An Elegant Mathematics Proof

One part of math that I love is the sense of astonishment when you discover something new in math. This “ah-ha” moment is why I love math so much. It drives my passion for math, and getting better at it. An “ah-ha” moment that I want to share with you is one relating to geometry. There are a lot of astonishing proofs in geometry, but this is just one. It is so simple and elegant, and understandable by anyone.

Here is the statement: Any angle inscribed in a semicircle (the two outer vertices are touching the corners of the semicircle, and the middle touches the curve) is a right angle! The illustration below will help visualize.

The proof for this statement is surprisingly simple. The next image explains it well.

Reflecting the semicircle makes a full circle. This shows that the shape formed will always be a rectangle. And a rectangle has only right angles…

If you were to reflect a semicircle (in this case across AB) then the ensuing shape will be a rectangle. The image above shows that happening. So no matter how you inscribe the angle (look it up if you still don’t get inscribed angles) the resulting angle will always be a right angle if you drew it in a semicircle.

How cool is that?

The Power of Apple

Around 700 million people use an iPhone. I am baffled by this number. I can’t even think straight when I think of 1 million, but 700 million? I can’t imagine the true scale of 700 million users. That means that a staggering 1/3 of all cell phone holders have an iPhone. But this astounding scale that Apple has gotten to was relatively overnight. the iPhone launched in 2007. In a mere decade, Apple has built an empire of services, software, and hardware. Apple has single handedly changed the entire world for the better.

Practically no one knows the true impact of the iPhone. It changed every single aspect of the world. Every single job has been speed up and completely turned sideways since the iPhone. A practical example is of a very common job, the store owner.

Before the iPhone, managing the item supply would be a very tough job. It still is today, but the iPhone massively streamlined this process. Before the iPhone, a store owner would have to call up every single supplier that they were getting supplies from and physically go to the supplier to get their goods. The iPhone enabled this same store owner to easily schedule all of this automatically. Automatic delivery, automatic scheduling, an automatic win. This would greatly impact the scale of all businesses, and not just store owners. The ease of communication allowed for much quicker operations.

When my dad would go on a road trip with my mom, it was a big planning and logistics ordeal. The route, the hotel, the restaurant that they would stop in, all of that would have to be meticulously planned to the dot. If this plan wasn’t executed, then all they had with them was their lousy GPS which didn’t work very well. After the iPhone, all of this was eliminated. The route could be planned along the way. The trip was a new experience, with less planning and more of having fun, which is what a road trip is all about.

There are other countless other ways in which the iPhone revolutionized the world and life as we know, and the above example are just a small sliver of what the iPhone did.

But how did the iPhone come to be? Back when the iPod came out around 2005, Apple was doing great. Their sales were through the roof, they were all making lots of money, and financially, Apple was a very happy company. But Steve Jobs, the iconic leader of Apple, wasn’t satisfied. He saw the potential for a new market in the cell phone industry, and the technology was rapidly growing at that time. If they didn’t make something to break the roof off of this industry, there were in big danger of being splashed out of the water.

So Apple engineers, including Steve Jobs and his partner, Steve Wozniak, worked to make a new phone that would completely shatter the expectations of that time. The concept of not only acting as a phone, but as an entertainment and even gaming machine was the Apple concept. With the help of the newer touchscreen research that Apple had been doing, they came up with the first iPhone in 2007. After the iPhone came out, Apple truly dominated the phone industry. Sales skyrocketed to even higher numbers, and Apple truly became what it is today.

The Apple Brand

Apple is a very dominant force in the hardware and software industry. With countless products such as the iPod, iPhone, and the Mac, Apple is a force of nature. But are Apple products really worth the cost?

The new iPhone X started at $1000! One thousand dollars for a phone?! That figure is mind blowing. But there is good reason for that price, and a reason that people are still willing to pay $1000 for a phone. It all comes down to the Apple brand.

Apple, as you may know from the previous post, “The Power of Apple”, launched the landmark iPhone in 2007. Since then, Apple has been dominating the industry. Even back then, the iPhone was almost like a status symbol that people would flash to each other to show that they were cool and techy. And right now isn’t any different. Back when the iPhone 6 came out, my dad would joke about flashing it out in his office as a show off gesture. Apple has become a symbol of a good, quality product. They first planted this idea on the ground with the iPhone, and then kept making the product better with newer versions of the iPhone.

Before I became acquainted with the quality of an Apple product, I thought that $1000 was way too high for a phone, and was surprised that the phone even sold well. But then my mother got a Mac Book. When she asked me to carry it, I was baffled by how light it felt. It was even lighter than the Chromebook I use for school, and that is about half the general size! I then opened the ultra thin lid to reveal a nice, glowing keyboard. The letters would glow slightly, making for easy typing. The keyboard also felt nice in my hands, it was a pleasure to type with it. But the one thing that impressed me the most wasn’t necessarily the biggest feature on there. One day when my mom made me charge the Mac Book, I guided the cable close to the Mac Book, and to my bewilderment, the charging cable was magnetic! Who knew that you could make the long lasting charging cable even better? The very fact that they had thought out the design so well, made it feel and look nice, it all made for a very satisfying experience while using the Mac Book.

The magnetic charging cable wasn’t necessarily a huge improvement. But it was unique, and elegant. The designers of this product must have perfected every nitpick of the design, down to the charging cable. This experience with the Mac Book showed me the quality of an Apple product. And although the Mac Book may not be totally revolutionary in its functionality, the very quality and feel of the product allows it to be a very dominant product on the market today.