Vizual math

Danilin Russians formulas

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Probability Reliability

Probability of an event and reliability of probability are given, and it is possible to calculate normal number of repetitions of similar cases in a row.
P - probability of event, for example 25% = 0.25 = 1/4 = ¼ = 25/100.
C - reliability of case, for example 99% = 0.99 = 99/100.
N - number of similar cases in a row.

Match and mismatch are interchangeable and probability of a match is one minus probability of a mismatch, then C=1-c and c=1-C and P=1-p and p=1-P and these formulas are valid for probability over 50%.

Probability multiplication including reliability         C+(1-P)^N=1             [23]
calculating degree, calculates number of similar cases in a row
N = LOG(1-C)/LOG(1-P)                     [24]

Example: reliability C=99% probability P=25%
Normal number of similar in a row N = LOG(1-0.99)/LOG(1-0.25) = 16
and it means that at a probability of 25% it’s normal not to match 16 times in a row
and real mathematicians understand difference between LOG and LN in formulas.

Simplified formula         N = 7+(5*(1/P-2))                         [25]
Example: P=0.1 and N=47 is normal and P=0.78 and N=4 is normal.

Inverse problem: calculate probability of a case if reliability is ensured for a given number of cases             P = 1 - (1-C)^(1/N)                         [26]

Let's calculate reliability of probability.

Each worker does 78% of work.
Determine number of employees to complete work by 99%.

Given: probability P=78% and reliability C=99%.
Formula: C+(1-P)^N=1.
N = LN(1-C)/LN(1-P) = LN(1-0.99)/LN(1-0.78) = 3.
Answer: 3 workers are needed.

Division of such cases in a row in reality: temperature seams of rails and change of seasons and change of time of day and breathing and sleep and wages, etc.

Percent

Having deposit money at P% for Y years, expect multiplier M. M = (1+P/100)^Y     [19]
Percentage         P = -100*(1 -M^(1/Y))                     [20]
or     Percent         P = 100*(M^(1/Y)-1)                     [21]
Years         Y = LOG(M)/LOG(1+P/100)                 [22]

 
Inflation

Inflation to power of years calculates price multiplier
Inflation ^ Years = Factor                     [5]
Inflation:             Inflation = Factor ^ (1 / Years)                     [6]
Years:             Years = LOG(Factor)/LOG(Inflation)                 [7]
Example: if multiplier is 12 over 15 years: inflation =12^(1/15) = +18%.
Example: if multiplier is 12 & inflation is 18%, years: =LOG(12)/LOG(1,18) = 15 years.



Salary

Suppose several workers receive a small salary and 1 manager receives a large salary and it is required to find number of employees to form a given average salary.

We denote: B = high pay and C = average pay and M = low pay and N = number of employees who receive little and mean
(B+M*H)/(H+1)=S                     [8]
H = (B-S) / (S-M)                         [9]

Excel compatible programs allow you to assign any names to cells to left of formula bar, and formulas in Russian are possible.

Example: B=300 and M=28 and C=45 and H = (B-S) / (S-M) = (300-45) / (45-28) =
= 255/17 = 15 people and check: =(300+28*15)/(15+1) = 45.

Math programs calculate N = (S-B)/(M-S).

Formulas in detail:

(B + M * H) / (H + 1) = C     
B + M * H = C * (H + 1)     
B+M*H = S*H+S
S*H-M*H = B-S     
H * (S-M) = B-S     
H = (B-S) / (S-M)

Derivative 

Derivative: tangent of slope of tangent of function graph.

Derivative 1st: minimax function.
Visual: horizontal tangent plot of a function.

Derivative 2nd: function inflection.
Visually: change in direction of angular movement of tangent of graph.

Path and derivative 1st: speed.
Derivative 2nd: acceleration and derivative 3rd: start.

 
Interpolation and extrapolation

Variables are ordered and a trapezoid x1 y1 y3 x3 is created and desired: y2 from x2.

y2=y1+(x2-x1)*(y3-y1)/(x3-x1)                     [1]

In Excel, formula can be inserted into standard cells Y2, etc.
Interpolation is possible: for 4 cells and tables entirely.
Interpolation and extrapolation are possible through graphical programs.
Coordinate in middle: arithmetic mean         y2 = (y1+y3)/2            [2]


Excel formulas

Example: Excel multiplication table and multipliers horizontally and vertically.

Product is calculated by formula                 =B$1*$A2                 [3]
where $ sign makes a row or column constant and formula expands.

If $ sign is assigned to a row and a column at same time:     =$B$1*1000        [4]
Cell with 2 $ signs: constant when copying and pasting.

Word automation

Word can automatically number ordinals, such as paragraph numbers or image numbers or formula numbers, and rearranging paragraphs provides numbers in order.

Word includes headers and footers for decorating repeating elements.
Word can automatically number pages and write number of pages.
Word formulas can be arranged in a table by setting invisible cell borders.

Word can replace characters using special characters, 
for example, multiple Enter in a row is replaced by 2 Enter.

Word or Excel can create a simple macro manually from actions on screen.
 

 
Combinatorics 

Number of combinations of 2 characters:
N = X * (X-1) / 2    

X=5     a b c d e     N=5*4/2=20/2=10     ab ac ad ae bc bd be cd ce de
X=4     a b c d     N=4*3/2=12/2=6         ab ac ad bc bd cd
X=5         a b c d e         N=5*4/2=20/2=10     ab ac ad ae bc bd be cd ce de

Connect of 2 points encrypt 2 bits of a 2-ry of form DA = 1001.
Transport task: number of connections by 2 points.
Combinations of 2 topics of this manual are possible.

Sorting halves

Considering: array includes elements no more and no less than average, bubble sorting is converted into several: from 2 to 8 small parts of array sorting 
and total sorting is accelerated up to 20 times and recursive algorithm 
sorts a million: 1,000,000 elements in 0.3 seconds.
 
Number of permutations former Z=N*(N-1)/2, decreases and improves to:
Z = log(N;2)*(N/log(N;2)*(N/log(N;2)-1)/2+2*N/log(N;2))         [41]

Definitions

Integral: sum of multiplications of quantity and quality.
Logarithm: number of multiply to desired result: "to what degree to raise".
Factorial: Multiply from 1 to desired number.
Derivative: tangent of slope of graph's tangent.

Percentage: 1 in 100 parts and permille: 1 in 1000 parts.
Probability: event opportunity 0 <= P=events/variants < 100%.

Integral: from word integer.
Logarithm: from word ratio.
Factorial: from word multiplying.

 
Integral

Constant value: time, seconds.
Specific value: speed, meters per second.
Time periods: 1 second.

Each square product: time seconds
multiplied by speed of meters per second and get path, meters.
 
Summing up entire path for a given time and distributing it evenly over time:

 
Vertical average speed of 3 meters per second, shows mean value of integral.


Guess number

computer guesses number X, guessed automatically or by a person,
for number of steps logarithmic         N = LOG(X;2)                     [12]
Guessing is optimal to start from middle of array.

For example, 1 out of a billion is equal to 1 out of X=10^9, guessed in N=30 moves:
N = LOG(10^9;2) = 30                    [13]

Dependency of 10-ary and 2-ary:         10^3 = 2^10                     [14]

Number of N guesses X: for each increase in X by 1000 times +10 guesses.

 
Binomiality

Binomial: each element is approximately 2 times next.
Binomial percentage distribution: 56 +24 +14 +6 = 100.

  

A square 10x10 = 100 cells divides point at location 2 into 3 cells
and whole zones are obtained 2x3=6 and 2x7=14 and 3x8=24 and 7x8=56.

Possibly specific to processes where a 2 by 2 table is created
and a pattern is noticeable: 24% is 4 times more than 6% and 56% / 14% = 4.

An array of thousands of random elements 1 0 1 0 1 creates similar chains in a row and same number of chains in a row is distributed binomially, of form:
1 in a row 64 pieces, then 2 in a row 32 pieces and 3 in a row 16 pieces, etc., which means that array under study, distributing chains in a row binomially: random and quantum random observe principles of binomial distribution.

Shared formula x+2x+4x+8x=100 & x=6,67 & relations 80%\20% & 70%\30%.

Pattern Excel

Events 2-ary controls multiplication of events, including =NOT() functions
X = (A1 | NOT(A1)) ... * … * (AN | NOT(AN)) = ( 0 | 1 )             [11]
Example: find an event like 1010011:
=A1 *NOT(A2) *A3 *NOT(A4) *NOT(A5) *A6 *A7 =         ( 0=no | 1=yes )
Explored big data is possible in non-consecutive cells.

Another pattern checks all numbers in a row from 1 to N are mixed:
for "2 4 5 3 1" = 2^2 +2^4 +2^5 +2^3 +2^1 = 62 = 2^( N+1)-2    [12]


Permutations and factorial

Given: 5 books to read in order.
Find number of options for reading books.

We distribute books according to list:

1st book can be read anywhere from 5
2nd book can be read anywhere from 4
3rd book can be read anywhere from 3
4th book can be read anywhere from 2
5th book can be read anywhere from 1

Multiplying: = 1 * 2 * 3 * 4 * 5 = 120 options for reading order of 5 books.

number of permutations calculates factorial of a number         N! = 1 * … * N     [28]

Mathematics

First, studied addition of integer digits.
further addition of integers numbers
further multiplication of integer digits
further multiplication of integers numbers
further addition of non-integer numbers
further multiplication of non-integer numbers: integral
further integer powers of integer digits
further integer powers of integers numbers
further integer powers of non-integer numbers
further non-integer powers of non-integer numbers: logarithm.

Logic

Deduction: reasoning from general maximum
to private minimal individual

Induction: reasoning from particular minimum
individual to general maximum

PI

Circumference     L = 2πR  = πD 
Area of circle    S = πR2  = πD²/4 

π = 3,1415926 ~ 3,1416 ~ 355/113 

Decision square

Options 2 quantitative and 2 qualitative 
create 4 combinations

Example: a transportation problem 
creates its own square of solutions.

Solution options:
if an event occurs: what will happen?
if event does not happen: what will happen?
if an event occurs: what will not happen?
if event does not happen: what will not happen?     

Excel proportion

3 cells A1, B1, A2 are filled
and formula is set in cell             B2 =B1*A2/A1                     [18]
Example: video scaling and screen resolution are calculated.

    A    B        Excel    A    B
1    A1    B1        1    16    9
2    A2    B2        2    1280    720

Integral geometrically

Integral: area of figure between verticals
Integral 2nd: figure area between graphs
integral 3rd: body volume between graphs
Integral N-th: includes qualitative features of mass type.

Knapsack 0-1 binary 

It is required to integrally assemble a set of many items,
having value and mass, limiting maximum mass,
in order to get most value.

Understanding: a thing in a set is either there or not, they are compiled
all possible combinations of N elements in amount =2^N,
for example, with N=5, ciphers from 00000 to 11111 are synthesized
and as a result, most valuable integral set is found.

Algorithm Knapsack 0-1 binary implemented in many languages
programming and is checked through internet by online compilers.

Division of similar events in a row

Modeling processes where several costs in a row are of passive type
and then 1 success of type of asset, as a result, a growing graph is created, 
similar to fractal and each time wave is calculated using a logarithm.
Scientific name: volatility clustering.

Speed

Speed from kilometers per hour to meters per second is translated by formula based on 36 km/h = 10 m/s             V(m/s) = V(km/h) / 3.6                     [15]
V(km/h) = V(m/s) * 3.6                        [16]
It is useful to translate speed of kilometers per minute into meters per second 
in your mind             1 km/minute = 60 km/h = 16.7 m/s                 [17]
Examples: 20 m/s = 72 km/h = 1.2 km/min or 8000 m/s = 28800 km/h = 480 km/min.

Sorting and Shuffling Excel

Excel sorting is possible for 1 column and for several related columns.
Excel shuffling occurs through ordering of random cell values created by a formula of form                         =randbetween(1;1000)                 [29]
Next, by copying a random array and inserting it into a free column as a «Special Insert – Values» and sorting linked list with data.

Olympic points

Points integrally take into account quantity and quality of medals.
Pyramid of points = gold *3 + silver *2 + bronze             [42]
Few countries get highest points and it is possible to unite countries.

Divisibility

Number is divisible by 2     if rightmost digit         is divisible by 2.
Number is divisible by 3     if sum of all digits     is divisible by 3.
Number is divisible by 5     if rightmost digit         is divisible by 5.
Number is divisible by 9     if sum of all digits     is divisible by 9.

Number is divisible by 11     if sum of all digits in even places
is equal to sum of all digits in odd places.

Lever balance

Lever balance is integral each of 2 sides through 
moment kg*m, equal to product weight P kg on shoulder L m

M1=P*L  and  M2 = P*L     [45]
 
If shoulder L m is reduced in 2 times, having become L/2 m, 
then balance will create mass 2*P kg in 2 times more

P * L = 2*P * L/2 = P * L     [46]

Path

Optimal path is chosen as maximum sum of benefit integrals 
and motion options are compared at each step.

If on a flat field are calculated benefits for each step, 
then visualization shows volumes: 
red maxima create optimal path.     

Squares

Maximum square includes c2 
and triangular fields I-IV, if rearranged differently,

obviously:  a2 + b2 = c2     [47]
and 
(a+b)2 = a2 + 2ab + b2       [48]

Duct section and Pipe diameter

Side of duct channel section d, m, knowing flow rate L, m3/h and air velocity v, m/s
d = root(L/(v*3600))                         [50]

Diameter d, m, knowing loads Q, Gcal/h and temperature of coolant T1 & T2 oC
or flow rate G, m3/h and setting water density r, kg/m3 and water velocity v, m/s
G = Q *1000 / (T1-T2)                         [51]
d = 2*root(G/(3,6*r*v*pi()))                     [52]

MathSolver

mathsolver.microsoft.com/ru/solve-problem/C+%7B%60left(1-P%60right)%7D%5E%7BN%7D%3D1
 

mathsolver.microsoft.com/ru/solve-problem/0.99%2B{`left(1-0.25`right)}^{N}%3D1
 

mathsolver.microsoft.com/ru/solve-problem/%60frac%7B%20%20%60left(%20B%2BM%20%60cdot%20%20H%20%20%60right)%20%20%20%20%7D%7B%20H%2B1%20%20%7D%20%20%20%3D%20%20S

 

 
 
WolframAlpha

wolframalpha.com/input/?i=solve+C%2B%281-p%29%5EN%3D1+for+N

wolframalpha.com/input/?i=solve+%28B%2BM*H%29%2F%28H%2B1%29%3DS+for+H

 

 
 
MathWay

 

 

 

 

 
 
MALmath Android

PhotoMath

 

Mathematica

 

 
Catch up

If coefficient K is set: bet multiplier M:         M=1+(1/(K-1))                 [43]
If coefficient is K=3, minimum bet multiplier is M=1+(1/(3-1))=1.5.


Savings Square

player is always in worst position relative to organizer of game
and advantage of player: refusal to play and refusal to raise stakes,
as well as lowering rates after winning,
moreover, by lowering stakes, they could win in classic “Queen of Spades”.

Savings square: rising rates vertically and equal rates horizontally,
always mathematically justified: gain absorbs loss of vertical,
and check approximate square of savings under a coefficient of 3 probability 1/3:

Bet 13, having won, absorbs lower ones: = + 13 * 3 -8 -5 -3 = +23 and rest of costs, and higher vertical, greater return on winning.

An example of cell movement, implemented on paper or programmatically or in Excel
 
Having gone into plus, it is possible to restore square of savings and start from beginning.
 

Paradoxes of Probability

Dividing similar cases in a row probably increases likelihood of success,
where number of similar cases in a row is less than normal.

Visually representing increase in probability of guessing,
a bet after signal to bet when probability has decreased.

 
Winning

Payoff integral: sum of products of probabilities and odds.
V = P1*K1 + P2*K2 + … + PN*KN                     [44]
Example: guessing probability of 6% coefficient 10 and guessing probability of 38% coefficient 1, then V = 0.06 * 10 + 0.38 * 1 = 0.98 = 98%.

Excel example: = 10*hypergeomdist(2;2;20;80) + 1*hypergeomdist(1;2;20;80) = 0.981.
Excel example: =2* hypergeomdist(1;1;18;37) =3* hypergeomdist(1;1;12;37) =0.973.

Number of Excel combinations: = 1 / hypergeomdist(5; 5; 5; 36) = 376992.
Note: It is convenient to compare many lotteries.

Winning Dependent Events

Determine winning of set of dependent events knowing coefficients.
Formulation: sum of coefficients divided by square of quantity 
must be greater than one         (K1 + K2 + ... + KN) / (N^2) > 1             [45]

Example: =(2.4+2)/(2^2) = 4.4 / 4 = 1.1 > 1 means winning
when betting on both events, provided that one of events wins.

Note: Dutching system, plus a better understanding of simple fractions, for example
of form 5/4>1 and 4/5<1, understanding: more than 1 or vice versa.

Side note: most logical application: betting on elections or
betting on events with many options like “winning 1st half and match”.

Winning Opposite Events

Determine winningness of several opposite events, which is usually called a «fork».

Formulation: product of coefficients minus each coefficient 
must be greater than zero.         K1 * K2 -K1 -K2 > 0                     [46]

Example: = 1.2 * 8 -1.2 -8 = 0.4 > 0 means there is a surebet and there is a win.
Note: mathematical curiosity for winnings if 2 options.

Queen of Spades and Risk Management

Story "Queen of Spades" tells how main character participated:

first = 47 -47 + 2*47 = 94
and further = 94 -94 + 2*94 = 188
and further = 188 -188 +0 = 0,

having lost his 47 thousand rubles and winnings.

If protagonist applied risk management
and participated mathematically lowering stakes by 2 times:

first = 47 -47 + 2*47 = 94
and further = 94 -23.5 + 2*23.5 = 117.5
and further = 117.5 -11.75 +0 = 105.75.

So main character, lowering stakes, would have received 105.75 thousand rubles
and even a loss on 2nd or 3rd move woulds have balance in plus.

Task 1

2 walkers walking towards each other at a distance of 10 km

Speed of 1st walker is 2 km/h.
Speed of 2nd walker is 3 km/h.
From 1st walker, a drone takes off at same height at a speed of 10 km/h.

Drone flies to walking walker and turns back.
Drone flew between walkers all time until walkers met.

Q: How far did drone fly?

Task 2

Airplane is at an altitude of 5000 m.
How much is distance to horizon?

Task 3

Plane flew south for 500 km.
Then plane flew east for 500 km and north for 500 km.
As a result, aircraft returned to its original starting point.

Q: Where did plane start from?
Hint: Solution in hemispheres of North and South.

Task 4

Given: 9 balls, 1 of which is lighter.
Determine light ball in 2 weighings.

Task 5

Bottle plastic transparent closed filled with water
by about half.

Determine in a second whether there is water of half in bottle
or more than water of half or less than water of half.

Task 6

Lliquid level shows horizontal.
Figure out vertical.
 

Reliability of probability Division of mismatches in a row

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