Because of the way mathematical ideas build on other mathematical ideas, the order in which these ideas were discovered/invented is often the same as the order in which they are taught. All humans typically start where humanity (is thought to have) started, with counting objects and identifying simple shapes. From there, we begin making calculations….
Tag: Math
Operations
You’re certainly already familiar with several mathematical operations, such as addition, subtraction, multiplication, and division. You may also be familiar with the logical operations AND, OR, NOT, XOR, and IF…THEN. More generally, an operation can be defined on any nonempty set. Operations have the following properties. Arithmetic operations We’ve already discussed a few of the…
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: ()()()(())()(),…
thoughts on mind 