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.
1,200 fundamentals
“'Is the same age as' is an equivalence relation — it sorts people into age groups.”
A relation can be reflexive, symmetric or transitive; all three make it an equivalence relation.
Memory trick: Reflexive+symmetric+transitive = equivalence.
“cos is even, sin is odd — their graphs mirror differently.”
Even functions satisfy f(−x)=f(x) (symmetric about y-axis); odd satisfy f(−x)=−f(x) (symmetric about origin).
Memory trick: Even = y-axis mirror; odd = origin flip.
“Logarithm undoes exponentiation, like subtraction undoes addition.”
Undoes a function; it exists only if the function is one-one and onto (bijective).
Memory trick: Only bijections are invertible.
“Distance can't be negative — that's the modulus in one idea.”
|x| gives the distance of x from zero, always non-negative.
|x|
Memory trick: |x| ≥ 0 always.
“[2.9] = 2 — it always rounds down to the nearest integer.”
[x] gives the greatest integer not exceeding x (the 'floor').
[x]
Memory trick: Floor: round down.
“Bacteria, viruses and compound interest all grow exponentially.”
f(x)=a^x (a>0) grows or decays by a constant factor.
a^x
Memory trick: Constant ratio per step.
“Logarithms turn hard multiplication into easy addition.”
log(mn)=log m+log n; log(m/n)=log m−log n; log(m^p)=p·log m.
Memory trick: Log turns × into +.
“These few values unlock most trigonometry problems.”
sin30°=½, sin45°=1/√2, sin60°=√3/2, sin90°=1.
Memory trick: Learn 0,30,45,60,90 cold.
“They let you shrink a doubled angle back to a single one.”
sin2θ=2sinθcosθ; cos2θ=cos²θ−sin²θ=1−2sin²θ.
sin2θ=2sinθcosθ
Memory trick: cos2θ has three handy forms.
“tan is the slope of the line at angle θ.”
tanθ = sinθ/cosθ; undefined at 90° where cos is zero.
tanθ=sinθ/cosθ
Memory trick: tan = sin/cos.
“'Co' in cosine literally means the complement of sine.”
sin(90°−θ)=cosθ and cos(90°−θ)=sinθ.
Memory trick: Swap sin↔cos at 90°−θ.
“Trig equations have infinitely many solutions — these formulas list them all.”
sinθ=sinα → θ=nπ+(−1)ⁿα; cosθ=cosα → θ=2nπ±α.
Memory trick: Add the periodic family, not one answer.
“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.