All Math Formulas

A math Guide for Students: Arithmetic, Algebra, Geometry, Trigonometry, Calculus, Probability & More

1. Arithmetic and Number Theory

• Order of Operations (PEMDAS/BODMAS):
  \( P \rightarrow E \rightarrow MD \rightarrow AS \) (Left-to-right within pairs)
• Prime Factorization: Any integer \( n \) can be factored as \( n = p_1^{a_1} p_2^{a_2} \ldots p_k^{a_k} \)
• Divisibility Rules and Tricks:
  A number is divisible by 3 if sum of digits is divisible by 3.
  By 4: If last 2 digits are divisible by 4.
  By 9: If sum of digits is divisible by 9.
• HCF/LCD:
  HCF (\(\gcd(a,b)\)): Highest Common Factor
  LCM (\(\mathrm{lcm}(a,b)\)): Lowest Common Multiple
• Percentage: \( \text{Percentage} = \frac{\text{Part}}{\text{Whole}} \times 100\% \)
• Simple Interest: \( SI = \frac{P \cdot R \cdot T}{100} \)
• Compound Interest: \( CI = P \left(1+\frac{R}{100}\right)^T - P \)
• Average/Mean: \( \bar{x} = \frac{1}{n}\sum_{i=1}^{n} x_i \)
• Sum of First \( n \) Natural Numbers: \( S_n = \frac{n(n+1)}{2} \)
• Sum of First \( n \) Squares: \( S_n = \frac{n(n+1)(2n+1)}{6} \)
• Sum of First \( n \) Cubes: \( S_n = \left[\frac{n(n+1)}{2}\right]^2 \)

2. Algebra

• Laws of Exponents:
  \( a^m \times a^n = a^{m+n} \)
  \( \frac{a^m}{a^n} = a^{m-n} \)
  \( (a^m)^n = a^{mn} \)
  \( (ab)^n = a^n b^n \)
  \( a^0 = 1 \)
  
• Quadratic Formula:
  \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)
• Algebraic Identities:
  \( (a+b)^2 = a^2 + 2ab + b^2 \)
  \( (a-b)^2 = a^2 - 2ab + b^2 \)
  \( a^2 - b^2 = (a+b)(a-b) \)
  \( (a+b)^3 = a^3 + 3a^2b + 3ab^2 + b^3 \)
  \( (a-b)^3 = a^3 - 3a^2b + 3ab^2 - b^3 \)
  \( a^3 + b^3 = (a+b)(a^2 - ab + b^2) \)
  \( a^3 - b^3 = (a-b)(a^2 + ab + b^2) \)
• Slope-Intercept Line Equation: \( y = mx + c \), slope \( m = \frac{y_2 - y_1}{x_2 - x_1} \)
• Distance Formula: \( d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2} \)
• Midpoint Formula: \( M = \left(\frac{x_1 + x_2}{2},\; \frac{y_1 + y_2}{2}\right) \)
• Arithmetic Sequence: \( a_n = a_1 + (n-1)d \),  Arithmetic Series: \( S_n = \frac{n}{2}[2a_1 + (n-1)d] \)
• Geometric Sequence: \( a_n = a_1 \, r^{n-1} \),  Geometric Series: \( S_n = a_1 \frac{r^n - 1}{r-1} \)
• Sum to Infinity of Geometric Series (\(|r| < 1\)): \( S = \frac{a_1}{1 - r} \)
• Binomial Theorem:
  \( (a+b)^n = \sum_{k=0}^{n}C_n^k a^{n-k}b^k \)

3. Geometry

• Perimeter of Square: \( P = 4a \)
• Perimeter of Rectangle: \( P = 2(l + b) \)
• Area of Square: \( A = a^2 \)
• Area of Rectangle: \( A = lb \)
• Area of Triangle: \( A = \frac{1}{2} b h \)
• Area of Parallelogram: \( A = b h \)
• Area of Trapezoid: \( A = \frac{1}{2} (b_1 + b_2) h \)
• Area of Circle: \( A = \pi r^2 \)
• Circumference of Circle: \( C = 2\pi r \)
• Pythagoras Theorem: \( a^2 + b^2 = c^2 \)
• Volume of Cube: \( V = a^3 \)
• Volume of Rectangular Prism: \( V = lwh \)
• Volume of Sphere: \( V = \frac{4}{3}\pi r^3 \)
• Surface Area of Sphere: \( SA = 4\pi r^2 \)
• Volume of Cylinder: \( V = \pi r^2 h \)
• Surface Area of Cylinder: \( SA = 2\pi r(r + h) \)
• Volume of Cone: \( V = \frac{1}{3}\pi r^2 h \)
• Area of Sector: \( A = \frac{\theta}{360^\circ} \pi r^2 \)
• Arc Length: \( L = \frac{\theta}{360^\circ} 2\pi r \)

4. Trigonometry

• Basic Ratios:
  \( \sin\theta = \frac{\text{Opposite}}{\text{Hypotenuse}} \) \( \cos\theta = \frac{\text{Adjacent}}{\text{Hypotenuse}} \) \( \tan\theta = \frac{\text{Opposite}}{\text{Adjacent}} \)
• Pythagorean Identities:
  \( \sin^2\theta + \cos^2\theta = 1 \)
• Co-Function Identities:
  \( \sin(90^\circ - \theta) = \cos\theta \), \( \tan(90^\circ - \theta) = \cot\theta \)
• Angle Sum/Difference:
  \( \sin(A\pm B) = \sin A \cos B \pm \cos A \sin B \)
  \( \cos(A\pm B) = \cos A \cos B \mp \sin A \sin B \)
• Double Angle:
  \( \sin 2A = 2\sin A\cos A \)
  \( \cos 2A = \cos^2 A - \sin^2 A \ = 2\cos^2 A - 1 = 1-2\sin^2A \)
• Law of Sines:
  \( \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \)
• Law of Cosines:
  \( c^2 = a^2 + b^2 - 2ab\cos C \)
• Area of Triangle (Sine formula):
  \( A = \frac{1}{2}ab\sin C \)
• Radian-Degree Conversion: \( 180^\circ = \pi \) radians

5. Calculus (Differential & Integral)

• Differentiation:
  \( \frac{d}{dx} x^n = n x^{n-1} \)
  \( \frac{d}{dx} \sin x = \cos x \)
  \( \frac{d}{dx} \cos x = -\sin x \)
  \( \frac{d}{dx} e^x = e^x \)
  \( \frac{d}{dx} \ln x = \frac{1}{x} \)
• Integration:
  \( \int x^n dx = \frac{x^{n+1}}{n+1} + C \), \( n\neq -1\)
  \( \int e^x dx = e^x + C \)
  \( \int \sin x dx = -\cos x + C \)
  \( \int \cos x dx = \sin x + C \)
  \( \int \frac{1}{x} dx = \ln|x| + C \)
• Definite Integral as Area:
  \( \text{Area} = \int_a^b f(x)\,dx \)
• Product Rule: \( (uv)' = u'v + uv' \)
• Quotient Rule: \( \left(\frac{u}{v}\right)' = \frac{u'v - uv'}{v^2} \)
• Chain Rule: \( \frac{dy}{dx} = \frac{dy}{du} \frac{du}{dx} \)
• Taylor Expansion (about \( x=a \)):  \( f(x) = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \cdots \)
• Limit Definition (Derivative):
  \( f'(x) = \lim_{h \to 0} \frac{f(x+h)-f(x)}{h} \)
• Fundamental Theorem:
  If \( F' = f \), then \( \int_a^b f(x)dx = F(b) - F(a) \)

6. Probability and Statistics

• Basic Probability: \( P(A) = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} \)
• Addition Rule: \( P(A \text{ or } B) = P(A) + P(B) - P(A \cap B) \)
• Multiplication Rule: \( P(A \cap B) = P(A) \cdot P(B|A) \)
• Conditional Probability: \( P(A|B) = \frac{P(A \cap B)}{P(B)} \)
• Mean: \( \mu = \frac{1}{n}\sum_{i=1}^n x_i \)
• Median: Middle value in ordered data
• Variance: \( \sigma^2 = \frac{1}{n}\sum_{i=1}^{n}(x_i - \mu)^2 \)
• Standard Deviation: \( \sigma = \sqrt{Variance} \)
• Binomial Theorem (repetition): \( P(k,n) = C_n^k \, p^k (1-p)^{n-k} \)
• Permutation: \( nPr = \frac{n!}{(n-r)!} \), Combination: \( nCr = \frac{n!}{r!(n-r)!} \)
• Z-Score: \( z = \frac{x - \mu}{\sigma} \)
• Normal Distribution:
  \( f(x) = \frac{1}{\sigma \sqrt{2\pi}} e^{-\frac12 \left( \frac{x-\mu}{\sigma}\right)^2} \)

7. Additional Topics & Math Tricks

• Logarithm Laws:
  \( \log_b(xy) = \log_b x + \log_b y \)
  \( \log_b \left(\frac{x}{y}\right) = \log_b x - \log_b y \)
  \( \log_b(x^r) = r \log_b x \)
• Complex Number Formulas:
  Standard: \( z = a+bi \)
  Modulus: \( |z| = \sqrt{a^2+b^2} \)
  Conjugate: \( \overline{z} = a - bi \)
• Math Tricks:
  Fast Square of Numbers Ending in 5: \( n5 \times n5 = n(n+1)25 \)
  Divisibility by 11: Alternate sum of digits is divisible by 11.
  Sum of n consecutive integers: \( S = \frac{n}{2}(\text{First term} + \text{Last term}) \)
• Useful Inequalities:
  AM ≥ GM ≥ HM (Means: Arithmetic ≥ Geometric ≥ Harmonic)
• Unit Circle Values (Radians):
  \( \sin 0 = 0, \sin \frac{\pi}{6} = \frac{1}{2}, \sin \frac{\pi}{4} = \frac{\sqrt{2}}{2}, \sin \frac{\pi}{3} = \frac{\sqrt{3}}{2}, \sin \frac{\pi}{2} = 1 \)