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Algorithm Notes and Its Formula

Mathematics encompasses a wide range of topics and concepts, so it's
difficult to provide a comprehensive list of algorithms and formulas
...
Euclidean algorithm: This algorithm is used to find the greatest
common divisor (GCD) of two integers
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2
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It states that:

c^2 = a^2 + b^2

where c is the length of the hypotenuse (the side opposite the right
angle), and a and b are the lengths of the other two sides
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Quadratic formula: This formula is used to find the roots (or
solutions) of a quadratic equation, which is an equation of the form:

ax^2 + bx + c = 0

The quadratic formula is:

x = (-b ± sqrt(b^2 - 4ac)) / 2a

where "±" means "plus or minus," and "sqrt" means "square root
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Fourier transform: This algorithm is used to decompose a function
(or signal) into its constituent frequencies
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5
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The formula for
Bayes' theorem is:

P(A | B) = P(B | A) P(A) / P(B)

where P(A | B) is the conditional probability of A given B, P(B | A) is
the conditional probability of B given A, P(A) is the prior probability of
A, and P(B) is the prior probability of B
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Depending on your field of study or application, you may
encounter many others
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The GCD is the largest integer that divides
both numbers without leaving a remainder
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"

Here's a step-by-step guide on how to use the Euclidean algorithm:

1
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Let's call them a and b
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Divide a by b to get the quotient q and remainder r
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3
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Otherwise, continue to step 4
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Replace a with b and replace b with r
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5
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6
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Here's an example of how to use the Euclidean algorithm:

Find the GCD of 42 and 56
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Let a = 56 and b = 42
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Divide 56 by 42 to get q = 1 and r = 14
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3
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4
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5
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That is, 42 = 14 x 3 + 0
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The final value of b is 14, which is the GCD of 42 and 56
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It has a time complexity of O(log
n), where n is the larger of the two numbers being compared
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It relates to the lengths of the sides of a right triangle,
which is a triangle with one angle equal to 90 degrees (a right angle)
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Here's the Pythagorean theorem in mathematical notation:
c^2 = a^2 + b^2
where c is the length of the hypotenuse, and a and b are the lengths of
the other two sides
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Imagine a square with sides of length c
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The four triangles will
have sides of length a, b, and c, as shown in the diagram below:
```

+-----a-----+
|

/|

|

/ |

|

/ |

b|

/

|c

| /

|

|/

|

|/

|

+-----a-----+
The area of the square is c^2, since it has sides of length c
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Therefore, the total area of the four
triangles is 2ab
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The Pythagorean theorem is used in many areas of mathematics,
science, and engineering
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The Pythagorean theorem states that in a right triangle, the square of
the length of the hypotenuse is equal to the sum of the squares of the
lengths of the other two sides
...

The Pythagorean theorem is often used to find the length of one of the
sides of a right triangle when the lengths of the other two sides are
known
...

To use the Pythagorean theorem to find the length of one of the sides
of a right triangle, we simply rearrange the equation to solve for the
unknown side
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Using the Pythagorean theorem, we have:
c^2 = 3^2 + 4^2
c^2 = 9 + 16
c^2 = 25
Taking the square root of both sides, we get:
c = sqrt(25)

c=5
Therefore, the length of the hypotenuse is 5 units
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It is also a fundamental concept
in geometry and is often used in higher-level math courses
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A
quadratic equation is of the form:

ax^2 + bx + c = 0

where x is the variable, and a, b, and c are constants (with a not equal
to 0)
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There are three possible cases for the roots of a quadratic equation:

1
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The roots are given
by the quadratic formula with the plus-minus symbol:

x = (-b ± sqrt(b^2 - 4ac)) / 2a

2
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Two complex conjugate roots: If the discriminant is negative, then
the quadratic equation has two complex conjugate roots, which are
given by the quadratic formula with the imaginary unit i:

x = (-b ± sqrt(b^2 - 4ac)i) / 2a

The quadratic formula is derived by completing the square of the
quadratic equation ax^2 + bx + c = 0
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The resulting equation can be solved for x using algebraic
manipulation
...
It is
also a fundamental concept in algebra and is often used in higher-level
math courses
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It allows us to express a function or signal as
a sum of sine and cosine waves of different frequencies
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The Fourier transform takes a function in the time domain (f(t)) and
transforms it into the frequency domain (F(ω))
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It tells us how much of each frequency component is
present in the signal
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It is given by:
f(t) = (1/2π) ∫ F(ω) e^(iωt) dω
where the integral is taken over all frequencies ω
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It has applications in music and audio processing, image
processing, and quantum mechanics, among others
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Some of these properties
include linearity, time shifting, frequency shifting, and convolution
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In summary, the Fourier transform is a mathematical technique that
allows us to express a function or signal as a sum of sine and cosine
waves of different frequencies
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Bayes' theorem
Bayes' theorem is a fundamental concept in probability theory and
statistics that allows us to update the probability of a hypothesis based
on new evidence
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The theorem states that the probability of a hypothesis H given some
observed evidence E is proportional to the product of the probability of
the evidence given the hypothesis and the prior probability of the
hypothesis:
P(H|E) = P(E|H) * P(H) / P(E)
where P(H|E) is the posterior probability of the hypothesis given the
evidence, P(E|H) is the likelihood of the evidence given the hypothesis,
P(H) is the prior probability of the hypothesis, and P(E) is the probability
of the evidence
...
The likelihood ratio P(E|H) is the relative probability of
observing the evidence under the hypothesis compared to the
probability of observing the evidence under some other hypothesis
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The denominator P(E) is the normalizing
constant that ensures that the posterior probabilities add up to 1
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It is used in medical diagnosis, financial forecasting,
machine learning, and many other areas
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Bayesian statistics is an alternative to classical (frequentist) statistics
that emphasizes the use of prior information and the incorporation of
uncertainty into statistical models
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In summary, Bayes' theorem is a fundamental concept in probability
theory and statistics that allows us to update our belief in a hypothesis
based on new evidence
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