• Clean, free, unlimited energy for allCLEAN, FREE, UNLIMITED ENERGY
  • Earth's renewable energyEARTH'S RENEWABLE ENERGY
  • Eliminate batteries, wind turbines and solar panelsELIMINATE BATTERIES, WIND TURBINES & SOLAR PANELS
  • 100% perfect medicines in minutesPERFECT MEDICINES IN MINUTES
  • Eliminate surface skin-frictionELIMINATE SKIN FRICTION
Clean, free, unlimited energy1 Earth's renewable energy2 Eliminate batteries, wind turbines and solar panels3 100% perfect medicines in minutes4 Eliminate surface skin-friction5
Useful Stuff Algebra Trig Functions Calculus Max-Min Differentiation Rules Differentiation Trig Differentiation Logs Integration Methods Standard Integrals Stiffness & Capacity Mohr's Circle Earth's Atmosphere Earth's Properties Stars & Black Holes Stars & The Gas Planets Laws of Motion Solar System Orbits Planetary Spin Core Pressure Earth's Magnetic Field Dark Matter? The Big Bang Astro-Lensing Rydberg Atom Planck Atom Classical Atom Newton Atom The Atom Newton's 'G' Coulomb's 'k' The Neutron E=mc² Gravity is Magnetism Relativity is Dead Quantum Theory is Dead Artificial Satellites Brakes and Tyres Vehicle Impacts Speeding vs Safety Surface Finish Pressure Classes Hardness Conversion Energy Electro-Magnetic Spectra Thermodynamics Steam (properties) Heat Work Energy Power Constants

Algebra Formulas

The following table contains alternative ways of expressing algebraic functions.

General

a-x = 1 / ax

a1/x = x√a

a-1/x = 1 / x√a

(a.b)x = ax . bx

(ax)y = ax.y

aˣ⁺ʸ = aˣ . aʸ
aˣˉʸ = aˣ / aʸ

e.g.: a²˙⁵ = a² . a⁰˙⁵
e.g.: a⁰˙²⁵ = a¹ / a⁰˙⁷⁵

xⁿ/yⁿ + 1 = (xⁿ + yⁿ) / yⁿ

(x - a) / A = x/A - a/A

(x² - a²) = (x - a).(x + a)

Simplify: (x + b)/(x² - a²)
Note: (x² - a²) = (x - a).(x + a)

(x + b)/(x² - a²) = (x + b) / [(x - a).(x + a)]
(x + b) / [(x - a).(x + a)] = A / (x - a) + B / (x + a) {where A & B are unknown}
(x + b) = A.[(x - a).(x + a)] / (x - a) + B.[(x - a).(x + a)] / (x + a)
(x + b) = A.(x + a) + B.(x - a)
Find A:
set x = a
(x + b) = A.(a + a) + B.(a - a)
(x + b) = 2.a.A
A = (x + b) / 2.a
Find B:
set x = -a
(x + b) = A.(-a + a) + B.(-a - a)
(x + b) = B.(-a - a)
B = (x + b) / 2.-a
(x + b)/(x² - a²) = (x + b)/[2.a.(x - a)] + (x + b)/[2.-a.(x + a)]
cancel (x + b):
1/(x² - a²) = 1/[2.a.(x - a)] - 1/[2.a.(x + a)]

Continue in order to prove the above:
1/[(x - a).(x + a)] = 1/[2.a.(x - a)] - 1/[2.a.(x + a)]
1 = [(x - a).(x + a)]/[2.a.(x - a)] - [(x - a).(x + a)]/[2.a.(x + a)]
1 = (x + a)/(2.a) - (x - a)/(2.a)
1 = [1/(2.a)] . [(x + a) - (x - a)]
2.a = [(x + a) - (x - a)]
2.a = x + a - x + a
2.a = a + a

Factorial: e.g. 5! = 5x4x3x2x1

Binomial

(a+b)n = an + nC1.a(n-1).b + nC2.a(n-2).b2 + ..... + nCr.a(n-r).br + ..... bn
Where:
nCr = n! / (n-1)!.r!
nC1 = n
nC2 = n! / (n-2)!.2! = n.(n-1) / 2!
nC3 = n! / (n-3)!.3! = n.(n-1)(n-2) / 3!
etc.
first few terms are as follows;
(a+b)n = an + n.a(n-1).b + n(n-1).a(n-2).b²/2! + n(n-1)(n-2).a(n-3).b³/3! + ..... + bn

If; 0 = ax² + bx + c
then; x = -b ± (b² - 4.a.c)½ / 2.a

Simple and Compound Interest

The following table contains formulas for calculating simple and compound interest.

Where: P = the principal sum, p = percentage interest, n = payment term (years), q = payments per year, I = interest paid over full term, m = amount of each payment & Pn = total amount paid over 'n' years

Simple

I = P.p.n

Pn = I + P

Compound

I = Pn - P

Pn = P.(1 + p/q)n.q

m = Pn / n.q

Net-Present-Value

V = Pn / (1+p/q)n.q

Discount(simple) = Pn(simple) - V

Discount(compound) = Pn(compound) - V

Progressions

The following table contains the formulas for arithmetic and geometric progressions.

Note: r = 2nd term ÷ 1st term, d = 2nd term - 1st term, n = number of terms

Arithmetic

nth term = 1st term + d.(n-1)

Σn terms = n.[2 . 1st term.(n-1)] / 2

Geometric

nth term = 1st term . r(n-1))

Σn terms = 1st term . (1 - rn) / (1 - r)      [r < 1]

Σn terms = 1st term . (rn - 1) / (r - 1)      [r > 1]

Logarithms

The following table contains alternative ways of expressing logarithmic functions.

Note: 'base' refers to the logarithmic base, which can be any positive number
The most common bases are 10 and 2.71828182845905 (the base for natural logs, normally written thus 'ln(x)')

e = 1 + 1/1 + 1/2 + 1/3 + 1/4 + 1/5 + 1/6 + 1/7 + ..... = 2.71828182845905

ex = 1 + x¹/1 + x²/2 + x³/3 + x⁴/4 + x⁵/5 + x⁶/6 + x⁷/7 + ....

e-x = 1 - x¹/1 + x²/2 - x³/3 + x⁴/4 - x⁵/5 + x⁶/6 - x⁷/7 + ....

y = Logbase(basey)

Log x = Logbase(x)
Anti-Log x = basex

logbase(x) = loga(x) / loga(base)
Where 'a' can be any number; 1, 2.71828182845905, 10, etc.

base(a+b+c) = base(a).base(b).base(c)
e.g.: exp(a+b+c) = exp(a).exp(b).exp(c)

Logbase(x) - Logbase(y) = log10(base).x/y

Logbase(xa) = a.Logbase(x)

Logbase(x) - Logbase(y) = z
Same as: x/y = basez

1/exp(x) = exp(-x)

eln(x) = x

Xn = Y
n = Log(Y) / Log(X)

Determinants

Determinants are a means of solving simultaneous equations, e.g.
a₁.x + b₁.y + c₁.z = 0
a₂.x + b₂.y + c₂.z = 0
a₃.x + b₃.y + c₃.z = 0

which can be written thus:
ǀa₁,b₁,c₁ǀ
ǀa₂,b₂,c₂ǀ
ǀa₃,b₃,c₃ǀ

The following table contains the procedure for solving determinants.

2ᴺᴰ Order

a₁.w + b₁.x = 0
a₂.w + b₂.x = 0

|a₁,b₁| = a₁.b₂ - a₂.b₁
|a₂,b₂|

To solve a 2ᴺᴰ Order equation you perform the calculation as shown above

3ᴿᴰ Order

a₁.w + b₁.x + c₁.y = 0
a₂.w + b₂.x + c₂.y = 0
a₃.w + b₃.x + c₃.y = 0

|a₁,b₁,c₁| = a₁.|b₂.c₂| - a₂.|b₁.c₁| + a₃.|b₁.c₁|
|a₂,b₂,c₂|         |b₃.c₃|       |b₃.c₃|         |b₂.c₂|
|a₃,b₃,c₃|

To solve a 3ᴿᴰ Order equation you convert to 2ᴺᴰ Order equations as shown above then solve 2ᴺᴰ Order equations

4ᵀᴴ Order

a₁.w + b₁.x + c₁.y + d₁.z = 0
a₂.w + b₂.x + c₂.y + d₂.z = 0
a₃.w + b₃.x + c₃.y + d₃.z = 0
a₄.w + b₄.x + c₄.y + d₄.z = 0

|a₁,b₁,c₁.d₁| = a₁.|b₂.c₂.d₂| - a₂.|b₁.c₁.d₁| + a₃.|b₁.c₁.d₁| - a₄.|b₁.c₁.d₁|
|a₂,b₂,c₂,d₂|         |b₃.c₃.d₃|       |b₃.c₃.d₃|         |b₂.c₂.d₂|        |b₂.c₂.d₂|
|a₃,b₃,c₃,d₃|         |b₄.c₄.d₄|        |b₄.c₄.d₄|         |b₄.c₄.d₄|        |b₃.c₃.d₃|
|a₄,b₄,c₄,d₄|

To solve a 4ᵀᴴ Order equation you convert to 3ᴿᴰ Order equations as shown above, then convert to 2ᴺᴰ Order equations then solve 2ᴺᴰ Order equations

The same procedure may be followed for all subsequent Order equations: 5ᵀᴴ, 6ᵀᴴ, 7ᵀᴴ, etc.

Graph with a positive slope

Fig 1. Finding x and y

Slope

A calculation method to find a point on a graph of positive or negative slope

Positive Slope

with reference to Fig 1

x = (x₂-x₁).(y-y₁)/(y₂-y₁) + x₁

y = (y₂-y₁).(x-x₁)/(x₂-x₁) + y₁

 

Negative Slope

Graph with a negative slope

Fig 2. Finding x and y

with reference to Fig 2

x = (x₂-x₁).(y-y₂)/(y₁-y₂) + x₁

y = (y₁-y₂).(x-x₁)/(x₂-x₁) + y₂

 

Slope

A calculation method to find a point on a graph of positive or negative slope

Geometry

Properties of a Sphere

Circumference: C = 2.π.r
Surface Area: A = 4.π.r²
Volume: V = ⁴/₃.π.r³
Distance between 'n' equally spaced points on the surface:
Arc: d = π.A / C.n = 6.π.V / A.n
Linear: ℓ = 2.r.Sin(½.d/r)
to find 'n':
n = π/Asin(½.ℓ/r)

If ℓ = r then:
n = π/Asin(½) = 6
In this special case; 'n = 6' is a constant, irrespective of the spherical radius

 

Further Reading

You will find further reading on this subject in reference publications(19)

      Go to our store
CalQlata™ Copyright ©2011-2019 CalQlata info@calqlata.com Site Map Terms of website use Our Store