#Math

This "math overhaul" is ridiculous. Sure, if students are struggling, we should make math more intuitive for them, but that doesn't me that we should dumb it down or even eliminate it. Firstly, the concept that gifted students can just be shoved in with struggling students and then learn the same "big ideas" at different levels will never work, and advanced students will only be held back. Speaking from personal experience, schools already love to find excuses to keep students who are gifted at math in the same classes as everyone else, and this would only make that issue worse. A student at calculus level and a student at algebra level will never be learning the same "big ideas," and trying to force this on them will only drag students down. If parents ask for their child to be place in a higher class, even if they have real evidence to back up the fact that their child is significantly advanced, they may simply be told that their child was already being taught at their level, even when they are being lumped in with average and even struggling students.  Secondly, this new curriculum will barely even teach math. The idea of keeping math in touch with reality has been taken too far. Students who end up actually pursuing a career in math will not be prepared for the abstract and theoretical. Moreover, this idea that there are "multiple roads" to calculus is ridiculous. If you want to learn calculus, you will need to understand algebra and trigonometry, not data science, computer science, or financial algebra. The topics of data science, computer science, and financial algebra aren't even topics in math! Data science and financial algebra are applications of math to business and economics, but they are by no means themselves areas of math, and working with computers will at best build some intuition for mathematical thinking in the form of coding, but computer science is not in and of itself math. (To be absolutely clear, theoretical computer science, which is the math behind the mechanisms of computation, is certainly an area of math, but this is at the very least college level, and between this and their focus on the real world, it is certainly not what is being referred to here.) Lastly, the example of a class activity given is especially concerning. Going on about how "everyone has their own strengths" and we are part of a "mathematical community" could possibly be somewhat inspiring, but is better suited for a team-building exercise, and not math class. This activity is a waste of time that will not do anything to improve mathematical intuition. That "mathematical community" won't help you if they're too busy drawing colorful lines to do actual math.

Many things are "debatable." Certainly when it comes to art many things are up to interpretation. Even in science, there may be reasonable controversy. A certain set of available data may seem to suggest multiple different underlying facts. Math, however, is unique. By those who know it well, it is sometimes referred to as a sort of logical game. It has certain rules, and when applied in the right sequence, those rules can lead to much more complex things. Nothing is left to interpretation. Everything is known for certain. Even so, controversy manages to sneak in, especially among those who are less experienced. When people run into certain counterintuitive concepts, they often assume that mathematicians must surely be mistaken. In this post, I will cover a few of these areas of controversy, and then discuss how they come about, and what is actually true. I will try to keep this at a level that everyone should be able to understand, but for those who want the more technical details, I'll be covering them in a second post.

This is undoubtedly the most notorious area of controversy in math. There are numerous debates on this on DART alone. Many people see this counterintuitive fact for the first time and assume it must be wrong. In reality, due to a flaw in our decimal system, not every number has a unique representation. Nothing magical is happening, these are just two different ways to represent the same number, not unlike 1 + 1 and 2. One could reasonably ask what defines a decimal representation of a number, and I will cover that in my second post. For the less mathematically inclined, if you are willing to accept that 0.99999... is a well-defined quantity, and that 10(0.99999...) = 9.99999..., the following argument should be enough to convince you that 0.99999... = 1 is a logical necessity:

x = 0.99999...

10x = 9.99999... = 9 + 0.99999... = 9 + x

9x = 9

x = 1

This sum clearly diverges to infinity. What is really meant by this statement is that if we extend the notion of summation to divergent series like this one, we will get -1/12. (I'll discuss how this "extended notion" works in my second post.) Unfortunately, many people take the clickbait thumbnails a little two literally, and start arguing against this, even though there is nothing to argue about.

A significant number of people are now saying that infinity must be purged from mathematics, because it is not truly a valid concept. Here are the main three points of contention that I have seen:

Infinity is a tool for math which is useful for the real world. No one is claiming that infinity "exists" in a physical sense. This would be like pointing out that in the real world, there are no perfect geometric figures, so we should not use geometry.

I'm not really sure where people get the idea that infinity is not well-defined. Perhaps it is because there are multiple different definitions for depending on the context, but in every context that it is used, it is very well-defined.

Every paradox out there, even seemingly "sharp" paradoxes like the liar's paradox, are simply things which humans find it difficult to understand. Just because something is difficult for humans to understand does not make it invalid.

This simple paradox is widely debating even among (in fact, especially among) professionals. The paradox goes as follows: Sleeping Beauty is put under anesthesia on Sunday. A coin is then flipped. Regardless of its outcome, she is awoken on Monday and asked to guess the outcome of the coin toss. Then, if the coin landed tails, she is put back to sleep and looses all memory of being awoken on Monday. She is then awoken on Tuesday and asked to guess the outcome of the coin toss. You're Sleeping Beauty and you have just woken up. What is the probability that the coin landed heads?

Clearly the answer is 1/2. A coin flip is always 50-50.

Monday and heads, Monday and tails, and Tuesday and tails are all equally likely. If Sleeping Beauty is told it is Monday, the coin could have landed heads or tails with equal probability, and if she is told that the coin landed tails, it could be Monday or Tuesday with equal probability. Because of this, the answer is 1/3. This comes down to the fact that she is woken up twice when the coin lands tails, and only once when it lands heads. Given information can affect a probability, and in this case we are given that Sleeping Beauty has just woken up.

In reality, the controversy here is not because of the math, but because of ambiguous wording. What is meant by "You're Sleeping Beauty and you have just woken up?" Is it irrelevant, (as the "halfer" sees it) or is it intended to convey the given information "Sleeping Beauty just woke up?" (as the "thirder" sees it)

I'll start things off with this simple little example:

1 is defined as s0. (sn means "the successor of n")

2 is defined as s1.

Addition is defined as follows:

a + 0 = a for any a by definition.

a + b for b > 0 is defined recursively by a + sb = s(a + b).

With definitions out of the way:

1 + 1 = 1 + s0 = s(1 + 0) = s1 = 2

Do you accept this proof as providing 100% certainty that 1 + 1 = 2? Why or why not?

The title of this post is a question often posed by flat earthers to challenge the idea of a round earth. Because of its association with flat earthers, this question is often laughed at. I find this to be a shame, because its a quite interesting question. It also gives me the chance to do math and show off how smart I am!

I am going to start by answering a similar but related question. The earth is orbiting the sun at 66,660 mph [1]. Why doesn't the sheer speed of that orbit leave us all behind in the vacuum of space? The answer is inertia. Inertia is described in Newton's First Law of Motion: "An object at rest tends to stay at rest and an object in motion tends to stay in motion, unless acted upon by an outside force." How does that answer the question? Well, think about an airplane. Commercial airliners fly at about 600 mph. Do the passengers spend the entire flight pressed back into their seats at 600 mph? If one of them stepped out into the aisle, would he be propelled to the back of the airplane at 600 mph? No. Why? Because while the plane is moving at 600 mph, so are the passengers, the air in the cabin, and everything else in the plane. No additional force is required to keep them moving at that speed because of inertia. They will continue moving at 600 mph until acted upon by an outside force, i.e., the plane slowing down. In fact, if the plane were to vanish, they would continue moving at 600 mph until coming to an unpleasant stop due to the outside force of the ground. Military bombers have to account for this when they drop bombs. The bombs will keep moving at the speed of the airplane, so they have to calculate how far ahead of the target they need to be for their bombs to hit the target.

However, returning to the airliner, the passengers will be pressed back into their seats when the plane initially accelerates to 600 mph. This is because of Newton's Second Law of Motion, which states that Force = Mass * Acceleration. This tells us that force is only present when there is acceleration, and vice versa. So while the plane is accelerating, force is exerted on the passengers. While the plane is at a constant speed, no force is exerted on the passengers, and they are free to move about the cabin and eat disgusting airline food. Importantly, these effects are the same no matter what the speed is. A train traveling at 40 mph or a car traveling at 70 mph will have the same effects. This continues to be true no matter how large the number is. Inertia holds true whether you are traveling at one millimeter per millenia or 100,000 miles per second.

Returning to the question (finally!), we don't fly off the earth even though it is orbiting at 66,660 mph because it is traveling at a constant speed. If it were accelerating in some direction, we would feel a force. However, it is traveling at a constant speed, so we don't notice it.

Now, five paragraphs in, to the main topic. (Internet knights just love the sound of their own voice, don't they?) If the earth is spinning, that introduces a new consideration: rotation. Due to inertia, an object traveling in a straight line will continue traveling in a straight line, not a circle. So if the earth is spinning at 1000 mph, it should have left us behind long ago, right? Wrong. The big number of 1000 mph (1037.5646 if you want to be pedantic) looks impressive and scary. But when you do the math, it's anything but. Yes, the earth is spinning, but only once a day. That's 360 degrees per day, or 15 degrees per hour, or 0.25 degrees per minute, or 0.00417 degrees per second. So if we take your current velocity to be moving at an angle of 0 degrees, you are currently moving at 1037 mph at 0 degrees. Next second, you will be moving at 1037 mph at 0.00417 degrees, then at 0.00834 degrees, then at 0.01251 degrees, and so on. Notice what's happening? You're traveling at a constant speed, and your direction is barely changing. Going back to the plane example, if the pilot turned the plane 0.25 degrees every minute, would you notice? Not in the least.*

*Technically, you could if you put a level on the floor. Since planes bank in order to turn, you would notice the floor was slightly off level. However, my point is that you wouldn't feel your direction changing.

Intuitively, it's already clear why we don't fly off the earth. It's spinning so slowly that we can't even feel our direction changing, although we can measure it. But, for completeness sake, let's calculate how much acceleration we're all undergoing due to the earth's rotation. Let's also do the calculation in metric, because it's easier. The equation for centripetal force if acceleration = velocity squared / radius [2]. The earth has a radius of 6,378,000 meters, and it is rotating at 1,669.8 km/h or 463.83 meters / second. This gives us a centripetal acceleration of 0.0337 meters per second squared. That giant 1000 mph figure turned into a fraction of a fraction.

For the fun of it, let's find out how much force it would take to hold your average overweight American to the earth if gravity didn't exist. Suppose the overweight American weighs 100 kg. We can use Newton's Second Law of F=ma to find the answer of 3.37 Newtons, or 0.758 pounds. A quarter inch polyester ropecan hold 400 pounds, which is over 400 times stronger than needed [3]. So our conclusion is this: if gravity didn't exist, you could counter the acceleration of earth's rotation indefinitely using nothing more than string.

1. https://www.thoughtco.com/speed-of-the-earth-1435093

2. https://openstax.org/books/physics/pages/6-2-uniform-circular-motion

3. https://denverrope.com/rope-strength-guide/