Cryptohedonology

Culturally we frequently use basic arithmetic to emphasize when a point is self evident:
“its as basic as 2+2=4.” We also use a little twist on it to point out when someone is being absurd: “next you are going to tell me 2+2=5.”

Ok. So no one really uses those phrases. But I think the idea is clear. It is just plain silly to challenge the idea that 2+2=4.

Or is it?

To math and computer nerds, the idea of using alternative number systems is quite familiar. In contrast to our usual base 10 system (probably rooted in the biological fact that we have 10 fingers, but some cultures like the Babylonians used alternative number bases), computers and their programers regularly utilize base 2 (1s and 0s — representing “on” and “off” states of a transistor aka a “bit”), octal (base 8 — because a convenient standard for information is 8 bits, aka a ”byte“), and hexadecimal (base 16 — used for a lot of different reasons, perhaps most commonly seen by non-ubergeeks in their Wi-Fi WEP keys).

The idea of different number systems is straightforward enough. As we learned in kindergarden, all numbers can be represented by a set of numerals, each less than 10, living in various “places”: the ”one’s place“, another less than 10 living the ”ten’s place“, another in the ”hundreds place“ etc. So, when you write something like the number 1232, you really mean to add these four numbers:
1 times 10^3 (where 10^3= ”ten to the third“ or 1000); thousands place
plus
2 times 10^2 (where 10^2=”ten squared“ or 100); hundreds place
plus
3 times 10^1 (where 10^1=”ten to the first power“ or just 10); tens place
plus
2 times 10^0 (where 10^0=”ten to the zeroth power“ or 1 by definition); ones place

The numerals out in front of each place (the numbers you actually write down on a typical day) must be between zero and 9.

To represent a number in another basis, you just have to follow the same pattern. In base 2, for example, you have similar slots for numbers, but the name for the places will be appropriate for the new base:
2^0 is still the ones place but now
2^1 is the twos place
2^2 is the fours place
2^3 is the eights place and so on

The numerals out in front of each place must be less than two. That leaves just 0 and 1. These then are numbers you write down if you are working in a base 2 system (binary). For example 1101. Of course, it is convenient to convert this back to base 10 so we can use our usual math thinking:
1 times 2^3 (or 1 times 8)
plus
1 times 2^2 (or 1 times 4)
plus
0 times 2^1 (or 0 times 2)
plus
1 times 2^0 (or 1 times 1)

this gives: 8+4+0+1=13

In short, the number 13 is represented by 1101 in base 2. A fun fact about different bases is that the numeral 10 in a given base always represents the number of the base. That is 10 in base 2:
1 times 2^1 (or times 2)
plus
0 times 2^0 (or 0 times 1)

gives: 2+0=2
So the numeral 10 in base 2 is just the number 2.

There is a famous geek joke that goes:
There are only 10 types of people in the world, those who understand binary and those who don’t.

Get it? “10 types of people…”? 10 in binary is 2. Anyone? Ok, sorry.

What about base 5? Sure, why not. Same game:
The numerals you write down must be between 0 and 4
Use powers of 5 to be your place holders
If you wrote down the number 42 working in base 5, it would actually be:
4 times 5^1 (4 times 5)
plus
2 times 5^0 (2 times 1)

or 22 (in base 10)

Notice again, as promised, that 10 written in base 5:
1 times 5^1 (1 times 5)
plus
0 times 5^0 (0 times 1)

gives you back 5 (in base 10).

If your base is larger than 10, you have to come up with a little twist to keep things in order. For example, base 16:
The numerals you write down must be between 0 and 15
Use powers of 16 to be your place holders

However, because you can go higher than 10 for given place, it is awkward. For example, a perfectly legitimate number in base 16 would be (15)(12):
15 times 16^1 (15 times 16)
plus
12 times 16^0 (12 times 1)

or 252 (in base 10).

But writing (15)(12) is misleading because you are using base 10 thinking to write down a number in base 16. What is done in practice is a single letter represents each numeral larger than 9: A=10, B=11, C=12, D=13, E=14, F=15. Now, our number (15)(12) in base 16 is represented by FC (a Unibomber reference? Not quite. From wikipedia: Inside the bombs certain parts carried the inscription “FC” — at one point reported to stand for “Fuck Computers” but later found to mean “Freedom Club”). To convert it to base 10, we use the same method above, substituting our letter-to-number conversion.

So, getting back to the fun part, is it possible to write down an apparently absurd statement where 2+2 is not equal to 4, but still have that statement be true? Sure, if you use another number basis. For example, 2+2=11 in base 3. How? Well, notice that to write a number in base 3, the numeral in front of the place holder must be less than 3. That is, between 0 and 2. So, in base 3, the number 2 (2 times 3^0) is still just the numeral 2. So it is perfectly fine to write 2+2. That means the same thing in base 10 and base 3. However, the answer is still the number 4. But the number 4 is not the numeral “4” in base 3. Confusing? Remember, there is no numeral 4 in base 3. The numeral is the thing you write down, while the number is the actual quantity it represents. So, how is the number 4 (base 10) written as a numeral in base 3? It turns out to be 11:
1 times 3^1 (1 times 3)
plus
1 times 3^0 (1 times 1)

or 3+1=4 (in base 10)

In summary, it is possible to play with the representation of ordinary arithmetic in different number bases to write down apparently absurd statements like 2+2=11 (in base 3, that is). This is, of course, an illusion because it still means the same thing mathematically as 2+2=4 in base 10. This game only really works visually when the numerals are less than 10, otherwise you get strange expressions like ”F+F=15 in base 25!!” Not quite the same impact as 2+2=11. Below is the output of a quick mathematica program outlining various fun combos and their ”truthiness” or ”false-iness” in different number bases.

What I’ve discussed is more of a cosmetic shift in how numbers are presented. But it may be worth noting that is possible to delve deeper into the mathematics and tweek the very axioms of arithmetic and algebra where 2+2 is not 4 (numerically) in any basis! For example, in the group theory of angular momentum addition in quantum mechanics, it is not uncommon to see expressions like: 1/2 times 1/2 = 3 plus 1 (it works because the expression doesn’t really represent the thing it seems to at a glance).

The real message of this post? Question assumptions. The next time someone says ”its as obvious as 2+2=4“ ask them ”perhaps, but what basis are we working in?“

List of various apparently “true and false” statements playing with number bases:
numberbases.pdf