Category: General

I recently got into a protracted discussion in which the other person insisted that the fact that the character 0 is used in place value notation is merely a place holder is evidence that zero is not a number, but rather a concept we use to indicate the lack of a number. I have long…

It has been known since at least Euclid’s time that there are an infinite number of prime numbers. Here is his basic proof: Imagine that there is a finite set of prime numbers, P. Let N be the product of all the elements of P, plus 1. N’s factors do not include any element of…

I’m exploring if it’s possible to create a function in GeoGebra that would take an integer as input and create a simplified radical as output. For instance, it would take \(20\) as input and return \(2\sqrt{5}\) as output. I don’t know a way, so if someone does, please tell me. (Edit: There is the SurdText…

Here’s something that seems to confuse many people: \[\frac{1}{0} \text{ is undefined}\\ \frac{0}{0} \text{ is indeterminate}\] If some number, any number at all, divided by zero is undefined, then why isn’t zero divided by zero likewise undefined? And what does “indeterminate” mean anyway? Let’s start with a more concrete question: What is division? Assume we…

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…

What does “ten” mean? Here are some dictionary definitions: The number 10. (MacMillan) The cardinal number equal to 9 + 1. (American Heritage) Equivalent to the product of five and two; one more than nine; 10. (Oxford) Superficially, these seem like comparably valid definitions: Ten is the number that comes after nine, that is, 10….

Pascal’s Triangle Pascal’s Triangle represents the coefficients of a binomial such as \(x + 1\) raised to a power. Row n of the triangle lists the coefficients of \((x + 1)^{n-1}\). Here are the first few rows of Pascal’s Triangle: \[\newcommand\cn[3]{\llap{#1}#2\rlap{#3}} \begin{array}{c} &&&&&&\cn{}{1}{}\\ &&&&&\cn{}{1}{}&&\cn{}{1}{}\\ &&&&\cn{}{1}{}&&\cn{}{2}{}&&\cn{}{1}{}\\ &&&\cn{}{1}{}&&\cn{}{3}{}&&\cn{}{3}{}&&\cn{}{1}{}\\ &&\cn{}{1}{}&&\cn{}{4}{}&&\cn{}{6}{}&&\cn{}{4}{}&&\cn{}{1}{}\\ &\cn{}{1}{}&&\cn{}{5}{}&&\cn{1}{}{0}&&\cn{1}{}{0}&&\cn{}{5}{}&&\cn{}{1}{}\\ \cn{}{1}{}&&\cn{}{6}{}&&\cn{1}{}{5}&&\cn{2}{}{0}&&\cn{1}{}{5}&&\cn{}{6}{}&&\cn{}‌​{1}{} \end{array}\] For instance, row 4 is…

This one strikes me, and apparently others, as highly counter-intuitive, but it’s true because of mathematics! Take any two places in the world; call these points A and B. Take any two paths between A and B that are the same distance; call these paths C and D. Let C(x) be as far down path…

Just for fun… Here’s the pseudocode for the method of building a binary number from a decimal number, based on the Russian peasant method of multiplication: function mybin(mydec) { mybin = “”; do while mydec > 0 { if mydec is odd: { mydec = mydec – 1; mybin = “1” + mybin; } else: mybin…

Russian peasants do too much work There is a method of multiplication called the Russian peasant method. I’ve seen it mentioned here and there, but I was not explicitly educated in the process; it struck me as being more trouble than it was worth, and I didn’t previously bother to dig farther into it. I…