Mathematics

Mathematics is ARYANbtw, if that matters

wun plus wun equil tree

>o algorithm

This page is a gem.

If you see a mathGOD, suck him off

The average care fan of mathematics.

Mathematics (or Math for short, or if you are British, Maths) is best known for being the study of Shit Nobody Cares About patterns, quantities, structures, etc. It is extremely important in science. Math starts from logic; you can find out yourself how it works but it's similar to saying

>if a equals b, then b equals a

or something. Math education in the United States generally involves climbing your way up through arithmetic, algebra/geometry, calculus, linear algebra, real/complex analysis, number theory, differential geometry and topology, set theory, and other gemmy things. It is different in other countries DOE.

Why it's gemmy

this is what the $\sqrt{}$ looks like

• Mathematics is only accessible to sigma aryan gigasigma gods with IQs higher than 100.
• It describes anything, from the movement of the heccin planet(s) to how many big macs a valid trans lesbian queen of color can eat before turning into a heccin valid plus sized trans lesbian queen of color.
• Only those who don't get it hate it.
• It can be used to write post filters with regular expressions.
• It can be used to prove the Holocaust is fake.^{[citation needed]}
• Pythagoras cares fan about it
• It is used to create programming languages

Mathematics in soy culture

• Thrembo
• Steganography
• Calculus
• Contour integration
• Programming
• AI
• The probability of getting a nameroll highly despised by everyone (i'm trans btw)
• Infinite dubs theory
• Analytics (rate the 'log)

Branches of Mathematics

Arithmetic

Arithmetic deals with the four major operations: addition, subtraction, multiplication and division, starting with whole numbers and proceeding to harder topics like fractions. Sometimes it covers roots, exponents and logarithms. It also covers word problems. It's taught from kindergarten to about 6th grade. It often has to be revised and drilled because some people are retarded.

Examples

$1+1=2$

$5-4=1$

$3\times 5=15$

$10÷5=2$

$(4\times 10)+1=41$

$3+\sqrt{81}=3+9=12$

$9!=362,880$

wat dis big number?

$\left(123456789^{10}\right)\times \left(987654321^5\right)+\left(111111111^{15}\right)-\left(999999999^7\right)=772996168176110156450737782449950650752890464012301823360161797559258061729364022175243678346032023684884573542136543753364353$

$\frac{\sqrt{\frac{1}{4}+\frac{1}{3}\cdot (2.\bar{6}-\frac{5}{7})}}{\frac{{(0.75)}^{-2}\cdot \sqrt[3]{\frac{8}{27}}}{{\left(\frac{2}{3}\right)}^{-3}}}-\frac{\frac{5}{8}+1\frac{1}{3}}{2-\frac{3}{4}}\cdot {(\frac{3}{7}-\frac{1}{2})}^{-1}=\frac{329}{15}+\frac{243\sqrt{1589}}{3584}$

Here's how to get the answer for the last example.

Comma or dot

For some reason, most countries prefer to use the comma (,) over the dot (.) as a decimal point, evendoe it makes no difference. In addition, some countries write 10,000 as 10.000 or 10 000.

Algebra

Algebra deals with abstract systems and expressions within those. Elementary algebra deals with functions and graphing them, alongside solving basic equations, modeling certain situations and events, etc. In America, it is split into Algebra 1 and Algebra 2.

Algebra 1

Algebra 1 covers the following:

• Concept of a variable, substitution, combining like terms, equivalent expressions,
• Linear equations:
  • With variables on both sides
  • With parentheses
  • Number of solutions
  • Unknown coefficients
• Linear inequalities, compound inequalities
• Working with complex units
• Two-variable linear equations, slope, horizontal/vertical lines, x/y-intercepts
• Slope-intercept form, graphing and writing slope-intercept equations, point-slope and standard form
• Systems of equations: solving by substitution/elimination, equivalent systems of equations, number of solutions to systems of equations (they are called simultaneous equations in the UK)
• Two-variable inequalities and their graphs
• Evaluating functions, inputs/outputs of a function, domain and range of a function, recognizing functions, maximum and minimum points, positive/negative/increasing/decreasing intervals, features of graphs, average rate of change, inverse functions
• Sequences: arithmetic and geometric
• Absolute value, piecewise functions
• Powers, radicals (or surds)
• Exponential growth/decay
• Multiplying expressions, factoring quadratics
• Parabolas, solving quadratic equations, vertex form, completing the square
• Irrational numbers

Other branches of algebra

Abstract algebra studies algebraic structures and is a shitload more complicated. You learn elementary algebra in middle school and take Algebra 1/2 in high school, yet probably won't touch on abstract algebra unless you're a mathematician.

Examples

• 
  The graph of a polynomial function

Geometry

Geometry is the study of shapes. You learn the most basic form of this in preschool but you study it more deeply in high school. The shapes inclde circles which have a diameter and radius that are calculated with a special number called pi (π, value of 3.14159265(a gorillion more digits)) to get the circumcision circumference (the outline of the circle) and area (what's inside of the circle). Polygons are shapes with straight lines like triangles, quadrileterals (squares, rectangles, rhombuses, trapezoids and others), pentagons, hexagons, thrembagons heptagons, octagons and so on. There's also angles which is what you get when putting two lines together, there's acute (<90°), right (exactly 90°, it's all the angles of a square) and obtuse (90°< x 180°). 180° is a flat plane and 360° just goes back again to 0°^{[marge, why 360?] <----[Because it's a number that can be easily divided by various numbers (2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20), which makes calculations easier]}. Real mathematicians however use radians - 1 radian is defined as the angle made where the radius is equal to the arc length created by it ($1\text{rad}={\frac{180}{\pi }}^{\circ }\approx 57.3^{\circ }$).

Trigonometry

Related to geometry, it's the study of the relationships between angles and side lengths, more specifically of right triangles. It uses the functions sine (sin), cosine (cos) and tangent (tan) (the are others; mostly just ratios of these functions on each other, but that's shit nosecant cares about).

There are also the hyperbolic trigonometric functions but it's not relevant to circles so we're not gonna talk about it here

Probability and Statistics

Probability is the study of how likely things are to happen, and statistics is the study of quantifying what happens. Statistics is where you learn about per capita. You learn about this in high school, though this class is usually optional.

Calculus

Calculus tries to find a solution to two universally encountered problems in mathematics: the area problem and the tangent problem.

The area problem

The Greeks tried finding answers to this thousands of years ago. To find the area of random shapes, they divided them into triangles and added up their areas. Nobaldi knew how to find the area of a curved shape, doe - but they managed to approximately calculate the area of a circle. To do this, regular polygons were inscribed into one; more sides meant a closer approximation - note how a 1000-gon already looks like a circle. Coincidentally this is how a limit works. While in ancient times the modern-day concept of a limit was never used, it was intuitively understood. Let A_n be the area of the inscribed regular n-gon. The pattern can be written formally as:

$A=\underset{n\to \mathrm{\infty }}{\lim }{A}_n$

The area problem is the gateway to integral calculus which may be used in practice to compute volumes of different solids, find lengths of curves, etc.

The tangent problem

The tangent of a function

Trying to find the tangent t of a function f(x) at a point A can also cause problems (see figure to the right). We can find the equation of it if we know its slope m, though. To get it we need to plot two points, but we want the slope at one point. We approximate again by picking another point very close to A, say B, then compute the slope of AB. Imagine that B gets closer and closer to A. Using limits we can also say that

$m=\underset{B\to A}{\lim }{m}_{AB}$

This problem gave birth to differential calculus. We consider Isaac Newton and Gottfried Leibniz as the fathers of calculus.

Isaac Newton as the apple fell on his head or something

Limit of a function

Let's now take a closer look at limits in general. We discussed their use cases for the tangent problem. Let's analyze the function $f(x)=x^2+2x+3$ for x near 4. Below is a table that shows values of the function as x gets closer to 4 from both sides.

x	f(x)	x	f(x)
3	18	5	38
3.5	22.25	4.5	32.25
3.8	25.04	4.2	29.04
3.9	26.01	4.1	28.01
3.95	26.5025	4.05	27.5025
3.99	26.9001	4.01	27.1001
3.995	26.950025	4.005	27.050025
3.999	26.990001	4.001	27.010001

Clearly then, as we approach 4, the function will give us something very close to 27. If you think otherwise, meds NOW, it is literally one of the basic conclusions from the table. To express this intuition formally, we say "the limit of the function $f(x)=x^2+2x+3$ as x approaches 4 is equal to 27", and the notation is:

$\underset{x\to 4}{\lim }{x}^2+2x+3=27$

because the word limit is too long or something. Generalised:

Definition. We may say "the limit of f(x), as x approaches a, is equal to L" and write

$\underset{x\to a}{\lim }f(x)=L$

if we can make the values of f(x) as close to L as we like by taking x to be close enough to a, on either side of a, but not equal to a.

There is also a second definition called the epsilon-delta definition; it is more precise algebraically, but the stated one should suffice for now (also nophono even understands epsilon-delta).

You can also write$f(x)\to L\text{ as }x\to a$ and it will mean the same thing.

Example 1. Estimate the value of $\underset{x\to 0}{\lim }\frac{\sqrt{x^2+16}-4}{x^2}$.

Solution. The table shows values of the function for x near 0.

x	$\frac{\sqrt{x^2+16}-4}{x^2}$
±1	0.123105...
±0.5	0.124515...
±0.2	0.124922...
±0.1	0.124980...
±0.05	0.124995...
±0.01	0.1249998...

We notice that the function seems to approach 0.125 as we get closer to 0, therefore we guess that

$\underset{x\to 0}{\lim }\frac{\sqrt{x^2+16}-4}{x^2}=\frac{1}{8}$. ■

What if we took smaller values of t? Let's use a calculator to compute them:

x	$\frac{\sqrt{x^2+16}-4}{x^2}$
±0.001	0.124999998...
±0.0001	0.12499999998...
±0.0000001	0

You might be wondering: "Marge, why is the limit 0 when I put in a smaller number? Shouldn't it be closer to our guessed value?" To answer this, we have to acknowledge that calculators are limited - they give false values because the inputs are simply too small. Note that $\sqrt{x^2+16}$ is almost exactly 4 for small values of x.

Example 2. Guess the value of $\underset{x\to 0}{\lim }\frac{\sin x}{x}$.

Solution. $\frac{\sin x}{x}$ isn't defined when x = 0, so we will use our table method. Note we're calculating values of sin x using radians.

x	$\frac{\sin x}{x}$
±1	0.8414709848...
±0.5	0.9588510772...
±0.2	0.9933466540...
±0.1	0.9983341665
±0.01	0.9999833334...
±0.001	0.9999998333...

Thus, we guess that $\underset{x\to 0}{\lim }\frac{\sin x}{x}=1$. ■

Example 3. Find the value of $\underset{x\to 0}{\lim }(x^3+\frac{\cos 5x}{10,000})$.

Solution. Create a table of values:

x	$x^3+\frac{\cos 5x}{10,000}$
1	1.0000283662...
0.5	0.1249198856...
0.2	0.0080540302...
0.1	0.0010877583...
0.01	0.0001008750...
0.001	0.00001...

Note that without the smaller values it looks like the function approaches 0. We conclude that:

$\underset{x\to 0}{\lim }(x^3+\frac{\cos 5x}{10,000})=0.0001$. ■

Other examples

1. If$f(x)=7x^3+3x^2+2x-1$, then $f^{\prime }(x)=21x^2+6x+2$

2. Find $\int x\sin xdx$

Suppose we choose $f(x)=x$ and $g^{\prime }(x)=\sin x$. Then $f^{\prime }(x)=1$ and $g(x)=-\cos x$. (For g we can choose any antiderivative of g'.) Thus, using the formula:

$\int f(x)g^{\prime }(x)dx=f(x)g(x)-\int g(x)f^{\prime }(x)dx$

we have

$\int x\sin xdx=f(x)g(x)-\int g(x)f^{\prime }(x)dx=x(-\cos x)-\int (-\cos x)dx=-x\cos x+\int \cos xdx=-x\cos x+\sin x+C$

Number Theory

Number theory is the study of the numbers themselves and their properties, probably the most useless branch as it's pure mathematics instead of applied.

Types of numbers

>Just because they're called "imaginary" numbers, doesn't mean they're fake

• Natural numbers, the basic numbers we use to count (1, 2, 3, 4, 5, 6, 7, 8 and so on). Because mathematicians tend to argue about everything, sometimes it's defined as $\mathbb{\mathbb{N}}=(0,+\mathrm{\infty })$instead.
  • Prime numbers, numbers that can't be divided by any number other than one and themselves. They're 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31 and so on. All other numbers can be factorized as a multiplication of prime numbers (1 is the exception); For example, 1488 is 2^4 × 3 × 31.
• Integers, the naturals but it also includes the negative numbers and zero.

• Rational numbers, includes all the fractions and decimals like 1/2, 3/4 and 14/88^{[ermm actually, that's the same as 7/44]}
• Real numbers, all the numbers inside the number line.
  • Irrational numbers, the real numbers that can't be expressed as a fraction. The most famous ones are $\pi ,e,\varphi ,\sqrt{2}.$
• Imaginary numbers, numbers that are multiples of i (the square root of -1, which is impossible with just the real numbers) and exist on a separate number line outside the real numbers.
  • Complex numbers, numbers that have a real part and an imaginary part. They can be on a 2D grid where the horizontal axis is the real part and the imaginary part is the vertical axis. (Example, 13+50i)
• Quaternions, numbers beyond complex numbers that require 4 dimensions, they're SNCA

Calculator

The calculator is the a24, atmospheric, slowburn, genre-defyining, character-driven video game adaptation of Mathematics^{[it just is, ok?]}. The controllers are the buttons with the numbers and operations on them. It's kino but they haven't fixed the "÷0" bug yet.

Wolfram Alpha

Wolfram Alpha is like a calculator but even gemmier. You can do literally hecking everything on it because it's powered by some supercomputers o algorithm. This means you can calculate literally anything to superhuman precision nophono will ever need or care for.

See also

• SNCA:Calculus
  • SNCA:Contour integration

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