Some Infinities are Bigger than Other Infinities

Hello, good day to you all!
I'm really sorry about not posting at all, I'm on vacation.
And now we're out of school, we can post more freely and you know, talk about our interests and discuss random things rather than focus on math.
So hello. It's Strix Spell here. I think Kazuma decided that she does not want to post anymore and Annis might, occasionally. I like to keep a schedule but well, that's near impossible now.
As well as talking about math, we can expand it to some more science as well.
Today, I'm focusing on the popular quote by John Green in his new book The Fault in Our Stars which I read and I'm going to review in my other blog.
Anyway, the quote is: "Some infinities are bigger than other infinities."
The thing about infinity, it is endless. It goes on and never stops, like a set of numbers (1,2,3,4...) So how are some infinities bigger than other infinities?
Hazel incorrectly explains it as: the set of numbers between 0 to 1 is a smaller infinity than the set of numbers between 0 and 2. While that is untrue, the statement some infinities are bigger than other infinities is still true.

That video explains a lot better than I can. But, if you are too lazy to watch the video, here is a random summary. While the set of numbers between 0 and 1 is the same number as the set of numbers between 0 and 2, we know that the set of numbers between 0 and 1 is bigger than the set of positive integers (1,2,3,4...). 

Okay, I can't explain so haha! You have to watch the video to understand. 

Anyway, basically, what I am trying to point out is that some infinities are bigger than other infinities. Yay!

Goodbye Answers

The answer key, also known as Tricky and Twisted Math has been reverted to draft because Kazuma, Annis, and I are doing a class thing and we need the students to figure it out on their own rather than cheat so, sorry!

Problem of the Day #10

Hooray, we are on 10!
So, more patterns, oh, and this has something to do with twin primes.

What comes next in this sequence? 4, 6, 12, 18, 30, 42, 60, 72, 102, 108,...?
Have fun solving!

Problem of the Day #9

Sorry for not posting yesterday.
There were some complications but, I'm here today!
I like patterns so here is another riddle for patterns.
1 = 3, 2 = 3, 3 = 5, 4 = 4, 5 = 4, 6 = 3, 7 = 5, 8 = 5, 9 = 4, 10 = 3, 11 = ?, 12 = ?
What do eleven and twelve equal?

Reminder!!

Just a reminder, on Friday, we are going back to the original schedule with Annis posting on Mondays and Tuesdays, me on Wednesdays, Saturdays, and Sundays, and Kazuma posting on Thursday and answering the problem on Friday. I guess it will be a weekly question rather than a daily problem. 

Problem of the Day #8

I'm covering for Kazuma because we usually work together on posts but I was late for the meeting. It is completely my fault.

Figure out the next part of this pyramid:

1
11
21
1211
111221
312211
13112221

So, what comes next? This will help you sharpen your pattern senses.

Tessellations

Kazuma is helping me again. :) Tessellations are a pattern of shapes that fit perfectly together. Tessellations are classified as regular, semi-regular, and other tessellations. 

Regular tessellations are repetitions of 1 polygon. There are only three types of regular tessellations. Triangles, squares and hexagons. 

Semiregular tessellations are made of two or more regular polygons. There are 8 semi-regular polygons. For pictures click on this link. 

Other tessellations include demi-regular tessellations but mathematicians disagree on what defines them. 

Some authors define demi-regular polygons as orderly compositions of three regular and eight semi-regular tessellations. Other authors define demi-regular tessellations as a tessellation having more than 1 transitivity class of vertices.
Then, there are tessellations that have curved edges and circles.
Escher is a tessellation artist. He created pieces of art made with objects that can tessellate like the flying horses above. You can make your own tessellation with polygons at this tessellation maker.

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Problem of the Day #7

How can you add eight 8s to get the number 1,000? (only use addition)

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Problem of the Day #6

Angela has 71 cents. What is the least number of coins she can have?

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Problem of the Day #5

Ok, I forgot to cover for Kazuma so I'm doing three problems today. Kazuma will be helping me.

The problem:

If one nickel is worth 5 cents, how much is half of one half of a nickel worth?

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e

e is a mathematical constant. A mathematical constant is any well-defined real number that is significantly interesting in some way. e has a value of about 2.718. e is an irrational number. It is named after Leonhard Euler and can also be called Euler's number because Euler proved e is an irrational number.
You can calculate e with this formula: 1 + 1/1! +1/2! +1/3! +1/4! +1/5! + 1/6! + 1/7! and so on so forth. if calculated out, the first few terms are 1 + 1 + 1/2 + 1/6 + 1/24 + 1/120 = 2.718055556. 
e is found in many interesting areas. To memorize e, remember this sentence:
to express e remember to memorize a sentence to simplify this.
(count the letters).
The value of e is used to find a nonlinear increase or decrease. e occurs naturally with some frequency so it is the base of natural logarithms.

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next time: logarithms

Back to Napier!

By the way, Annis and I are collaborating! :D
You probably figured out this post is about John Napier, but he's a good mathematician. Today, you'll learn about his invention of logarithms!! A logarithm is a quantity representing the power in which a base must be raised to in order to produce another number. The formula is log b( x). Well, it doesn't look like that because the b is below log and x. Anyways, the b is base and the x is the number to be produced. For example, log 3 (9) is 2 because 3 to the power of 2 or 3 squared is 9! Get it??

Well, there is also what is called common logs. Common logs are regular logarithms just by 10s. So, the equation for common logs is log x. So, log 100 is 2 because 10 squared is 100.  Now, you might just be thinking, why the heck do I need logarithms because they seem ridiculous.When they were invented, they were used to calculate big and long numbers because they didn't have calculators, which is really funny because Napier helped invent the calculator. Now, they are used to calculate pH scales, the Richter scale, and astronomers use logarithms too. Engineers use logarithms.
Logarithms are everywhere!
You might be thinking at this point, logarithms seem to be similar to exponents. Well, not exactly. Exponents of a number says how many times that number is multiplying itself. Logarithms are the opposite. It asks the question, what exponent produced this number?

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P.S: I how you enjoy our recent collaborations,because we might just have more!!

Pythagoras

Kazuma has already done a post about the Pythagorean Theorem. Who was behind this equation?
Why, Pythagoras, of course.
Pythagoras is recognized as the first mathematician. It was probably Pythagoras who said that numbers can uncover the secrets of the universe.
Pythagoras is known the best for his Pythagorean Theorem where  .
This formula is used to find the approximate length of the hypotenuse of a right triangle. Note, this only works on right triangles.
Pythagoras paved the paths for all aspiring mathematicians.

Next Time: e

More Euler, Less History!

Ok, well first of all, Euler is awesome.However I couldn't really find history on his really cool formula, so I figured I'd just blog normally. Anyways, I thought I'd blog about Euler's formula. Euler's formula, first of all is V-E+F=2  or V+F-E=2. V for vertices, E for edges and F for faces. Basically, this means the number of vertices and faces of a convex polyhedron(Click here for a definition!) will always be 2 more than the number of edges of the same polyhedron. Try it out! If a convex polyhedron(in this case a cube) has 8 vertices, 6 faces and 12 edges, what does the formula look like? That would be 8+6-12=2. You may find that some numbers don't work, but that's probably because your solid is just not a convex polyhedron. However, V+F-E can also equal 1and/or other values, so Euler's characteristic was made. He's pretty smart, huh? Anyway, the general formula is V+F-E=x.That is where Euler's characteristic comes in. Umm... so this is an ultra short blog, read up and  things I already explained here! Because Math is Fun! ;D See Ya!

Look it's Euler!!

The Math of Rock, Paper, Scissors

Yes, scientists have studied how to never lose at rock paper scissors. Really.
A Chinese mathematician, Zhijian Wang, studied random people playing rock, paper, scissors. He noticed that the winners tend to stick to their winning strategy and losers try different methods.
The two players start by using random strategies. For example, if Player A chooses rock and Player B chooses scissors, Player A wins. Player A will most likely use rock again while Player B will try something else. If Player A used rock, Player B can assume that Player A will use rock again, therefore using paper to beat Player A. After that, Player B can assume that Player A will use the nxt strategy in the sequence, scissors and use rock to beat Player A, again.
To always beat someone at rock, paper, scissors, is quite simple. You must use the next strategy in the sequence. Easy.
Next time you go two out of three, remember what I have told you.

Next Time: Pythagoras

Problem of the Day #4

Chiyo is picking between 7 different candy canes to put on her Christmas tree. However, she only wants 3 to go on the tree. What is the number of combinations of candy canes Chiyo can pick if she is only picking 3 candy canes at a time?

The Invention of the Calculator

  I am so sorry for not posting yesterday, but I am posting now. Anyway, you might be thinking, Why are you doing this again??? You already did the abacus, so they are similar. And yes, that's correct, but it's kind of cool, because it relates to another mathematician I blogged about. Blaise Pascal. Pascal first created a sort of calculator. Pascal created it as a way to help his dad handle taxes, not knowing what might become of his invention. Later, Gottfried Leibiniz picked it up, and added the addition, subtraction, multiplication, and divide button on the calculator for easy use. John Napier also added a set of metal rods he called, Napier's Bones, for the multiplication table.  That's basically it. I'll post later today, so do not worry. So, for now, See Ya!

Problem of the Day #3

The cylinder's diameter was 4 inches and it was 8 inches tall. Shiro and Jamie were trying to find the volume of the cylinder Shiro says that you need to do 3.14 times 2 squared times 8. Jamie says that you need to do 3.14 times 4 squared times 8. Who's right? Explain.

Comment your answers!

The Two Types of Numbers

There are two types of numbers. Solitary and Friendly/Amicable numbers.
The Solitary numbers don't have friends. Ok, it isn't exactly like that but it's similar.
To define a solitary number, we must first know what a friendly number is. A friendly number, also known as an amicable number, is a number in a friendly pair. The pair of numbers are so related that the sum of one of the number's proper divisors is the other number.
A Solitary number is not in a pair like that. There are many numbers that are thought to be solitary but proving it is hard.
It is hard to find a friendly pair. In fact, most numbers are solitary, including all primes and prime powers.
One of the famous unanswered questions is, is 10 a solitary number?
Well, if you can find a pair for ten, please comment and tell us.

next time: the math of rock paper scissors

Problems of the Day #2

Jake bought two rectangular containers of coffee powder. The decaf coffee container is 1/4 feet long, 2 inches wide, and 7 inches tall. The regular coffee container is 3 inches long, 1/4 feet wide, and 3/4 feet tall. Which coffee container would hold more coffee powder?

Comment your answers!