## Archive for August 8th, 2022

### Not just for mathematicians? Are most of us so dumb to figure out daily math concepts?

Posted on: August 8, 2022

# Mathematical Concepts that All Should Understand

Joshua Reingold

July 24, 2022

One of the older fields offered in college today is that of Mathematics.

Math has been taught (in some shape or form) since at least the time of Plato and Socrates and it dates back much earlier in the Middle-East civilization.

.Math is usually considered a rather difficult subject taught to students or as a major and few find it easy. Fewer still suspect that it will ever be useful in their lives.

A time-honored question that is asked quite often is “When will I ever need to use this

I think that Math does have a place and that there are lessons one can learn from Math. This story will attempt to contain Mathematical concepts that all individuals (regardless of whether they’re mathematicians or not) should understand

# Mapping

In Matrices or Set Theory, one of the types of Operations that can be done is that of Mapping.

In mapping, one takes a certain collection of the data, applies a function to at least some of the input, and there emerges a new collection.

A simple example might be when one has a simple 1×3 matrix of the values [3, 2, 1]. Now, let’s say that we apply a function of multiplying each value by 2 and adding 1 to all input in that matrix. As so:[3, 2, 1] * 2 + 1

We will end up with a 1×3 matrix of the values [7, 5, 3]. A Mapping Function can be applied to a lot of different matrices of data sets.

One might not initially see the importance until they think of real-life scenarios. I.E.

Just as a Matrix can be mapped, so can details about an event be mapped. Imagine a certain criminal trial or case on the news which manages to garner a certain amount of attention.

Or conversely one can imagine a certain trial that hardly garners any attention at all. One might be puzzled as to why this is the case until they examine the circumstances of the case.

By circumstances, I mean The Perpetrator(s) of the crime, The Victim(s) of the crime, The crime committed, The Date of the crime, The setting of the crime, The motives of the crime, And perhaps other details

Now, imagine that these details could be changed slightly, and this might change the level of interest one might have in a certain trial.

By changed, I mean that one’s initial interest might either go up or down depending on the way the facts are presented to them.

There are many instances where this might happen. On the much less serious side, this is probably done all the time on reality television shows. I.E.

A small incident is blown out of proportion by the production crew changing certain details surrounding the incident. One could say that they “mapped” the incident.

Reduction Image

The next entry, somewhat similar to mapping, is that of Reduction. Technically, Reduction is a type of Mapping in which one takes the initial data and aggregates them through some means to get data of different cardinality.

For example, we could use the matrix used in the prior section of [3, 2, 1]. Now, we will attempt to reduce this matrix by finding the average of all entries in it. I.E.(3 + 2 + 1)/3

This will give a single data entry of [2]. The concept of Reduction can be applied to many different data sets and via many different functions. Such also presents itself as something used in the real world.

As unrelated as it might sound, I recall an answer written on the “Question and Answer” website Quora a time ago about the character Sgt Barnes from Platoon.

Image Credit: https://historica.fandom.com/wiki/

Bob_BarnesIn such an answer, it was stated (and I’m paraphrasing) that Sgt Barnes was not so much a character as a trope; I.E., a character made up of many different characters all aggregated into one.

This probably happens more than one would care to admit. I.E. One takes a collection of the data and comes up with a composite or aggregate way to look at it

Frequency

When one begins studying the higher levels of Algebra or Precalculus, they will begin to examine different functions (I.E. Equations with not just one variable but perhaps two or more variables).

If we were to examine a very simple function of “y = x + 1”, then we have one variable which is considered the independent variable x and one variable which is considered the output variable y.

For this function in particular, the output is just one more than the value of the input.

A certain amount of the possible functions that one can examine are not continuous. By not continuous, I mean that a certain output is generated a finite number of times. E.G.

Looking at the equation of “y = x + 1”, then the output value of 10 is only generated when the input variable is 9. One also has certain functions that are considered continuous; Which means that a certain output could hypothetically occur an infinite number of times or with an infinite number of input variable values.

Perhaps one of the simplest examples I can think of would be the sinusoidal function of “y = sin(x)”. If one plots this function on a graph, they get something that resembles a wave (always returning to the same spots)

As it relates to frequency, frequency is just the inverse of the “Period” of a function; the period being the time it takes for something to repeat itself.

The concept of continuous functions and frequencies exists in the Engineering world as well. E.G.

There is one type of current called “Alternating Current” that takes the form of a sinusoidal function rather than a constant current applied to a circuit.

But does this have any place for non-STEM majors? I would say yes.

The reason is that many different events in our lives do not simply occur once but rather multiple times. For a few examples, I will consult the world of fiction twice.

The first example occurred in the 2001 War film Enemy at the Gates and the second occurred in the 1996 Action Thriller The Rock (Spoiler Alerts for both movies).

In Enemy at the Gates, the protagonist must take out some German soldiers via his Sniper rifle despite being outnumbered. His solution is to take his shot precisely when an artillery shell lands, thus, using the ambient noise to hide the shooting of his rifle.

It’s possible that he simply listened to the artillery shell before taking his shot but also could’ve timed his shot precisely to when the next shell landed (E.G. If there was a 10-second delay from the time the current shell landed to when the next shell landed then he would simply wait 10 seconds).

In The Rock, a character must gain entry into a locked room by going underneath the incinerator. Underneath the incinerator lies an opening just large enough for an individual to lie prone and crawl through or roll through.

As the movie makers couldn’t make it too easy, the incinerator has flame bursts and large metal gears (independent of each other) that periodically rotate to the opening beneath it. The character claims to have “memorized the timing” and he has.

While one flake burst or metal gear rotates through, the character remains still underneath the previous ones (only moving after the coast was clear).

Although fictional, these examples prove the value of understanding frequency. In the event something happens repeatedly, they should be able to time these events and base their response around the proper timing of the original event.

If one studies math long enough, a term that they might end up hearing is that of “infinity”. As any studious mathematician will surely say, infinity is not so much a number but rather a concept.

The concept is difficult for me to explain (as I am not a mathematician), but I will explain it as the eternity of numbers. The most obvious example of where this presents itself is when one looks at the number line and begins to trace the integers.

Even the individuals with the longest attention spans or computers with the most computational ability will not be able to reach the end because there simply is no end.

Our number line goes on forever or is “infinite”. In this way, one perhaps takes a certain number and either continually adds to it or multiplies it forever.

This is just one way infinity can present itself. I.E. Just as a number can be multiplied an infinite number of times, a number can also be divided (by any value really) an infinite number of times. E.G. 40/2 = 2020/2 = 1010/2 = 55/2 = 2.52.5/2 = 1.25…

Why does this matter? Because many times the nature of humans being somewhat flawed and prone to error means that the number (in any arbitrary application) that is considered “real” is too much for us to handle.

If that sounds too cryptic, I shall use music as an example.

When we play a piece of music, there is a certain tempo that we play the piece at. Such usually corresponds to a certain number of beats per minute. However, a tempo that is too fast will mean that the player will have a difficult time playing, and (to practice), it might be better for them to play it at a slower pace.

So, they might use some simple math and divide their normal tempo in half to get the playing tempo at which to practice the piece. If that also is too fast, they can divide it once more and continue until arriving at a tempo suitable for them.

Technically for this example, one will usually need to round up or down if their current tempo divides to a decimal number, but the idea remains the same

.Perspective

In the field of mathematics and science, there are certain units used to quantify something. E.G. Weight is measured in pounds or Newtons. Mass is usually measured in grams (or grams multiplied by a certain scaler). Distance is measured in meters, feet, or some other unit of length. Time is measured in seconds. Temperature is measured in Fahrenheit, Celsius, or Kelvin. Speed is measured in velocity over time .Loudness is measured in decibels .And plenty of other examples.

There do exist, however, some numbers that need not have any units attached to them. E.G. The number “4” can exist by itself without any units or with units. These numbers can exist on their own or as scalers for other metrics. Why does this matter for most ordinary people?

In the most literal sense, one can design something small (like a house) to see how this same design will behave when big

Past this, one looks at the dimensions of something and determines a way to create something similar but of slightly reduced dimensions. An odd example I can think of would be that of paintball or airsoft.

Before the rise of Simunition, one knew about the possibility of getting into a gunfight, and soldiers or police officers practiced for it, but few were willing to practice for a gunfight with actual gunfire; More than likely, the scenarios practiced were something of a kata (E.G. A Soldier would shoot three times and then move to a different position).

When we examine gunfighting in particular, the only attribute we care about is the speed at which bullets are flying. Such involves bullets flying at fast enough velocities to inflict bodily harm or even death. But what if we could take these bullets and make them fly at a lesser velocity (one less likely to inflict such damage)?

This was the idea behind using Simunition and (in the literal sense) one is taking a smaller perspective of what will happen in real life and practicing with this.

The concept of Perspective also shows up in the field of Psychology. Although I am not a psychologist, I would imagine that an individual might have problems with a certain endeavor or activity if looking at it from the wrong perspective

In the first case, they simply look at the whole picture (instead of a small portion) and get overwhelmed.

In the second case, they simply look at too small the picture and cannot see why their actions are important.

If one thinks hard enough, one can come up with other examples.

When I searched for the mathematical concept of similarity, it referred to the notion of certain shapes (E.G. Triangles) being considered similar to other shapes. This section here, however, has more to do with some events being similar in general.

If one was to talk about the concept of “Similarity”, they would have to examine what defines something (either an event, action, or product) and then compare it to something else. If one’s perspective was strictly mathematical in nature, then the things that define something would be everything that could be quantified by the field of science. To use apples as an example, one might examine the Weight (or mass) Shape/Size/Color of the apple and use this as a means of comparing it to another apple.

Some of these variables (like the color or perhaps the weight) could be said to be discrete in that they cannot be divided any further. Other variables (like the weight and size) could be continuous in that one can always find a smaller fraction by which to judge them.

When comparing it to another apple, one will take a combination of all the different attributes and use them to categorize the apple (the same way the other apple will also be categorized based on its attributes). There could be multiple methods for aggregating these attribute values.

Why does this matter? Because in the real world, things are seldom how they appear in a textbook. Life gets messy and small differences exist that will prohibit something from falling neatly into a little category. Consider the many different punches that can be thrown by a boxer.

If one finds a book on boxing basics, they will no doubt notice many different types of punches including but not limited to Jabs/Crosses/Straight Punches (the kind one throws when one’s stance is squarer) Descending Punches/UppercutsHook Punches/ Overhand Punches/ …

The purpose of this section will focus more on uppercuts and hooks though.

The way I was taught to throw an uppercut or hook punch, was to position my forearm some 90° away from my upper arm (thus making one’s arm appear like a hook) before throwing the punch. The two punches are different in a few different ways

The uppercut one throws will (ideally) be straight down the centerline of the individual. Conversely, the hook will be thrown anywhere from their arm being 90° perpendicular to their torso to perhaps being 45 away from their torso (depending on their height, their opponent’s height, and one part of their opponent’s body they intend to hit).

The nature of throwing the punch at different angles means that one’s rotator cuff will be in a different position

For the uppercut, one’s footwork will usually be such that their lead leg is in line with their punch or (if they intend to punch with the reverse hand) off by half the length of their body; the stance taken will usually be the traditional or southpaw stance from boxing.

For the hook, one’s feet will usually be further apart (as it would be difficult to generate rotational power with one’s feet close together).The nature of throwing the uppercut along one’s centerline means that one’s feet can remain in the same general position. For the hook, one must pivot their body and therefore their feet; thus, they begin the punch facing one way and can rotate as much as 90° another way.

For the uppercut, one’s fist will usually be such that their knuckles are facing down (towards the ground) whereas for the hook, one’s knuckles will either be positioned away or up towards the sky.

And these are all the differences between the two punches; there is a third hybrid punch (called the Smash Punch) that I will discuss later on. As such, one could look at them as distinctly different punches. They do, however, share the commonality of a similar arm position (as far as their forearm’s position to their upper arm), and therefore (from the perspective of basic trigonometry) the punch is thrown from the same radial distance.

As one understands from the section on infinity, the degrees possible for which the punch can be thrown are technically infinite. The integer degrees are finite but there is siempre a fractional degree positioned in between two integer degrees. This can further be divided up multiple times.

If one wished to view a punch distinctly based on the angle at which it was thrown, there would technically be an infinite number of different punches (all of them completely unique), and this does not even count slight tweaks to the punch’s radial distance.

This means that if one treated each punch as unique, then for them to train all the different punches, they would have to practice an infinite number of different punches.

A different option is to categorize all the different punches based on the angle at which it is thrown; one could denote the angle as “a” and consider a punch thrown in the third quadrant. E.G.45 < a <= 90: Hook punch30 <= a < 45: Smash Punch30 > a: Uppercut

If one chooses the above categorization, then there could be multiple different “hook punches”, but they all share enough similarity to be considered the same type of punch, and practicing one will not be radically different from practicing another.

If one has no interest in boxing, this presents itself as useful in many environments where a certain scenario (or group of scenarios) need to be trained. E.G. A company could try to prepare their employees for every possible scenario that comes their way, or they could find a finite few number of generic scenarios and use these as the training scenarios.

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