A magical land where rounding equals truncation

The previous post discussed complex numbers, dual numbers, and double numbers. All three systems are constructed by adding some element to the real numbers that has some special algebraic property. The complex numbers are constructed by adding an element i such that i² = −1. The dual numbers add an element ε ≠ 0 with […] The post Matrix...

The complex numbers are formed by adding an element i to the real numbers such that i² = − 1. We can create other number systems by adding other elements to the reals. One example is dual numbers. Here we add a number ε ≠ 0 with the property ε² = 0. Dual numbers have […] The post Euler’s formula for dual numbers and double numbers first appeared...

The simplest trick for converting from one base to another is grouping. To convert between base b and base bk, group numbers in sets of k and convert one group at a time. To convert from binary to octal, for instance, group bits in sets of three, starting from the right end, and convert each […] The post Tricks for radix conversion by hand first...

The previous post started by saying that rounding has a surprising amount of detail. An example of this is double rounding: if you round a number twice, you might not get the same result as if you rounded directly to the final precision. For example, let’s say we’ll round numbers ending in 0, 1, 2, […] The post Double rounding first appeared on...

Abramowitz and Stegun has quite a few intriguing plots. The post will focus on the follow plot, Figure 9.4, available here. We will explain what the plot is and approximately reproduce it. The plot comes from the chapter on Bessel functions, but the caption says it is a plot of the Hankel function H0(1). Why […] The post Duplicating Hankel plot...