A periodic function has at least as many real zeros as its lowest frequency Fourier component. In more detail, the Sturm-Hurwitz theorem says that has at least 2n zeros in the interval [0, 2π) if an and bn are not both zero. You could take N to be infinity if …

If y is a quaternion, how many solutions does x² = y have? That is, does every quaternion have a square root, and if so, how many square roots does it have? A quaternion can have more than two roots. There is a example right in the definition of quaternions: …

Sometimes you don’t have all the math functions available that you would like. For example, maybe you have a way to calculate natural logs but you would like to calculate a log base 10. The Unix utility bc is a prime example of this. It only includes six common math …

The smaller the base you write numbers in, the smaller their digits sums will be on average. This need not be true of any given number, only for numbers on average. For example, let n = 2021. In base 10, the sum of the digits is 5, but in base …

A few days ago I wrote about the expected number of roots in a random polynomial where each coefficient is drawn from a standard normal, i.e. a Gaussian distribution with mean 0 and variance 1. Another class of random polynomials, one that comes up in applications to physics, draws each …

If you want to add ∞ to the real numbers, should you add one infinity or two? The answer depends on context. This post gives several examples each of when its appropriate to add one or two infinities. Two infinities: relativistic addition A couple days ago I wrote about relativistic …

The radio spectrum is conventionally [1] divided into frequency bands that seem arbitrary at first glance. For example, VHF runs from 30 to 300 MHz. All the frequency band boundaries are 3 times a power of 10. Why all the 3’s? Two reasons: 3 is roughly the square root of …

Let c be a positive constant and define a new addition operation on numbers in the interval (-c, c) by This addition has several interesting properties. If x and y are small relative to c, then x ⊕ y is approximately x + y. But the closer x or y …

Dec. 29, 2020, 12:45 p.m.

William J. Milne [1] attributes the following theorem to Galileo: The area of a circle is a mean proportional between the areas of any two similar polygons, one of which is circumscribed about the circle and the other is isoparametric with the circle. So imagine a polygon P and let …

The previous post looked at finding the expected number of real zeros of high degree polynomials. If you wanted to see how many real roots a particular high degree polynomial has, you run into several difficulties. If you use something like Descartes’ rule of signs, you’re likely to greatly over-estimate …

Suppose you create a 100th degree polynomial by picking coefficients at random from a standard normal. How many real roots would you expect? There are 100 complex roots by the fundamental theorem of algebra, but how many would you expect to be real? A lot fewer than I would have …

Dec. 27, 2020, 10:44 p.m.

A matrix is said to have a saddlepoint if an element is the smallest element in its row and the largest element in its column. For example, 0.546 is a saddlepoint in the matrix below because it is the smallest element in the third row and the largest element in …

David Shelupsky [1] suggested a generalization of sine and cosine based on solutions to the system of differential equations with initial conditions αs(0) = 0 and βs(0) = 1. If s = 2, then α(t) = sin(t) and β(t) = cos(t). The differential equations above reduce to the familiar fact …

You can define a topology on the positive integers by choosing as an open basis sets the series of the form an + b where a and b are relatively prime positive integers. Solomon Golumb defines this topology in [1] and proves that it is connected. But that paper doesn’t …

Pick a positive integer k and take the product of k consecutive integers greater than k. Then the result is divisible by a prime number greater than k. This theorem was first proved 128 years ago [1]. For example, suppose we pick k = 5. If we take the product …

Logic vs C I recently made a mistake because I interpreted the symbol ^ in the wrong context. I saw an expression of the form a ^ b and thought of the ^ as AND, as in logic, but the author’s intent was XOR, as in C. How else might …

Applications of statistics often require working with for large x and fixed α and β. The simplest approximation to this ratio of gamma functions is You can do a little better [1] with where c = (α + β – 1)/2. To get more accuracy we’ll need more terms. Tricomi …

You can uniquely represent a large number by its remainders when divided by smaller numbers, provided the smaller numbers have no factors in common, and carry out arithmetic in this representation. Such a representation is called a Residue Number System (RNS). In the 1960’s people realized RNSs could be useful …