📐 Algebra to Calculus
Mathematics Complete Cheatsheet
Algebra, geometry, trig, calculus, probability and statistics in one place.
01
Algebra Fundamentals
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Commutative
a+b=b+a, a×b=b×a
Associative
(a+b)+c=a+(b+c)
Distributive
a(b+c)=ab+ac
BODMAS
Brackets→Orders→Div/Mul→Add/Sub
Exponent rules
aᵐ·aⁿ=aᵐ⁺ⁿ, (aᵐ)ⁿ=aᵐⁿ, a⁻ⁿ=1/aⁿ
Zero product
If ab=0 then a=0 or b=0
MATHLinear equations
ax + b = c → x = (c-b)/a # Simultaneous (elimination) 2x + 3y = 12 4x - 3y = 6 ────────────── 6x = 18 → x=3, y=2
02
Quadratics
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MATHQuadratic formula
ax² + bx + c = 0
-b ± √(b² - 4ac)
x = ─────────────────────
2a
Δ = b²-4ac:
Δ > 0 → 2 real roots
Δ = 0 → 1 repeated root
Δ < 0 → no real roots
Sum of roots
r₁+r₂ = -b/a
Vieta's formula
Product of roots
r₁×r₂ = c/a
Vieta's formula
Difference of squares
a²-b² = (a+b)(a-b)
Factoring shortcut
Perfect square
(a±b)² = a²±2ab+b²
Expansion
03
Geometry
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Rectangle
A=lw, P=2(l+w)
l=length, w=width
Triangle
A=½bh, P=a+b+c
b=base, h=height
Circle
A=πr², C=2πr
r=radius
Cylinder
V=πr²h, SA=2πr(r+h)
r=radius, h=height
Sphere
V=4/3πr³, SA=4πr²
r=radius
Cone
V=1/3πr²h
h=perpendicular height
Cuboid
V=lwh, SA=2(lw+lh+wh)
l,w,h=dimensions
Pythagoras
c²=a²+b²
Right-angled triangles only
04
Trigonometry
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MATHTrig ratios
Opposite Adjacent Opposite
sin θ = ───────── cosθ = ───────── tanθ = ─────────
Hypotenuse Hypotenuse Adjacent
Key angles:
θ 0° 30° 45° 60° 90°
sin 0 ½ √2/2 √3/2 1
cos 1 √3/2 √2/2 ½ 0
tan 0 1/√3 1 √3 ∞
Pythagorean
sin²θ+cos²θ=1
Most important identity
Sine rule
a/sinA = b/sinB = c/sinC
Non-right triangles
Cosine rule
c²=a²+b²-2ab cosC
Non-right triangles
Double angle
sin2θ=2sinθcosθ
Radians
π rad=180°
rad = degrees×π/180
05
Derivatives
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MATHDifferentiation rules
Power rule: d/dx[xⁿ] = nxⁿ⁻¹ Constant: d/dx[c] = 0 Product rule: d/dx[fg] = f'g + fg' Quotient rule: d/dx[f/g] = (f'g-fg')/g² Chain rule: d/dx[f(g(x))] = f'(g(x))·g'(x) d/dx[eˣ]=eˣ d/dx[lnx]=1/x d/dx[sinx]=cosx d/dx[cosx]=-sinx d/dx[tanx]=sec²x
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Set f'(x)=0 to find critical points (maxima/minima).
06
Integrals
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MATHIntegration rules
∫ xⁿ dx = xⁿ⁺¹/(n+1) + C (n ≠ -1) ∫ 1/x dx = ln|x| + C ∫ eˣ dx = eˣ + C ∫ sinx dx = -cosx + C ∫ cosx dx = sinx + C ∫ sec²x dx = tanx + C Definite integral: ∫[a→b] f(x) dx = F(b) - F(a) Integration by parts: ∫ u dv = uv - ∫ v du
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Always write +C for indefinite integrals!
07
Probability
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MATHProbability
P(A) = favourable / total 0 ≤ P(A) ≤ 1 P(A') = 1 - P(A) Addition: P(A∪B) = P(A)+P(B)-P(A∩B) Multiplication: P(A∩B) = P(A)×P(B) [independent] Conditional: P(A|B) = P(A∩B)/P(B) Permutation: P(n,r) = n!/(n-r)! Combination: C(n,r) = n!/[r!(n-r)!] Binomial: P(X=k) = C(n,k)pᵏ(1-p)ⁿ⁻ᵏ
Mutually exclusive
P(A∩B)=0, so P(A∪B)=P(A)+P(B)
Independent
P(A∩B)=P(A)×P(B)
Normal dist
68% within 1σ, 95% within 2σ, 99.7% within 3σ
08
Statistics
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Mean
x̄ = Σxᵢ/n
Average
Median
Middle value when sorted
Better for skewed data
Variance
σ²=Σ(xᵢ-x̄)²/n
Average squared deviation
Std Dev
σ=√variance
Same units as data
IQR
Q3-Q1
Robust to outliers
Z-score
z=(x-μ)/σ
Standard deviations from mean
Correlation
r: -1 to +1. |r|>0.7=strong, 0.4-0.7=moderate
Central Limit Theorem
Sample means → normal distribution as n→∞
Skewness
Right-skewed: mean>median. Left-skewed: mean
09
Linear Algebra
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MATHLinear algebra
# Vectors v=[a,b,c], u=[d,e,f] v+u=[a+d,b+e,c+f] v·u=ad+be+cf (dot product) |v|=√(a²+b²+c²) # Matrix 2×2 A=[[a,b],[c,d]] det(A) = ad-bc A⁻¹ = (1/det) × [[d,-b],[-c,a]] # Multiplication: A(m×n)×B(n×p)=C(m×p) # NOT commutative: AB ≠ BA
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Matrix multiplication is NOT commutative: AB ≠ BA
10
Mini Quizzes
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❓ Quiz 1
What does discriminant b²-4ac tell you?
Δ=b²-4ac: Δ>0→2 real roots, Δ=0→1 repeated root, Δ<0→no real roots.
❓ Quiz 2
What is d/dx[sinx]?
d/dx[sinx]=cosx. And d/dx[cosx]=-sinx. d/dx[tanx]=sec²x.
❓ Quiz 3
For independent events A,B: P(A and B)=?
Independent: P(A∩B)=P(A)×P(B). Addition rule P(A)+P(B)-P(A∩B) is for P(A OR B).