How Long Should Your Phone Number Be?

Arranging a carpool recently I was surprised to discover that the father of my daughter's friend has a very suspicious and easy to remember number ending with "9999." Inquiring whether he is anyone very important to be assigned such a number, I received a straight answer from his 9-year old daughter: "He got this number as a payback after being very mad at his cell service."

Most of us cannot keep more than 4 digits in a memory for too long. Why then our phone numbers should be so long we have to bribe or threaten  phone providers to get an easy-to-remember number?

As you may have guessed the answer is in math, simple and applicative.  Exactly the type of math that many argue should be taught at school. Blessed with a free afternoon this week, I decided to try the following on my 9 and 12 year old kids.

Imagine that we have just 10 people spread around the world. Just 10 people, no one else. And a fully functioning cell phone system. Do these people really need 10-digit phone numbers to dial each other?
Since there are only 10 of them, we can give them 1-digit numbers:  0, 1, 2, 3, 4, 5, 6, 7, 8, 9.

But as soon as eleventh person is born and grows mature enough to get her/his phone number, we need to use number 10. And we add to all the other numbers 0 in front: 00, 01, 02, 03, 04, 05, 06, 07, 08, 09, 10.
Then another person (twelfth) grows and gets number 11.
We can continue adding up more phone customers till a hundredth person gets number 99.

So, we see that:
1 digit is enough for 10 people.
2 digits - 100 people.
continuing this we get:
3 digits - 1,000 people.
4 digits - 10,000 people.
etc.

How many people live in our home state of MA?  6.5 million.  Let's see how many digits do we need so that everyone can get a unique phone number.
5 digits are enough for 100,000 people.
6 digits - 1,000,000 people. Still less than we need.
7 digits - 10,000,000 people.

So, using 7 digits in a phone number will allow us to create 10 million various phone numbers that will be more than enough for all the people of MA.  And guess what, check your home phone number.  Not counting the 3 area code digits, it is exactly seven digits long!  Someone thought it through.

Now, imagine that you want to establish a new cell phone service in Russia. How long should your numbers be?  I let my kids to Google Russian population (160 million) and come up with a 9 digit answer.

Then we tried China with 1 billion 344 million people and realized that we need 10 digits.

Now think car license plates or zip codes. Imagine yourself planning a Mars colony. How many digits in their phone numbers, flying saucer license plates and zip codes would you need?

Top image by Ed Yourdon, distributed under the CCL.