2 + 2 = 10300

An adventure!

We will begin with base 10. Look at a number, like 457. That’s four hundreds, five tens, and seven ones. Or, another way to put it:

\(4 * 10^2 + 5 * 10^1 + 7 * 10^0\)

Because 10⁰ is just one. To belabor the point, big numbers are like this too. Like, when you read 32,406 you naturally interpret it as:

\(3 * 10^4 + 2 * 10^3 + 4 * 10^2 + 0 * 10^1 + 6 * 10^0\)

To use another “base”, like base 8, is just to swap out “10” for some other number. Sometimes bases can be represented with little subscripts, like so: 154₈ means “154 in base 8” and would be interpreted as:

\(1 * 8^2 + 5 * 8^1 + 4 * 8^0\)

So, translating into base 10, we’d have:

\(64 + 5 * 8 + 4 = 64 + 40 + 4 = 108\)

In other words: 154₈ = 108₁₀.

You can use just about any base. The main rule is in any base, you can only use numerals up to that base. If you do, you can represent any integer in any positive base in exactly one way, just like in base 10. So for example, “19” isn’t a thing in base 8, because it has a 9 (which is more than 8), but “77” is.

Any positive base, I said? Well. What about negative bases?

2 + 2 = 130

Consider base -4. Now things are getting fun. You are having fun now. Trust the process.

Specifically, let’s consider 130_-4:

\(1 * (-4)^2 + 3 * -4 + 0 = 16 - 12 = 4\)

Okay. So “130” represents “4”. Thus, 2 + 2 = 130. Crazy! Can this possibly work? Sure! Not only can you do ordinary arithmetic in a negative base, but you don’t need the pesky negative sign for it; you can represent any negative or positive integer without it. Though you do get some zany inequalities, like:

\(100 > 130\)

100 is 16, and 130 is 4, so in fact 100 is much bigger than 130. In general, numbers with odd numbers of digits are positive in negative bases, and numbers with even numbers of digits are negative.

Doing addition can be quite an adventure in negative bases. You have to work out or memorize a few simple additions, much like a multiplication table in positive bases. For example, just like 2 + 2 = 130, 2 + 3 = 131. For adding multiple-digit numbers, you just carry both extra digits as needed. Like, say:

12 + 12 = 10 + 10 + 130 = 130 + 20 = 100 + 1310 = 1000 + 10 + 13000…

(If you are enjoying this post at all, try writing it out on a piece of paper and you’ll understand the next observation much better.)

Wait a minute! This is infinite repetition! We just keep adding ever greater 130s and 10s! And if we think about it, it makes sense these perpetually cancel out, because “130” translates to 4, and 10 translates to -4.

So anyway, we’re left with just the residual 10 at the end. 12 + 12 = 10. Is that right?

Yes! 12_-4 is -2₁₀, and 10_-4 is -4₁₀. Again, trust the process. 12 + 12 = 10. You just have to notice that you’ll be carrying the 130 for all eternity, and stop.

2 + 2 = 10300

We can go deeper. Negative numbers, schmegative numbers. It is time to go back in time with me. The year is 2004. I am 12 years old, at my grandparents’ house. That once forbidden fruit, Spongebob Squarepants, is no longer enough to entertain me. But there is a treasure trove of printer paper, and pencils, and boredom.

I have entertained myself with negative bases for some time. These have tormented my poor parents, who struggled through much worse explanations than this post, delivered with excitement that boils over into frustration and condescension when not immediately understood. But perhaps I can befuddle them even more. With… imaginary bases.

Like, say, what’s base 2i like?

The good news first: if you allow yourself to use a decimal point, you can express any complex number¹ in base 2i. Moreover, you can add, subtract, multiply, and divide complex numbers using the rules of good old fashioned arithmetic! The bad news: since the complex numbers are best represented on a two-dimensional plane rather than a number line, you have to allow yourself more digits. Specifically, you have to square the number of digits that you’d need in a real number base.

In other words, in base 2i, you have to let yourself use the numerals 0, 1, 2, and 3, as opposed to base 2 or -2, when only 0 and 1 would suffice.

Anyway! On to the good stuff! What’s 10300 in base 2i?

\(10300 = 1 * 2i^4 +3 * 2i ^ 2 = 16 + 3 * -4 = 4\)

Thus, 2 + 2 = 10300. I think I’ll stop here.

1. Okay, fine, any complex number with integer real and imaginary components. Or any complex number with rational real and imaginary components, if you allow infinite repetitions.