Burton MacKenzie

I heard some wisdom last week that got me thinking: It's more important to listen to people who disagree with you rather than those who do agree.

Those who agree with you will never encourage you to examine the beliefs for correctness/validity; there is little corrective feedback for mistaken ideas. Those who disagree with you will help you illuminate the validity/correctness of things you believe to be true, and help you discard the ideas you hold that fall apart under a different light. Those who disagree with us propel us towards knowledge, or we propel them, incrementally driving each other towards knowledge.

But there's no need to be a dick about it.

Burton MacKenZie www.burtonmackenzie.com

What if 7 was the same quantity as 34? And 142.001529 as well? That seems like an impractical crazy number system, but those can be the most fun, so I'm going to show you one.

The number system we most commonly use is a base 10 place-value system. A number is described by digits from 0 to 9, and the position of each digit denotes the exponent of the base, multiplied by the value of the digit. For instance, 341 = 3*10² + 4*10¹ + 1*10⁰. In binary (base 2), each digit represents a power of two, e.g. 2⁵ 2⁴ 2³ 2² 2¹ 2⁰. 2^-1 etc. With these bases, each quantity has only a single representation. For instance in base 10, 2 is equal only to 2 [1] and no other finite number [2].

This is what we consider normal and natural. However, if we choose an irrational number such as the golden ratio as the base of a number system, we get something entirely different.

In an earlier post I derived the golden ratio, showing its origin from the equation φ = 1 + 1/φ. Rewrite it as φ² = φ + 1. Since the digits in base φ represent exponent multiples of φ, all numbers take the form of φ² φ¹ φ⁰, etc. In base φ, the equation above can be directly represented as numbers, stating

1* φ² + 0*φ¹ + 0* φ^{0 =} 0*φ² + 1*φ¹ +1*φ⁰

or simply, 100_φ = 11_φ, or even more simply 100 = 11 (assuming base φ). This demonstrates that a number in base φ has non-unique finite representations! One number (100) can be represented as another (11). Multiplying or dividing the original equation by φ shows that we can apply this rule universally. Just as 100 = 11, 1000 = 110, 10 = 1.1, 0.001 = 0.00011, etc.

The next question is how many representations does each number have in base φ? Look at the case of 100 again. 100 can be represented as 11, 11 can be represented as 10.11, 10.11 can be represented as 10.1011, 10.1011 can be represented as 10.101011, ad infinitum. Not only are finite numbers in base φ representable in non-unique ways, there are an infinite number of ways to finitely represent any number in base φ!

Now that's a fun number system!

Burton MacKenZie www.burtonmackenzie.com

[1] Even for large values of 2!

[2] strictly speaking, 2 can also be represented in base 10 as an infinite series, 1.9999999....

Alternative energy generation is a multi-faceted problem. I strongly believe that we should have as many energy-generation technologies as we possibly can in operation at any given time. A wide-ranging diversity of energy sources is as important as genetic diversity of crops; if some fall to a blight or the cold, other varieties will fare all right. Andean potato farmers do this. It ensures survival by maintaining a stable food supply. Energy generation now means survival for a significant chunk of the world, especially those in hostile environments. We need it to be diverse to maintain a stable energy supply.

Ben Peterson from VictoryGasworks.com made the following video of him building a gassifier, which supplies a portable 6kW generator that runs an arc welder. The gassifier runs on yard waste such as woodchips or compressed pellets. I am impressed.