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Advanced Maths Calculator

🧮 Comprehensive Mathematics Learning Center

Master essential math concepts with our interactive tools! This comprehensive calculator suite covers fractions, percentages, equations, geometry, statistics, and more. Each tool includes step-by-step solutions, visual diagrams, and educational content to help you understand the underlying mathematics.

7 Tools
Interactive Calculators
Formulas
Complete Reference
Examples
Step-by-Step Solutions
Diagrams
Visual Learning

📚 Complete Mathematics Topics Index

1. FOUNDATIONS

2. ARITHMETIC

  • • Basic Operations (PEMDAS/BODMAS)
  • • Decimals & Fractions
  • • Estimation & Rounding
  • • Time, Speed, Distance Problems
  • • Work Rate Problems
  • • Percent & Ratio Applications
  • • Word Problems & Real-World Applications

3. ALGEBRA

  • • Expressions & Equations
  • • Simplifying, Expanding, Factorizing
  • • Linear, Quadratic, Higher Degree Equations
  • • Rational & Radical Equations
  • • Systems of Equations (Substitution, Elimination, Matrices)
  • • Functions: Linear, Quadratic, Polynomial, Rational
  • • Exponential, Logarithmic, Trigonometric Functions
  • • Domain, Range, Inverses, Composition
  • • Sequences & Series: Arithmetic, Geometric, Harmonic
  • • Binomial Theorem

4. GEOMETRY

  • • Plane Geometry: Angles, Lines, Triangles
  • • Quadrilaterals, Circles
  • • Congruence & Similarity
  • • Circle Theorems
  • • Coordinate Geometry: Distance, Midpoint, Slope
  • • Line Equations
  • • Conics: Circle, Parabola, Ellipse, Hyperbola
  • • Trigonometry: Ratios, Identities, Laws
  • • 3D Geometry: Solids, Surface Area, Volume
  • • Bearings & Directions

5. CALCULUS

  • • Differential Calculus: Limits, Continuity
  • • Derivatives & Applications
  • • Chain Rule, Product Rule, Quotient Rule
  • • Integral Calculus: Definite & Indefinite Integrals
  • • Applications of Integration
  • • Differential Equations: First & Second Order
  • • Multivariable Calculus
  • • Partial Derivatives, Multiple Integrals
  • • Vector Calculus

6. STATISTICS & PROBABILITY

  • • Probability: Events, Sample Space
  • • Conditional Probability, Independence
  • • Counting: Permutations, Combinations
  • • Distributions: Binomial, Normal, Poisson
  • • Statistics: Data Collection, Tables, Graphs
  • • Mean, Median, Mode
  • • Variance, Standard Deviation
  • • Correlation, Regression
  • • Hypothesis Testing, Confidence Intervals

7. DISCRETE MATHEMATICS

  • • Logic & Proofs
  • • Combinatorics & Pigeonhole Principle
  • • Graph Theory: Vertices, Edges, Paths
  • • Boolean Algebra: AND, OR, NOT, XOR
  • • Number Theory: Modular Arithmetic, Congruences
  • • Algorithm Analysis
  • • Recursive Relations

8. ADVANCED MATHEMATICS

  • • Complex Numbers & Complex Analysis
  • • Abstract Algebra: Groups, Rings, Fields
  • • Linear Algebra: Vectors, Matrices, Eigenvalues
  • • Real Analysis & Topology
  • • Measure Theory
  • • Functional Analysis
  • • Differential Geometry
  • • Mathematical Modeling
  • • Optimization & Linear Programming
  • • Operations Research
  • • Control Theory
  • • Mathematical Economics
  • • Signal Processing
  • • Chaos & Dynamical Systems
  • • Numerical Methods

Click on any Foundation topic above to explore detailed content

Each topic includes comprehensive notes, step-by-step examples, visual diagrams, and built-in calculators for solving problems with complete working shown.

Mathematical Foundations - Complete Guide

📊 Complete Number System Hierarchy
Set Notation: ℕ ⊂ ℤ ⊂ ℚ ⊂ ℝ ⊂ ℂ
Progression: Natural → Whole → Integers → Rational → Real → Complex
Prime Numbers: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47...
Fundamental Theorem: Every integer > 1 has unique prime factorization
Composite Numbers: 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20...
🔢 Interactive Number Sets Explorer

Natural Numbers (ℕ)

1, 2, 3, 4, 5, 6, 7, 8, 9, 10...
Counting numbers starting from 1
Properties: Positive, Infinite set, Closed under addition & multiplication

Whole Numbers (ℕ₀)

0, 1, 2, 3, 4, 5, 6, 7, 8, 9...
Natural numbers + zero
Properties: Non-negative, Includes additive identity (0)

Integers (ℤ)

..., -3, -2, -1, 0, 1, 2, 3, ...
Positive & negative whole numbers
Properties: Closed under +, -, × but not ÷

Rational Numbers (ℚ)

1/2, -3/4, 0.25, 1.333...
Numbers that can be expressed as fractions p/q
Properties: Includes terminating & repeating decimals

Irrational Numbers

π, e, √2, √3, φ (golden ratio)...
Non-repeating, non-terminating decimals
Properties: Cannot be expressed as simple fractions

Real Numbers (ℝ)

All rational + irrational numbers
Numbers on the continuous number line
Properties: Complete ordered field, no gaps

Complex Numbers (ℂ)

a + bi where i = √(-1)
Real numbers + imaginary numbers
Properties: Algebraically closed field
🔍 Prime Numbers & Factorization
Prime Definition: A natural number > 1 with exactly two factors: 1 and itself
Composite Definition: A natural number > 1 that is not prime
Fundamental Theorem of Arithmetic: Every integer > 1 is either prime or can be uniquely factorized into primes
Prime Factorization Example: 60 = 2² × 3 × 5
Sieve of Eratosthenes: Ancient algorithm to find all primes up to a given number

🧮 Prime & Factorization Calculator

📐 Powers, Roots & Scientific Notation
Exponent Rules:
• a^m × a^n = a^(m+n)    • a^m ÷ a^n = a^(m-n)
• (a^m)^n = a^(mn)    • a^0 = 1 (a ≠ 0)
• a^(-n) = 1/a^n    • (ab)^n = a^n × b^n

Root Properties:
• ⁿ√(ab) = ⁿ√a × ⁿ√b    • ⁿ√(a/b) = ⁿ√a / ⁿ√b
• ⁿ√a^m = a^(m/n)    • √(a²) = |a|

Scientific Notation: a × 10^n where 1 ≤ |a| < 10
Examples: 3,450 = 3.45 × 10³, 0.0067 = 6.7 × 10⁻³

⚡ Powers & Roots Calculator

📊 Ratio, Proportion & Percentages
Ratio: Comparison of two quantities (a:b or a/b)
Proportion: Equality of two ratios (a/b = c/d)
Cross Multiplication: If a/b = c/d, then ad = bc

Percentage Formulas:
• Percentage = (Part/Whole) × 100%
• Part = (Percentage/100) × Whole
• Whole = Part ÷ (Percentage/100)
• Percentage Change = [(New - Old)/Old] × 100%
• Compound Percentage = (1 + r/100)^n × Principal

📈 Ratio & Percentage Calculator

: = :

🔍 Advanced Number Classification Tool

🧠 Mathematical Logic & Proof Methods
Logical Operators:
• ∧ (AND): P ∧ Q is true when both P and Q are true
• ∨ (OR): P ∨ Q is true when at least one of P or Q is true
• ¬ (NOT): ¬P is true when P is false
• → (IMPLIES): P → Q is false only when P is true and Q is false
• ↔ (IF AND ONLY IF): P ↔ Q is true when P and Q have the same truth value

Proof Techniques:
• Direct Proof: Assume P, use logical steps to prove Q
• Proof by Contradiction: Assume ¬Q, derive a contradiction
• Contrapositive: Prove ¬Q → ¬P instead of P → Q
• Mathematical Induction: Base case + inductive step proves for all n

🔬 Logic & Truth Table Generator

🔲 Set Theory Fundamentals
Set Notation:
• A = {1, 2, 3, 4, 5} (roster notation)
• B = {x | x is an even integer} (set-builder notation)
• ∅ or {} (empty set)
• ℝ, ℤ, ℚ, ℕ (standard number sets)

Set Operations:
• A ∪ B (Union): All elements in A or B or both
• A ∩ B (Intersection): Elements common to both A and B
• A' or Aᶜ (Complement): Elements not in A
• A - B (Difference): Elements in A but not in B
• A × B (Cartesian Product): All ordered pairs (a,b) where a∈A, b∈B

Set Properties:
• |A| = cardinality (number of elements in A)
• A ⊆ B means A is a subset of B
• A ⊂ B means A is a proper subset of B

🔢 Interactive Set Operations Calculator

Enter sets above to see Venn diagram
🔄 Modular Arithmetic
Definition: a ≡ b (mod m) means m divides (a - b)
Examples: 17 ≡ 5 (mod 12), 23 ≡ 3 (mod 10)

Properties:
• If a ≡ b (mod m) and c ≡ d (mod m), then:
  - a + c ≡ b + d (mod m)
  - a - c ≡ b - d (mod m)
  - a × c ≡ b × d (mod m)

Applications: Cryptography, Computer Science, Number Theory

🧮 Modular Arithmetic Calculator

mod

Arithmetic Operations & Applications

Arithmetic Fundamentals
Order of Operations: PEMDAS/BODMAS
Fractions: a/b + c/d = (ad + bc)/(bd)
Decimals: 0.25 = 25/100 = 1/4
Percentages: % = (Part/Whole) × 100
Scientific Notation: a × 10ⁿ where 1 ≤ a < 10
25% Visual Representation

4 out of 10 blocks = 40%

Percentage Quick Facts

  • 50% = 1/2 = 0.5
  • 25% = 1/4 = 0.25
  • 75% = 3/4 = 0.75
  • 10% = 1/10 = 0.1
  • 100% = 1 = 1.0

Real-World Percentage Applications

💰 Sales Tax: 8% tax on $50 = 0.08 × $50 = $4
🛍️ Discounts: 30% off $80 = $80 - (0.30 × $80) = $56
📊 Statistics: 65% of 200 students = 0.65 × 200 = 130 students
💡 Tips: 18% tip on $45 meal = 0.18 × $45 = $8.10
📈 Growth: 15% increase from $100 = $100 + ($100 × 0.15) = $115

Algebraic Expressions & Equations

Algebraic Essentials
Linear: ax + b = c → x = (c-b)/a
Quadratic: ax² + bx + c = 0 → x = [-b ± √(b²-4ac)]/2a
Functions: f(x) = y, Domain & Range
Sequences: aₙ = a₁ + (n-1)d (Arithmetic)
Exponential: f(x) = abˣ, Logarithmic: logₐ(x)
Linear Equations: A linear equation is an equation that can be written in the form ax + b = c, where a, b, and c are constants and a ≠ 0. The graph of a linear equation is always a straight line. These equations model many real-world situations like calculating costs, distances, and rates.

Solving Linear Equations - Memory Device: "UNDO"

U - Undo addition/subtraction first (inverse operations)
N - Next, undo multiplication/division
D - Do the same operation to both sides
O - Opposite operations cancel each other out
🎯 Goal: Isolate the variable on one side
x + =

Example 1

2x + 3 = 7
2x = 7 - 3 = 4
x = 4 ÷ 2 = 2

Example 2

-x + 5 = 2
-x = 2 - 5 = -3
x = -3 ÷ (-1) = 3

Calculus - Derivatives & Integrals

Calculus Fundamentals
Limit: lim[x→a] f(x) = L
Derivative: f'(x) = lim[h→0] [f(x+h) - f(x)]/h
Integral: ∫f(x)dx = F(x) + C
Fundamental Theorem: ∫[a to b] f'(x)dx = f(b) - f(a)
Chain Rule: (f∘g)'(x) = f'(g(x)) · g'(x)
Calculus Applications: Calculus studies rates of change (derivatives) and accumulation (integrals). Used in physics for motion, economics for optimization, and engineering for modeling continuous systems.

Multiplication Patterns & Tricks

  • Any number × 0 = 0 (Zero Property)
  • Any number × 1 = itself (Identity Property)
  • Even × Even = Even, Odd × Odd = Odd
  • ×9 trick: digits sum to 9 (9×7=63, 6+3=9)
  • ×11 trick: 34×11 = 3[3+4]4 = 374
  • Numbers ending in 5: always end in 0 or 5

Times Tables Memory Strategies

🔢 2s: Double the number (2×8 = 8+8 = 16)
✋ 5s: Count by fives or half of 10s table
🔟 10s: Just add a zero to the number
⭐ 9s: Use finger trick or subtract from 10s table
🔄 Squares: 6² = 6×6 = 36 (visual: 6×6 grid)

Discrete Mathematics & Combinatorics

Discrete Mathematics Essentials
Combinatorics: C(n,r) = n!/(r!(n-r)!), P(n,r) = n!/(n-r)!
Graph Theory: Vertices, Edges, Paths, Cycles
Boolean Algebra: AND, OR, NOT, XOR operations
Number Theory: Modular arithmetic a ≡ b (mod m)
Pigeonhole Principle: n+1 objects in n boxes → overlap

GCF Example

GCF(12, 18) = 6
12 = 6 × 2
18 = 6 × 3

LCM Example

LCM(4, 6) = 12
4: 4, 8, 12, 16...
6: 6, 12, 18, 24...
LCM & GCF Applications: GCF simplifies fractions to lowest terms. LCM finds common denominators for fraction addition. In real life, GCF helps divide items equally, while LCM determines when recurring events coincide (like buses meeting at a stop).

Prime Factorization Method

Step 1: Break numbers into prime factors
Step 2: GCF = multiply common factors
Step 3: LCM = multiply highest powers of all primes
Example: 12=2²×3, 18=2×3² → GCF=2×3=6, LCM=2²×3²=36

Geometry Calculator

Geometry Formulas
Circle: Area = πr², Circumference = 2πr
Rectangle: Area = l×w, Perimeter = 2(l+w)
Triangle: Area = ½×base×height
Square: Area = s², Perimeter = 4s
Common Shapes
r
Circle
w l
Rectangle
h
Triangle
Rectangle
Circle
Triangle
Square
Geometry Fundamentals: π (pi) ≈ 3.14159 represents the ratio of a circle's circumference to its diameter. Area measures the space inside a shape (square units), while perimeter measures the distance around it (linear units). Understanding these concepts helps in real-world applications like calculating floor space, fence lengths, and material costs.

Geometry in Real Life

🏠 Home: Calculate room area for flooring (length × width)
🎾 Sports: Tennis court area = 2,808 sq ft (78' × 36')
🍕 Pizza: 12" pizza area = π × 6² = 113 sq inches
🏡 Yard: Circular garden perimeter = 2πr for fencing
📐 Construction: Right triangle tool for square corners

Statistics Calculator

Statistics Formulas
Mean = Sum of all values ÷ Number of values
Median = Middle value when data is ordered
Mode = Most frequently occurring value
Range = Maximum value - Minimum value
Sample Data Visualization

Data Set: 3, 5, 8, 6, 4, 7

Mean Example

Data: 2, 4, 6, 8
Mean = (2+4+6+8)÷4 = 5

Median Example

Data: 1, 3, 5, 7, 9
Median = 5 (middle value)
Statistical Insights: Mean (average) is affected by outliers and represents the center of data. Median (middle value) is better for skewed data. Mode shows the most frequent value. Range indicates data spread. These measures help analyze trends in business, science, and everyday decision-making.

When to Use Each Measure

📊 Mean: Best for normal distributions, salary averages (without outliers)
📈 Median: Ideal for skewed data, house prices, income studies
🎯 Mode: Useful for categorical data, most popular product
📏 Range: Shows variability, temperature fluctuations
🔍 Standard Deviation: Measures how spread out data is from the mean

Test Scores Analysis

Scores: 85, 90, 78, 92, 88, 95, 82
Mean = 87.1 (overall performance)
Median = 88 (middle student)

Business Application

Daily Sales: $200, $250, $180, $300, $220
Mean = $230 (average daily income)
Range = $120 (sales variability)

Advanced Mathematical Concepts

Advanced Mathematics Overview
Complex Numbers: z = a + bi, |z| = √(a² + b²)
Linear Algebra: Ax = b, det(A), eigenvalues λ
Real Analysis: Continuity, convergence, topology
Abstract Algebra: Groups, rings, fields
Differential Equations: dy/dx = f(x,y)
Mathematical Fields Hierarchy

Pure Mathematics

Analysis, Algebra, Topology
Number Theory, Logic

Applied Mathematics

Mathematical Modeling
Numerical Methods

Computational

Algorithms, Optimization
Signal Processing
Advanced Applications: Complex analysis for engineering, linear algebra for computer graphics, differential equations for physics modeling, abstract algebra for cryptography, and topology for data analysis.

📚 Mathematical Reference Guide

Essential Mathematical Formulas

  • Distance: d = √[(x₂-x₁)² + (y₂-y₁)²]
  • Slope: m = (y₂-y₁)/(x₂-x₁)
  • Quadratic: x = [-b ± √(b²-4ac)]/2a
  • Pythagorean: a² + b² = c²
  • Compound Interest: A = P(1 + r/n)^(nt)
  • Standard Deviation: σ = √[Σ(x-μ)²/N]
  • Circle: A = πr², C = 2πr
  • Sine Rule: a/sin(A) = b/sin(B) = c/sin(C)

Number Properties & Rules

  • Commutative: a + b = b + a
  • Associative: (a + b) + c = a + (b + c)
  • Distributive: a(b + c) = ab + ac
  • Identity: a + 0 = a, a × 1 = a
  • Inverse: a + (-a) = 0, a × (1/a) = 1

Common Mathematical Conversions

  • Decimal to %: multiply by 100
  • % to decimal: divide by 100
  • Fraction to decimal: divide numerator by denominator
  • Degrees to radians: × (π/180)
  • Radians to degrees: × (180/π)

Mathematical Study Strategies

  • Practice problems daily for fluency
  • Understand concepts before memorizing
  • Use visual aids and diagrams
  • Check answers with different methods
  • Teach concepts to reinforce learning
  • Connect math to real-world applications
  • Work backwards from answers to understand
  • Use estimation to verify reasonableness

Advanced Mathematical Concepts

  • Logarithms: log₁₀(100) = 2
  • Exponentials: e^x growth/decay models
  • Trigonometry: sin²θ + cos²θ = 1
  • Derivatives: rate of change (calculus)
  • Integrals: area under curves
  • Probability: P(A) = favorable/total outcomes
  • Matrices: array of numbers in rows/columns
Common Mathematical Mistakes to Avoid:
• Forgetting order of operations (PEMDAS/BODMAS)
• Dividing by zero or undefined operations
• Not simplifying fractions to lowest terms
• Confusing circumference and area formulas
• Using wrong units in final answers
• Not checking if answers make sense in context
• Mixing up positive/negative signs in algebra
• Forgetting to apply operations to both sides of equations
💡 Pro Tip: Mathematics builds upon itself. Master the basics thoroughly before advancing to complex topics. Use these tools regularly to strengthen your mathematical foundation!