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Advanced Math Formulas
1 / 20
What is the conjugate of a complex number a + bi?
Conjugate of a+bi is a-bi. Multiplying a complex number by its conjugate gives a real number: a²+b².
2 / 20
What is the formula for the foci of an ellipse x²/a² + y²/b² = 1 (a > b)?
For an ellipse: c² = a² - b², where c is the distance from center to each focus.
3 / 20
What is the eccentricity of a circle?
A circle has eccentricity = 0. Ellipse: 0<e1.
4 / 20
What is the formula for the focus of a parabola y = ax²?
For parabola y = ax², the focus is at (0, 1/(4a)). The directrix is y = -1/(4a).
5 / 20
What is the formula for the sum of a geometric series?
Sum of geometric series: Sₙ = a₁(1 - rⁿ)/(1 - r), where r ≠ 1.
6 / 20
What is (a+bi)(a-bi) equal to?
(a+bi)(a-bi) = a² + b². Multiplying conjugate pairs eliminates the imaginary part.
7 / 20
What is the formula for e (Euler's number) as a series?
e = Σ(1/n!) = 1 + 1 + 1/2 + 1/6 + 1/24 + ... ≈ 2.71828. It is the base of natural logarithms.
8 / 20
What is the change of base formula for logarithms?
Change of base: logₐ(b) = log(b)/log(a). Useful for computing logs with any base on a calculator.
9 / 20
How do you add two vectors and ?
Vector addition: + = . Add corresponding components.
10 / 20
What is the magnitude of vector ?
Magnitude |v| = √(a²+b²). It represents the length of the vector.
11 / 20
What is the formula for a geometric sequence with first term a₁ and ratio r?
Geometric sequence: aₙ = a₁ × r^(n-1). Multiply first term by ratio raised to (n-1).
12 / 20
What is De Moivre's Theorem?
De Moivre's Theorem: [r(cosθ+i sinθ)]ⁿ = rⁿ(cos(nθ)+i sin(nθ)). Used to raise complex numbers to powers.
13 / 20
What is a rational function?
A rational function is p(x)/q(x) where p and q are polynomials and q(x) ≠ 0.
14 / 20
What is the formula for the common ratio of a geometric sequence?
Common ratio r = any term divided by the previous term: r = aₙ/aₙ₋₁.
15 / 20
What is a unit vector?
A unit vector has magnitude 1. To create one, divide a vector by its magnitude: û = v/|v|.
16 / 20
What is the formula for the product of a matrix and a scalar k?
Scalar multiplication: multiply every element of the matrix by k. (kA)ᵢⱼ = k × Aᵢⱼ.
17 / 20
What does it mean if f(-x) = f(x) for all x?
If f(-x) = f(x), the function is even — symmetric about the y-axis. Example: f(x) = x².
18 / 20
What is the directrix of the parabola y = (1/4p)x²?
For parabola y = (1/4p)x², the directrix is y = -p and focus is at (0,p).
19 / 20
What is the degree of a polynomial?
The degree is the highest exponent. Degree determines end behavior and maximum number of zeros.
20 / 20
What is the recursive formula for the Fibonacci sequence?
Fibonacci: F(n) = F(n-1) + F(n-2), with F(1)=1, F(2)=1. Each term is the sum of the two before it.
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