Negative Bases

And now, for something silly.

In general, number bases are expected to be positive integers greater than one. The most widely used are decimal (because we have ten fingers and ten toes), binary (how computer data is stored), hexadecimal (a more convenient way of writing binary), and octal (base eight), but, mathematically speaking, there’s no reason why any positive integer greater than one can’t be a base.

It’s also possible to use non-integers, even transcendental numbers, as bases, but those are of far more limited utility (the most useful is probably base ϕ). Base one is also an option, but I wouldn’t recommend it.

The basis for bases in the first place is that each position, starting from the rightmost digit (for integers), represents the number shown times bn, where b is the base and n is the number of slots to the left of the units place. So, in decimal, 4516 = 4 * 103 + 5 * 102 + 1 * 10 + 6. In hexadecimal, 4516 = 4 * 163 + 5 * 162 + 1 * 16 + 6, which is 17686 in decimal.

So we can also have negative bases, although they lead to number series that are difficult to follow. In base negadecimal (-10), 4516 = 4 * -103 + 5 * -102 + 1 * -10 + 6 = -4000 + 500 – 10 + 6 = -3504. This means that any integer with an even number of digits is, by necessity, negative. However, we can still get all the positive integers: It just means that all positive integers have an odd number of digits. (Incidentally, it also means that there’s no need for negative numbers, since negative integers have positive integer representations.)

Let’s count from one to fifty in base negaternary: 1, 2, 120, 121, 122, 110, 111, 112, 100, 101, 102, 220, 221, 222, 210, 211, 212, 200, 201, 202, 12020, 12021, 12022, 12010, 12011, 12012, 12000, 12001, 12002, 12120, 12121, 12122, 12110, 12111, 12112, 12100, 12101, 12102, 12220, 12221, 12222, 12210, 12211, 12212, 12200, 12201, 12202, 11020, 11021, 11022.

Let’s count from one to twenty in base negabinary: 1, 110, 111, 100, 101, 11010, 11011, 11000, 11001, 11110, 11111, 11100, 11101, 10010, 10011, 10000, 10001, 10110, 10111, 10100.

And yes, some people have worked in imaginary bases as well, because they just don’t know when to quit. Which suggests the possibility of quaternion and octonion bases. If you’re looking for a PhD dissertation topic in mathematics and don’t particularly value your sanity.

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