In computer science, an algorithm might be classified as either lazy or eager. A lazy algorithm delays carrying out computations until the last minute, while an eager algorithm attempts to perform computations as soon as possible. Both approaches save time in different ways, and are appropriate in different situations. For example, if a program tends…
Category: Math
Metric space
We take for granted that we can measure distances. Most people take for granted that we can represent physical space mathematically and calculate distances between pairs of points. We generally think of ourselves as living in R3, or 3D Euclidean space. Since this is how we represent physical space, it’s generally the most useful (though…
Stories of math notation (part 2): The worst notation in mathematics
Sometimes notation is confusing. One way this can happen is when the same notation is used for multiple different concepts. Usually, if the same notation is used for different things, they occur in different contexts which allows them to be distinguished. In at least one case, the same notation is used for different things in…
Stories of math notation (part 1): y should come before x
Or rather, perhaps both the symbols and positions for x and y should be switched. The convention of writing coordinates as (x, y) is so deeply ingrained that, in my opinion, it causes confusion in situations where x doesn’t naturally come first. Reasons for x first As I see it, the primary reason for writing…
A different view of the multiplication and division algorithms
The “traditional” methods for multiplying or dividing numbers have generally been taught by rote memorization or with minimal explanation. By reframing these processes, I hope to make it more clear what is happening. Column method of multiplication This is similar to the column methods of addition and subtraction, although multiplication is slightly more complicated. Here’s…
Pathological mathematics
Logic sometimes breeds monsters. Henri Poincaré In math, we often speak of things being “nice,” “well behaved,” “elegant,” or even “beautiful.” The flip side of this is those things in math which often remain in shadow: the unfriendly, the unruly, the awkward, and the ugly. In some cases, inelegant solutions are apparently unavoidable. In other…
Probability estimation vs probability calculation
If I take the bus downtown today, what are the chances I’ll run into my friend (let’s call her Alice)? Suppose I’m on the bus for an hour. According to the local metro, there are approximately 31 people using a single bus per hour of operation. Assume that during the hour I’m on the bus…
The language of balanced parentheses
Grammatically, parentheses must be balanced, meaning every open parenthesis ‘(‘ must eventually be followed by a close parenthesis ‘)’. Parentheses can be nested, meaning there can be parentheses inside parentheses. If we were to take a piece of writing and remove all letters and punctuation except parentheses, we might end up with something like: ()()()(())()(),…
How compression works and why JPEG artifacts look like that
Do I look like I know what a JPEG is? In a computer, all information is represented as bits, strings of zeros and ones. However, bits can more or less be interpreted as generic numerical data written in base 2 instead of as decimals. In other words, in a computer, all information is represented as…
The state of math in the world today
I went to the grocery store today and at the self checkout, one of my items was not recognized by the barcode scanner. It scanned successfully, but it wasn’t in the system and couldn’t give a price. An employee came over to help. He was young, no older than his 20s, though he looked like…
thoughts on mind 