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Advanced Math Formulas
1 / 20
What is the imaginary unit i equal to?
The imaginary unit i = √(-1), so i² = -1. Complex numbers are written as a + bi.
2 / 20
What is a hole (removable discontinuity) in a rational function?
A hole occurs when a factor cancels from both numerator and denominator. The function is undefined at that x-value but has no asymptote.
3 / 20
What is the formula for the sum of first n cubes: 1³ + 2³ + ... + n³?
Sum of cubes = [n(n+1)/2]² = (sum of first n integers)². Example: 1³+2³+3³ = 36 = 6² ✓
4 / 20
What is the degree of a polynomial?
The degree is the highest exponent. Degree determines end behavior and maximum number of zeros.
5 / 20
What is the formula for the nth term of a recursive sequence?
A recursive sequence defines each term using one or more previous terms (e.g., Fibonacci: aₙ = aₙ₋₁ + aₙ₋₂).
6 / 20
What is the magnitude of vector ?
Magnitude |v| = √(a²+b²). It represents the length of the vector.
7 / 20
What is the formula for the argument of a complex number a + bi?
The argument (angle) of complex number a+bi = arctan(b/a). Adjust for the correct quadrant.
8 / 20
What is (a+bi)(a-bi) equal to?
(a+bi)(a-bi) = a² + b². Multiplying conjugate pairs eliminates the imaginary part.
9 / 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).
10 / 20
What is the modulus (absolute value) of complex number a + bi?
Modulus of a+bi = √(a²+b²). It represents the distance from the origin in the complex plane.
11 / 20
What is the transpose of a matrix?
Transpose A^T swaps rows and columns: the element in row i, column j moves to row j, column i.
12 / 20
What is a polar coordinate (r, θ)?
Polar coordinates (r,θ): r is the distance from the origin (pole), θ is the angle from the positive x-axis.
13 / 20
What is the formula for the nth term of the Fibonacci sequence approximately (Binet's formula)?
Binet's formula: F(n) = φⁿ/√5 rounded to nearest integer, where φ = (1+√5)/2 ≈ 1.618 (golden ratio).
14 / 20
What is the binomial theorem expansion of (a+b)²?
Binomial expansion: (a+b)² = a² + 2ab + b². The middle term 2ab is critical — don't forget it!
15 / 20
What is the standard form of an ellipse centered at the origin?
Ellipse: x²/a² + y²/b² = 1. The semi-major axis is the larger of a and b.
16 / 20
What is mathematical induction used for?
Mathematical induction proves a statement P(n) is true for all positive integers by: (1) proving P(1), (2) assuming P(k) and proving P(k+1).
17 / 20
What is the golden ratio φ?
The golden ratio φ = (1+√5)/2 ≈ 1.618. It satisfies φ² = φ + 1 and appears throughout nature and art.
18 / 20
What does the floor function ⌊x⌋ do?
Floor function ⌊x⌋ rounds DOWN to the nearest integer. ⌊3.9⌋ = 3, ⌊-1.2⌋ = -2.
19 / 20
What is the formula for the sum of a finite geometric series?
Geometric series sum: Sₙ = a₁(1-rⁿ)/(1-r). Use when you know first term, ratio, and number of terms.
20 / 20
What is the Rational Root Theorem?
Rational Root Theorem: possible rational roots = ±(factors of constant term a₀)/(factors of leading coeff aₙ).
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