Fundamentals
The core facts every aspirant should own — each a titled nugget with a real-world story, the concept in plain words, and a memory trick. Works even when the internet doesn't.
300 fundamentals
“Your calculator's 'shift sin' finds the angle back from a ratio.”
sin⁻¹x returns the angle whose sine is x, with a restricted principal range.
Memory trick: Mind the principal-value range.
“Surveyors measure a mountain's height without ever climbing it.”
Angles of elevation and depression let us find inaccessible heights using trigonometry.
Memory trick: Draw the right triangle first.
“Complex numbers are arrows — polar form gives their length and direction.”
z = r(cosθ + i sinθ), where r is the modulus and θ the argument.
Memory trick: r = length, θ = direction.
“It turns powers of complex numbers into simple angle multiplication.”
(cosθ + i sinθ)^n = cos nθ + i sin nθ.
Memory trick: Raise power → multiply the angle.
“These three special numbers sit at the corners of an equilateral triangle.”
The three cube roots of 1 are 1, ω, ω², with 1+ω+ω²=0 and ω³=1.
1+ω+ω²=0
Memory trick: They sum to zero.
“It's how you test if two equations secretly cross at the same point.”
Two quadratics share a root when a specific determinant of their coefficients vanishes.
Memory trick: Set up the condition, don't guess.
“The sign of 'a' tells you smile or frown at a glance.”
The parabola opens upward if a>0 (minimum) and downward if a<0 (maximum).
Memory trick: a>0 smile, a<0 frown.
“Young Gauss summed 1 to 100 in seconds using this pairing trick.”
1+2+…+n = n(n+1)/2.
Σn = n(n+1)/2
Memory trick: Pair first+last.
“A handy closed form that saves adding term by term.”
1²+2²+…+n² = n(n+1)(2n+1)/6.
n(n+1)(2n+1)/6
Memory trick: Memorise the /6 formula.
“The sum of cubes is the square of the sum — a beautiful coincidence.”
1³+2³+…+n³ = [n(n+1)/2]².
[n(n+1)/2]²
Memory trick: = (sum of first n)².
“Musical harmonics and lens formulas hide harmonic progressions.”
A sequence whose reciprocals form an AP.
Memory trick: Flip terms → get an AP.
“Splitting the difference evenly is the arithmetic mean.”
The AM of a and b is (a+b)/2.
(a+b)/2
Memory trick: Just the average.
“The biggest coefficients cluster in the middle of the expansion.”
For (a+b)^n, there's one middle term if n is even, two if n is odd.
Memory trick: Count n+1 terms.
“Put x=1 and the whole expansion collapses to 2^n.”
The coefficients of (1+x)^n add up to 2^n.
ΣnCr = 2ⁿ
Memory trick: Set x=1 to sum them.
“It's how mapmakers place a town a certain fraction along a road.”
A point dividing a segment in ratio m:n has coordinates ((mx₂+nx₁)/(m+n), (my₂+ny₁)/(m+n)).
Memory trick: Weighted average of endpoints.
“The shortest hop from a point to a line is always perpendicular.”
d = |ax₀+by₀+c| / √(a²+b²).
d=|ax₀+by₀+c|/√(a²+b²)
Memory trick: Use the perpendicular distance.
“The formula quietly reveals parallel (θ=0) and perpendicular (denominator 0) cases.”
tanθ = |(m₁−m₂)/(1+m₁m₂)| from their slopes.
Memory trick: Slopes → angle.
“A wheel touches the road along a tangent line.”
A tangent touches a circle at exactly one point and is perpendicular to the radius there.
Memory trick: Tangent ⊥ radius.
“Planets orbit the Sun in ellipses, not perfect circles.”
x²/a² + y²/b² = 1 describes an oval with two foci.
x²/a²+y²/b²=1
Memory trick: Sum of focal distances is constant.
“A ship's LORAN navigation traced hyperbolas of constant time-difference.”
x²/a² − y²/b² = 1 is an open two-branch curve.
x²/a²−y²/b²=1
Memory trick: Difference of focal distances is constant.
“It's a quick measure of how 'wide' a conic opens at the focus.”
The chord through a conic's focus, perpendicular to its axis.
Memory trick: Chord through focus ⊥ axis.
“Sine and cosine chase each other's derivatives in a four-step cycle.”
d/dx(sin x)=cos x and d/dx(cos x)=−sin x.
Memory trick: cos has the minus.
“e^x is the only function that is its own derivative.”
d/dx(e^x)=e^x and d/dx(ln x)=1/x.
Memory trick: e^x is self-derivative.
“The order in the numerator matters — top-derivative first.”
d/dx(u/v) = (u'v − uv')/v².
(u/v)'=(u'v−uv')/v²
Memory trick: (low·dHigh − high·dLow)/low².