduced at the Worlds Fair. Public service announcements would be produced and disseminated, educating the public about the new hexadecimal service, much like how they did when dial service was introduced in the real world.

You could introduce new dial tones, ring tones and busy tones to alert the caller that she's on the old exchange or the new hexadecimal system.

Using Hexadecimal is more efficient than decimal for a telephone system that uses digital logic to connect two telephones together based on a phone number.

This is a simplified example, but lets say a decimal digit in my phone number selects a choice for some relays in my telephone system, and those choices are being controlled by digital logic, then I need 4 bits to represent that choice. But digits 0-9 only use 10 of the 16 possible choices for those bits. I have wasted hardware because those 4 bits could be used to turn on one of 16 relays, but I am only using them to control 10 relays. My system becomes larger and more costly to support the same number of customers compared to using hexadecimal phone numbers.

One might argue that the development of telephone systems was a complex task at the time it was first undertaken. In your world the engineers needed to squeeze every last drop of efficiency out of the system to make things fit within the constraints of size, weight, power, and cost. In that case hexadecimal was chosen for the sake of efficiency.

Now let's go into why it's more efficient. Early telephone systems were completely analog, with switchboard operators manually connecting people by plugging wires into different holes.

The next phase in the evolution of the telephone system is replacing the human telephone switchboard operator with an automatic system that connects the different telephone circuits. That would likely involve relays. Relays are open/closed, on/off. Given that relays are the state-of-the-art technology of the time, and the phone companies are already using them, any other logic that decodes the phone numbers to control the signal relays would likely be composed of more relays. The relays are being used to make digital electronics.

For electrical engineers who are designing digital computer equipment, number systems that have a power-of-two as the base are typically more efficient. Even more efficient is power-of-two bases where the number of bits used to create that base is also a power of two (1, 2, 4, 8, 16, 32, or 64-bit numbers).

Some examples are below.

• 2^0 bits = base 2 = straight binary.
• 2^1 bits = base 4 (not commonly used anymore)
• 2^2 bits = base 16 (hexadecimal)
• 2^3 bits = base 256 (the common 8-bit byte)
• 2^4 bits = 16-bit numbers (common register size for 16-bit processors)
• 2^5 bits = 32-bit numbers (common register size for 32-bit processors)
• 2^6 bits = 64-bit numbers (common register size for modern 64-bit processors)

Of the above choices, straight binary take a lot of digits to write numbers (about 10 bits for every 3 decimal digits) and is therefore not very compact. Creating a symbol for each of the choices in an 8, 16, 32, and 64 bit number would result in way too many symbols for humans to memorize.

That just leaves 2-bit and 4-bit numbers. The 4-bit numbers only have 16 symbols to remember, and one can express numbers fairly compactly (about 5 hex digits for every 6 decimal digits). That of course explains the ubiquitous use of hexadecimal in computer programming and electrical engineering.

Digital is invented before the telephone.

Hexadecimal is very common in computer science and virtually unknown outside of it. Base 16 is great to work with in a digital system. So if digital (not necessary computers as we know them) is invented first and the telephone system is built on that, base 16 would make sense.

That doesn't mean computers must be invented first. Maybe just the concept of digital computation and the first phone switches work with binary trees or something.

When telephone was first created, there were few telephones, which means few telephone numbers, and so telephone numbers were relatively short.

As the ubiquity of phones increased, on Earth we gradually added more digits to phone numbers to cover the larger number of phones.

On your fictional Earth, instead of increasing the length of phone numbers, they decided to increase the number of possible digits. So, maybe at some point, phone numbers consisted in six digits 0-9, for a number of possible phones of 10^6. On Earth we first extended the number to eight digits 0-9 for a number of possible phones of 10^8, then later to ten digits 0-9, for a number of possible phones of 10^10; while in your fictional world they first decided to extend to six hexits 0-9A-F, for a number of possible phones of 16^6 ≈ 10^7, then later to ten hexits 0-9A-F, for a number of possible phones of 16^10 ≈ 10^12.

One advantage of extending to hexits is that already-existing phone numbers don't look too different from new phone numbers. If old phone numbers consists of six digits 0-9, and new phone numbers consist of six hexits 0-9A-F, then old phone numbers and new phone numbers can cohabit with no one complaining that old phone numbers have a different length.

Also note that on Earth, every country has its own numbering plan (and the actual length of a phone number varies from one country to the next). In your fictional world, they might have come up with a global numbering plan from the start.

• That would also require that early phone technology was different. For both rotary phones and switchboards, adding digits is relatively easy, changing the numbering system requires all the hardware to be updated or thrown out and exchanged.
– Tom
Jun 28, 2023 at 14:26

The extra digits could be used us a checksum, to reduce mis-dials.

For example, the classic number

086 543 932


could become something like

086F 543B 932C


where the trailing F, B and C characters are a hash of the preceding three numbers, and can be used by the phone to check that the numbers have been entered correctly.

Why are people mis-dialing so much? And why is it so important to prevent mis-dials? I don't know, it's your story.

Some real-world examples of related concepts:

• But that has nothing to do with Base16. This would be possible with hexadecimal as well, but isn't widely used Jun 26, 2023 at 8:47

It's simple: in your alternative timeline the Anti-Digit Dialing League won. This is a real organisation that really did exist and their goal was fighting against "creeping numeralism" in general and the switch to numeric area codes in particular.

But in your universe they gained widespread popular support (rather than being seen as a couple of weird guys charging at the windmill of technology, and "numeric" area codes were banned.

Of course the reasons behind introducing numeric area codes haven't gone away, so while there was a ban on all-digit area codes, hex was a kind of legal loophole.

(Yes, this whole answer is just an excuse to mention the ADDL.)

• +1 simply for mentioning this. I never knew that was a thing.
– Tom
Jun 28, 2023 at 14:34

This is a Frame Challenge

I can't imagine this shift occuring.

1. Technological inertia is very hard to overcome. Consider how long it took to discontinue including an RS-232 port on the back of computers.

2. There are many dialing protocols in the world. We in the U.S. too often think the whole world uses the 3-digit area code, 3-digit central office code, and 4-digit line number — but it simply isn't true.

3. How the world's phone number formats developed had everything to do with the development of telephony technology. It's anything but arbitrary — and yet you're trying to make an arbitrary change.

We could assume some overwhelming change occured that required more numbers. That's the only practical reason for changing the number base (or anything about the number format). More numbers comes about from more population or more ubiquitous use of communication. But the Internet (originally called the ARPAnet, started in 1969) and cell phone technology (launched in japan in 1979) are competing technologies that would create a technological drive that would, frankly, never embrace a base-16 solution.

Organically, it's more likely that you'd have one of the following shifts.

1. A never-ending extension of the decimal system. Longer and longer base-10 numbers.

2. A base-36 solution based on all English letters and numbers (and that's an arrogant statement) where the "phone number" becomes more compressed to continue to make numbers memorizable.

3. An earlier adoption of the speed-dial/cellphone/DNS solution, where human-readable names are mapped to protocol-dependent addresses (phone numbers).

Conclusion

You're asking us to modify known history to adopt a numbering base without any recognition of the history of communication technology. I can't imagine such a solution believably occuring.

• Base 36 is unlikely because when people write the numbers down, O/0, I/1 are very easily confused, G/6, Z/2, S/5, U/V somewhat less so. Eliminating G, I, O, S, V, and Z leaves 30 reasonably-distinct characters. Jun 25, 2023 at 14:59
• @MontyHarder Point well taken, and I'd also assume to avoid capitals-vs-lower case. But I hope my point was clear. There are a lot of ways to extend what we already have that don't require shifting to a 4x4 (or, more likely, a 4x5 to preserve * and #) keypad.
– JBH
Jun 25, 2023 at 15:59
• What is an RS-232 port and how long did it take lol Jun 25, 2023 at 17:11
• @AzorAhai-him- Tell me about it! I was once on a call with a bunch of store managers and the question was asked how to get music in the stores without paying for expensive services. I knew one of the vendors all the stores worked with sold a CD/MP3 capable player that could be used adequately and very cheaply. After suggesting it the phone call went dead quiet until one of the more smart aleck managers asked, "what's a CD?"
– JBH
Jun 25, 2023 at 19:34
• @AzorAhai-him- My apologies! I found RS-232 ports on off-the-shelf retail computers through about 2012. The RS-232 protocol had been functionally obsolete since the late 1980s. They continued to exist until USB mice becaue cheap as water and the number of RS-232-dependent devices seriously declined. There are still a handful of RS-232 devices out there (like the color scanners used to match paint at hardware stores), but once RS-232-to-USB converters were invented, that was the last nail in the coffin.
– JBH
Jun 26, 2023 at 0:04

Here is a very plausible scenario: For this to happen the time difference between telegraph and telephone should be larger, or, telegraph technology got more usage than what we had. This leads to more telegraph adoption, which in turn leads to automatic telegraph switching.

Early switching technology uses relays which are binary in nature. Thus with operators being comfortable with codes, it would be easier to handle hexadecimal addresses which does not require mathematical base conversion. Then once the phone is invented, it would use the same switching network with hexadecimal addressing for phone numbers.

1. They went digital early on in the development of the phone system. Digital doesn't have to mean electronic computers. Even relays are digital - they are either on or off.
2. They kept running out of phone numbers. An easy mistake in allocating numbers is not to allow for growing towns. You can allocate so many digits for each phone number, only to run out after a few years as the town grows.
3. Rather than re-number all the existing phones with an extra digit, they switched to hexadecimal. Decinal is inefficient in a binary system, as each digit only uses the binary numbers 0000 to 1001. That leaves 1010 to 1111 unused. Bringing those digits into play gives you a whole load of new numbers to allocate without doing anything to the existing ones.

### No numbers, at all

Why have human-readable numbers at all? You have a computer in your pocket that remembers it for you, even posts it in the cloud so all your current devices and future phones know them too.

Replace phone numbers with bar codes, QR codes, NFC tokens, hyperlinks, metadata, or whatever machine-readable solution you want. Then the user just has to tap a button on a webpage, snap a picture of a poster, hover their phone over the front door, and poof, their phone reads the number and saves it.

And if for some reason you ever need to write a number on a piece of paper, you could always ask your phone to generate an HEX string representing its number (or the SIM's number more likely), which someone could then type into their phones, and would then store without either of you having to remember it ever.

To be honest, I don't see a good reason to write out phone numbers in a story, so decimal or hexadecimal or whatever else doesn't matter much there.

For a visual medium, you could replace a number with a machine-readable logos. We already have companies mixing their logo with QR codes, so really we're already encoding contact information into company logos, if wouldn't take much to generalise it.

# Make the change before introducing direct dialing

The main problem with switching from decimal to hexadecimal phone numbers is that millions/billions of existing telephones would instantly become obsolete. You'd have to introduce new dials or keypads with the extra digits.

However, if you don't have to modify consumer hardware, the change is relatively straightforward for the phone company to implement.

So, make the change in the very early days of telephony, when end-users didn't dial phone numbers themselves, but had to contact a human operator at a switchboard to give a phone number. The switchboards themselves would be expanded so that each exchange could handle up to 65,536 phone lines (numbered 0000 through FFFF) instead of the 10,000 (0000 through 9999) that became standard in our timeline. And that would be the justification for the change: To allow more phone numbers.

Of course, another way to justify hexadecimal dialing is to make your civilization use base-sixteen numbers (perhaps by giving people 16 fingers), thus making hexadecimal phone numbers the obvious choice in the first place.

### Your phones don't use phone numbers

"Phone numbers" are identifiers in a namespace defined by E.164.

However, while nowadays many apps build upon them (to use the phone Contacts), it doesn't need to be so. You could call someone by their ICQ#, Skype username, Threema ID...

Or perhaps, just email addresses (maybe using Jingle over XMPP)

At one point in your world, a different system for calling replaced the traditional phones, and everyone moved to e.g. Skype. So people use phones (the devices), but the calls actually use a different protocol -originally fueled by a company- that uses a different naming ("numbering") schema.

You want this by the 1990s, which is a bit early. There were already solutions like Skype, but they required a computer (non-withstanding the "Skype phones" hardware). I would recommend moving your story timeline later, or accelerating the introduction of smartphones so they happen a decade earlier. If a company introduced an affordable (maybe subsidized?) smartphone a decade earlier, they could have monopolized the market, introducing whatever numbering calling system they wanted e.g. a BBPN (BlackBerry phone number) replacing phone numbers.