32 terms

# Theta

Theta junk you need to know
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|a+bi|
√[(a^2)+(b^2)]
Number of digits in x^y

(Ex. 7^43)
y(logx)+1

43(log7)+1
36.339...+1
37.339...
37 (Eliminate the decimal part)
"k"th Triangular Number
T(k)=[k(k+1)]/2
"n"th Pentagonal Number
P(n)=[n(3n-1)]/2
Sum of First "n" Perfect Squares
[n(n+1)(2n+1)]/6
Sum of First "n" Perfect Cubes
[(n/2)(n+1)]^2
What is the determinant of a singular matrix?
0
That is why singular matrices have no inverses.
For any square matrix A,
A • A^(-1)
The Identity Matrix
| 1 0 |
| 0 1 | For a 2x2 matrix,

| 1 0 0 |
| 0 1 0 |
| 0 0 1 | For a 3x3 matrix, etc.
For any square matrix A,
|A| - |A^T|
(A^T is the transverse of the matrix A)
0
The determinant of any square matrix is equal to the determinant of its transverse matrix.
Area of a Triangle Using Matrices
1 | x(1) y(1) 1 |
±— | x(2) y(2) 1 |
2 | x(3) y(3) 1|

1/2 is indicated as ± so that you can multiply whichever number would result in a positive value, depending on the situation, because area must be positive.

x(n) or y(n) means x "sub-n" or y "sub-n"
Arithmetic Mean of a and b
(a+b)/2

For a, b, and c, the arithmetic mean is
(a+b+c)/3 etc.
Geometric Mean of a and b
√(ab)

For a, b, and c, the geometric mean is
∛(abc) etc.
Harmonic Mean of a and b
(2ab)/(a+b)

For a, b, and c, the harmonic mean is
(3abc)/(a+b+c)
Base Changing
Sorry, I don't know how to explain this one on here you just have to know it.
Number of Positive Integral Factors of a Number (Ex. 3240)
Find the prime factorization of the number and multiply the exponents of the prime factors after adding one to each of them.

3240 = 2^3 • 3^4 • 5
(3+1)(4+1)(1+1)
4 • 5 • 2
40
Binomial Theorem
Sorry, again, I don't know how to explain this one on here you just have to know it.
Circular Permutations (keyring)
(n-1)!/2
Circular Permutations (non-keyring)
(n-1)!
√n+√(n+√(n+...=x
Square both sides
Set the radical junk equal to x
Solve the quadratic equation
The positive solution is the solution to the entire equation
Find A+B / Find AB
√(8-2√15)=√A+√B
Square both sides
8-2√15=A+B+2√AB
A+B=8
AB=15
Sum of the Roots of a Quadratic Equation
ax^2 + bx + c = 0
-b/a
Product of the Roots of a Quadratic Equation
ax^2 + bx + c = 0
c/a
Sum of the Reciprocals of the Roots of a Quadratic Equation
ax^2 + bx + c = 0
-b/c

Sum of Roots/Product of Roots
(-b/a)/(c/a)
-b/c
Sum of the Squares of the Roots of a Quadratic Equation
ax^2 + bx + c = 0
(b^2 - 2ac)/a^2
Sum of the Squares of the Reciprocals (or the Reciprocals of the Squares) of the Roots of a Quadratic Equation
ax^2 + bx + c = 0
(b^2 - 2ac)/c^2
Sum of the Roots of Other Polynomial Equations
Ax^n + Bx^(n-1) + Cx^(n-2) +...+ Jx + K=0
-B/A
Product of the Roots of Other Polynomial Equations
Ax^n + Bx^(n-1) + Cx^(n-2) +...+ Jx + K=0
If n is even, K/A
If n is odd, -K/A
Sum of the Reciprocals of the Roots of Other Polynomial Equations
Ax^n + Bx^(n-1) + Cx^(n-2) +...+ Jx + K=0
-J/K
Sum of the Squares of the Roots of Other Polynomial Equations
Ax^n + Bx^(n-1) + Cx^(n-2) +...+ Jx + K=0
(B^2 - 2AC)/A^2
Number of Zeroes at the End of a Specific Number Factorial
Ex. 1000
Divide the number by the largest integral power of 5 less than the original number
Continue dividing by decreasing integral powers of 5 until you reach 5
Add the quotients

Ex.
1000÷625=1
1000÷125=8
1000÷25=40
1000÷5=200
1+8+40+200=249
249 zeroes
"p"th Even Perfect Number
(2^p - 1)•(2^(p-1))
Where p is a prime number
In a rectangle with the dimensions x by y, the number of distinct paths from one corner to the other
(x+y)!/x!y!