Part 1Part 2 In the previous post, we created an approximation to i by iteratively solving Sum[j=0,n]{x_(j)*x_(n-j)} = 1 to construct a number which squares to our base 2 representation of -1, …111. The first few terms are: … (35/128) (5/16) (3/8) (1/2) (1) Some observations: it is tempting to simplify this to the standard… Continue reading Numeric approximations to the imaginary number: divergent series
Numeric approximations to the imaginary number: the imaginary value
Part 1 With a number for -1, we can calculate the value of i. In base 2, …111 =-1, so we need a number x such that x^2 = …111. We can multiply by term order to iteratively solve. For the first term: 1 = x0*x0 So x0=1 Second term: 1 = x1*1 + 1*x1… Continue reading Numeric approximations to the imaginary number: the imaginary value
Numeric approximations to the imaginary number: representations of negative numbers
The biggest barrier to approximating i, the imaginary number, as we approximate pi is the lack of a number -1. It is not clear how to take the square root of a negative symbol. Interestingly, there are numeric representations of -1 in the left infinite numbers, the numbers that can continue to arbitrary large terms… Continue reading Numeric approximations to the imaginary number: representations of negative numbers
Collapsing demographic wave
The small city was clearly past its prime. The houses I walked past were magnificent, once. Large, beautifully balanced architecture covered in peeling paint. Hedges were overgrown, with many weeds around their bases and throughout the yards. The neighborhood school felt undersized for the surrounding neighborhood, but oversized for the students it had. Many of… Continue reading Collapsing demographic wave
Taking frequency shift seriously
It is important to take observations seriously. Many discussions of light frequency shift phenomena focus on the transformations, with more than a little implication that any observer’s stationary frame is more real than the others. So let’s take frequency shift seriously, starting with black holes. Objects near a black hole are very large. As things… Continue reading Taking frequency shift seriously
Certainty and risk in human behavior
Why slack off at work? Because you want to have value in your future. Let’s start with your paycheck. Paychecks are very predictable. For example, the probability of my next paycheck not happening is much less than 1 percent. I can therefore claim all the value of that paycheck now, and regularly do so in… Continue reading Certainty and risk in human behavior
Risk, present and future value
Lower risk increases the present value of future opportunities, but reduces the opportunity gradient. You can imagine an opportunity gradient as a plot of your value at a series of points in your future. The greater the gradient, the steeper the increase in that curve between any two future points. A result is that low… Continue reading Risk, present and future value
Stretching Insights
I just touched my toes during a sitting “touch your toes!” stretch for the first time … in probably ever. As it turns out, the key was not to directly touch my toes, but to understand stretching. On prior stretches, I would do as directed. When the instructor said “Touch your toes!”, I would bend… Continue reading Stretching Insights
A thing is worth what someone else would pay.
Prices are usually set by what the next person would pay. In any transaction negotiation, there are 4 important numbers: buyer’s value, buyer’s next best offer (NBO), seller’s value, and seller’s NBO. The buyer’s or seller’s value is the benefit that each party receives from possessing a thing, measured in terms of money. This is… Continue reading A thing is worth what someone else would pay.