Numbers
What is a 20% tip of $35.00?
$7.00
0.20 times 35.00 = 7.00
State the number
pi
to at least 2 decimal places.
3.14
A function
f(x)
is a machine that takes in an x-input and spits out an f(x) output. The machine must NOT spit out a different f(x) output provided that the same x-input was used. Note the parenthesis here indicate "input" rather than multiplication. This is purely symbolic.
In Stein's Gate, Okabe was testing a microwave out. Isolating all other variables, Okabe placed an input "Banana" into the microwave. The resulting output of the microwave spat out a heated banana.
Okabe repeated the exact same situation with the same type of banana and controlled variables. However, this time, the output was not a heated banana but a "gelbanana". Would this microwave be considered by definition a mathematical function?
No. For a single input "banana" there were two different outputs: A heated banana and a gelbanana. In the series, it was revealed that the experiment was not fully "controlled" and the rotation of the microwave plate was changed. Hence, Okabe made a "spoiler" assumption based on the microwave not acting as a proper "function".
The Pythagorean Theorem
a^2+b^2=c^2
is used to calculate the length of the sides of a right triangle where a,b are the legs of the right triangle (doesn't matter which) but c represents the hypotenuse of the right triangle (largest side).
Calculate the hypotenuse of a right triangle given both legs are 1 ft long.
sqrt{2}
There are claims that after a math cult meeting, Pythagoras killed a devoted follower by tying the follower to a bed. Later leaving to individual to drift in the ocean during his sleep all because the follower proposed/discovered this number.
This is a number with decimals that neither terminates nor repeat
using Pythagorean theorem with a=1, b=1. At the time, the existence of such number was not fathomable.
The claim is a popular story amongst mathematician however, there are no supported evidence since Pythagoras is dated in 500BC.
Guess this opening song.
Sword Art online
Is
0.\overline{999}=1?
Yes,
0.9999 is NOT 1, however, the idea of
the ending decimals as repeating is equivalent to:
\frac{9}{10}+\frac{9}{100}+\frac{9}{1000}+... =\sum_{n=1}^{\infty}\frac{9}{10^n}
This converges to 1 as a geometric series. This means that mathematically we can show,
0.\overline{999} \leq 1 and 0.\overline{999}\geq1
In mathematics, we say two expressions are "equal" if we can write both expressions greater than or equal to AND less than or equal to the other expression. The formal proof of this statement is outside the scope of this event.
What symbol do we associate mathematically with
sqrt{-1}
i
We call this the imaginary number/component. This was constructed since there were various "quadratic" equations that were NOT solvable with real numbers. It was proven that all quadratic equations must have at least 1 solution (Fundamental Theorem of Algebra), hence the construction of
i
Let
f(x) = 5x+2
Calculate the output
f(-1)
given
x=-1
is our input.
-3.
We can calculate this as follows:
f(-1) = 5(-1)+2 = -5+2 = -3.
Logic plays a big role in mathematics. The validity of a declarative statement is either true or false.
In an "if... then..." statement, we say that "if.. then.." entire statement is TRUE all the time except only when the cause statement is TRUE AND the effect statement is FALSE.
Determine the validity of this entire statement as True or False:
"If Waley is gay, then the world is round."
The entire statement holds TRUE.
Since the initial statement "Waley is gay" is false, and the effect statement "the world is flat" TRUE, the entire conditional holds true.
The "if... Then..." statement is always TRUE UNLESS the first part is TRUE and the second part is FALSE. In the case that "Waley is gay" was a TRUE statement, the entire statement will still hold TRUE since "the world is round" is TRUE.
Guess this opening song.
Code Geass
Follow/finish the pattern:
2^5 = 2 \times 2 \times 2\ times 2\times 2 = 32
2^4 =2 \times 2 \times 2\ times 2 = 16
2^3 =2 \times 2 \times 2 = 8
2^2 = 2 \times 2 = 4
2^1 = 2
2^0 = ?
1.
If we work backwards, we can divide each of these output values by 2. Then
2^1=2 \implies 2^0= 2\div2 = 1
\mu
What is this greek symbol called? Hint: It is a combination of two English letters combined. In the Pokemon series franchise, a popular Pokemon was named after this symbol.
mu or "mew".
Note the symbol can be viewed as a combination of m and u creating the "mew" sound.
Given
f(x)=x^2+3x+1
, What is
f(\beta)
, provided beta is in the domain of f?
\beta^2+3\beta+1
Note that I never specified any information about
beta
This means that the best we can do is replace all the input x with our new input.
Modular arithmetic is a system developed in 1801 that had innovative in the tech world in the recent two decades. The idea is that we can treat numbers as "classes"/groups that cycle similar to how a clock does.
For example on a 12 hour clock ignoring am and pm, it is legal to say [1]+[11] = [0] where [?] represents the hours on a 12 clock system and [0] represents the "12th" hour. Algebraically, we call this math space
\mathbb{Z}_12
"Z-mod 12" where numbers gets placed into a class [0],[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11]. Then the number 20 in this space is:
20 \equiv [8]
What is 2+2 in
\mathbb{Z}_4
"Z mod 4" ?
[0].
Guess this opening song
D.N.Angel
In a game of Catan, participants are required to roll two 6-sided fair dice and use the sum of the top face number as the result for the turn.
What is the probability (fraction or percent is fine) that a user will roll a sum of 7?
\frac{6}{36} = \frac{1}{6} = 16.67%
One of these objects is not like the other. Which object is NOT like the other and why?
a, e, \pi, \sqrt{2}, -1, 0.10100100010001...
the variable/letter
a
since a is not
a
number.
The number
e \approx 2.71828...
is known as Euler's number and has various uses in the world of finances, population dynamics, and chemistry.
Given
f
a function, we say that
f^{-1}
is the inverse function that swaps output and inputs from the original function
Example: Given
f(x) = x+1
, if I use input
x=3
, the corresponding output will be
4
The inverse function
f^{-1}(x) = x-1
takes the input
4
and spits out
3
What is
f^{-1}(f(10)) =?
Note: f(10) is the OUTPUT of f given x=10 as the input.
10.
10 is the original input of
f
that will get a certain output. Whatever that output is, the inverse function
f^{-1}
will take that output and spit out the original input 10!
We can treat the inverse function as the "reverse" process as the original function.
The Induction Hypothesis:
Suppose I have a set of domino tiles placed adjacent next to each other. Assume if I knock one tile down, the sequential tile will ALWAYS fall down for any tile.
This part of the induction process is not enough to ensure all domino tiles will fall down. What additional statement do I need to ensure ALL domino tiles will fall down?
I need to ensure the FIRST domino will fall. This is known as the Base Case of the inductive hypothesis.
When proving theorems in mathematics, we must prove the theorem for ALL possibilities (may be infinite proofs needed). This process shortens the proof since we only need to prove 2 claims instead: An initial base case, and that I can ensure the my proof will always work for sequential claims.
Guess this ending song
Blue Lock
A collection of objects is "countable" if we can at least match/pair every object in the collection with the numbers
\mathbb{N} ={1,2,3,...}
without "missing" an object.
Example: Jma's household members are countable because we can match/pair:
Jma -> 1 , Aileen -> 2, Ryan ->3, Aaron -> 4, Linda ->5, and Dustin ->6.
The collection of numbers
\mathbb{Z}={...,-3,-2,-1,0,1,2,3,...}
are "countable". Find a way to match this collection of numbers in an organized fashion with
\mathbb{N}
We can break up the integers
\mathbb{Z}={...,-3,-2,-1,0,1,2,3,...}
into positive and negative values.
Next we can break up
\mathbb{N}={1,2,3,...}
into even and odd numbers. After matching up
0 \to 1
we can match each even number with positive values and odd number with negative values.
This means
0 \to 1, 1\to2, -1\to3, 2 \to4, -2\to5,3 \to 6, ...
We are essentially counting
\mathbb{Z}
starting from the center, then alternating going to the right from center and then to the left from the center.
The symbol Sigma
\sum
(Yes, get your sigma jokes out the way) is used to calculate "sum" where the bottom expression represents the starting index of your value you start the "sum". The top expression represents the final index you must "sum" up to.
Calculate:
\sum_{n=1}^{n=3}2n
12.
During the late 1700s, at the age of 8, Gauss was told by a teacher to add up all the numbers from 1 to 100 as a passing time for class. Gauss answered the teacher immediately in a couple of minutes. The teacher then later asked him to add 1-1000. He was still able to answer it within a few minutes.
Gauss had constructed the formula for counting using "sum" as follows:
\sum_{n=1}^{n=100} n= \frac{100(100+1)}{2}
In trigonometry, we create special functions
sin(\theta), \cos(\theta), \tan(\theta)
to relate the angle measurement
\theta
within a right triangle to the sides of the right triangle.
The convenient mnemonic device
S\frac{O}{H}C\frac{A}{H}T\frac{O}{A}
bridges the connection between the sine, cosine, and tangent function with the sides ratio where "O" represent side OPPOSITE to the angle theta, "A" represents the side ADJACENT to the angle, and "H" is the hypotenuse. These function takes in the input as angles and outputs the side of the right triangle as a fraction.
The inverse function
\cos^{-1}(x)
takes in the side ratios and outputs the angle measurement; the reverse of the original. In any calculator
\cos^{-1}(2)
where
x=2=\frac{2}{1}
will output "ERROR". Why does the calculator output error?
the input "2" in the inverse cosine function represents the side ratio "2/1". However, the original cosine function outputs
\frac{Adjacent}{Hypoten use}
For the inverse function, this means that I am working with a right triangle whose adjacent side is length 2 and the hypotenuse is length 1. This is a contradiction since hypotenuse is the longest side of the triangle.
Suppose a contestant joins a game where the host displays 100 different suitcases. It is verified that 99 of these suitcases are completely empty and only 1 suitcase has $1,000,000. Before starting the game, the contestant gets to choose a single suitcase to handcuff to their arms without opening the case. This will remain handcuffed until near the end of the game.
During the game, the contestant and the host slowly eliminates the other non-cuffed suitcases as empty. After eliminating 98 suitcases empty, only two total suitcase are remaining. One tied to the contestant, and the other suitcase yet to be verified to eliminate or not.
The host then offers the contestant the option in choosing between cuffed suitcase or to trade the cuffed suitcase with the last suitcase yet to be verified. If the contested holds the suitcase with the $1,000,000 after the trade/no-trade, then the participant will win the game.
In terms of probability, which suitcase would be favorable for the contestant to keep?
The contestant should ALWAYS trade the suitcase.
This is the Monte Hall famous question with 3 doors exaggerate into 100 suitcases. When the contestant first chooses the suitcase to handcuff, the probability the contestant has chosen the correct suitcase is 1/100 or 1%. The other suitcase also has a 1/100 chance. However, the elimination of the suitcases are ONLY to the non-cuffed suitcase. This means the suitcase cuffed will remain at 1/100 chance while the one yet to be verified will increase odds starting from 1/100 -> 2/100 -> 3/100 -> ... ->99/100 =99% chance of being the correct suitcase.
The probability of the suitcase handcuffed being the correct choice is completely independent of the elimination of other suitcases while the probability of the other suitcase will increase as more suitcases gets eliminated.
Guess this ending song.
Katekyo Hitman Reborn ending.